Introduction
A pulsar timing array (PTA) is a galactic-scale gravitational-wave (GW) detector. It searches for low-frequency (nanohertz) GWs by precisely monitoring the arrival times of pulses from a set of pulsars [1 ] . Gravitational waves (e.g., from inspiraling supermassive black-hole binaries in the centers of merging galaxies) influence the pulse arrival times in a way that is correlated between different pulsars. The mean correlation between a pair of pulsars depends upon the angular separation γ \gamma γ [2 ] .
The mean correlation was calculated by Hellings and Downs [3 ] for a unit amplitude, isotropic and unpolarized GW background. It has the simple analytic form [4 ]
μ u ( γ ) = 1 3 − 1 6 ( 1 − cos γ 2 ) + 1 − cos γ 2 ln ( 1 − cos γ 2 ) , (1) \mu_u(\gamma)=\frac{1}{3}-\frac{1}{6}\left(\frac{1-\cos\gamma}{2}\right)+\frac{1-\cos\gamma}{2}\ln\left(\frac{1-\cos\gamma}{2}\right), \tag*{(1)} μ u ( γ ) = 3 1 − 6 1 ( 2 1 − cos γ ) + 2 1 − cos γ ln ( 2 1 − cos γ ) , (1) where γ \gamma γ [5 ] .
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.
Several groups are searching for such correlations. These groups report strong statistical evidence for fluctuations in the individual pulsar arrival times that share the same “red” spectrum [6 , 7 , 8 , 9 ] that a GW background is expected to produce [10 ] . However, there is currently little evidence for the Hellings and Downs angular dependence, which makes it difficult to claim that a GW background is responsible.
Could the lack of evidence for the Hellings and Downs angular dependence be a statistical fluctuation? To answer this, it is important to understand what variations away from the Hellings and Downs predicted mean might be expected. The size of such fluctuations is quantified by the variance of the Hellings and Downs correlation.
Total variance, pulsar variance, and cosmic variance
Recent work [11 ] calculates this variance for several GW source models, neglecting all sources of noise. The most important model contains N N N [12 , 13 , 14 ] provided that the amplitude of the individual sources vanishes as N → ∞ N \to\infty N → ∞
The (total) variance σ t o t 2 \sigma_{\mathrm{tot}}^2 σ tot 2 [11 ] is a sum of “pulsar variance” and “cosmic variance” σ c o s 2 \sigma_{\mathrm{cos}}^2 σ cos 2 [15 ] . Pulsar variance arises because different pairs of pulsars separated by the same angle γ \gamma γ [11 ] computes the total variance (there denoted by σ 2 \sigma^2 σ 2 single randomly selected pulsar pair separated by angle γ \gamma γ after the correlation has been averaged over all possible locations/orientations of the pair. This corresponds to employing an infinite number of pulsar pairs separated by angle γ \gamma γ pulsar averaging .
In the confusion-noise model (Sec. III A in [11 ] ), cosmic variance arises because, even after pulsar averaging, the correlation depends upon the relative phases of the GW sources. Each realization of the Universe has different phases, and thus exhibits different pulsar-averaged correlations. The cosmic variance is the amount by which the pulsar-averaged correlation curve is expected to differ from the Hellings and Downs prediction in an ideal world containing an infinite number of noise-free pulsars. Unlike the pulsar variance, it cannot be reduced: the cosmic variance is a fundamental limit to the precision with which the Hellings and Downs predicted mean might be observed at a particular angle γ \gamma γ
Figure 1 illustrates the total and cosmic variance, as offsets away from the mean μ u \mu_u μ u γ \gamma γ γ \gamma γ γ \gamma γ γ \gamma γ In this paper, we study the transition between these two limits, when a finite set of pulsars at specific sky locations are used to estimate the correlation. This reflects observational reality because PTA pulsars are nonuniformly distributed on the sky, so the pairs formed from them have no separation angles in common.
Figure 1. The mean Hellings and Downs correlation μ u \mu_u μ u γ \gamma γ μ u ± σ t o t \mu_u \pm\sigma_{\mathrm{tot}} μ u ± σ tot μ u ± σ c o s \mu_u \pm\sigma_{\mathrm{cos}} μ u ± σ cos σ t o t 2 \sigma_{\mathrm{tot}}^2 σ tot 2 σ c o s 2 \sigma_{\mathrm{cos}}^2 σ cos 2 R 4 / h 4 = 1 / 2 \mathcal{R}^4/h^4 = 1/2 R 4 / h 4 = 1/2 h 2 = 1 h^2 = 1 h 2 = 1 σ \sigma σ
The plots in Figure 1 assume that the GWs arise from the incoherent sum of many weak sources, giving rise, via the central-limit theorem, to a Gaussian ensemble [12 , 13 , 14 ] . For such sources, the scaling relation between the (squared) mean and the variance is described in Appendix B; these plots take R 4 / h 4 = 1 / 2 \mathcal{R}^4/h^4 = 1/2 R 4 / h 4 = 1/2 h 2 = 1 h^2 = 1 h 2 = 1 [11 ] ), where source frequency is assumed to be commensurate with the inverse observation time.
Variance of the Hellings and Downs correlation for an arbitrary set of pulsars
We assume that the reader is familiar with Ref. [11 ] , and extend that analysis in five ways:
(i) First, we define the Hellings and Downs correlation [3 ] for many pulsar pairs. We have found only one sensible way to generalize the standard “single-pulsar-pair” definition. We partition the angular range γ ∈ [ 0 , 180 ∘ ] \gamma\in[0,180^\circ] γ ∈ [ 0 , 18 0 ∘ ] [ γ l o w , γ h i g h ] [\gamma_{\mathrm{low}},\gamma_{\mathrm{high}}] [ γ low , γ high ] ρ a b \rho_{ab} ρ ab n p a i r s n_{\mathrm{pairs}} n pairs γ l o w < γ a b < γ h i g h \gamma_{\mathrm{low}} < \gamma_{ab} < \gamma_{\mathrm{high}} γ low < γ ab < γ high a a a b b b γ a b \gamma_{ab} γ ab
What type of average should we employ? As shown in Figure 2 , a uniform average, where the correlation of each pulsar pair gets the same weight, 1 / n p a i r s 1/n_{\mathrm{pairs}} 1/ n pairs ρ a b \rho_{ab} ρ ab
Figure 2. The bin at angular separation γ \gamma γ n p a i r s = 3 n_{\mathrm{pairs}}=3 n pairs = 3 γ ∼ γ 12 ∼ γ 38 ∼ γ 47 \gamma\sim\gamma_{12} \sim\gamma_{38} \sim\gamma_{47} γ ∼ γ 12 ∼ γ 38 ∼ γ 47 ρ n a i v e = ( ρ 12 + ρ 38 + ρ 47 ) / 3 \rho_{\mathrm{naive} }=(\rho_{12}+\rho_{38}+\rho_{47})/3 ρ naive = ( ρ 12 + ρ 38 + ρ 47 ) /3 ρ o p t = 0.3 ρ 12 + 0.3 ρ 38 + 0.4 ρ 47 \rho_{\mathrm{opt}}=0.3\rho_{12}+0.3\rho_{38}+0.4\rho_{47} ρ opt = 0.3 ρ 12 + 0.3 ρ 38 + 0.4 ρ 47 ρ 47 \rho_{47} ρ 47 ρ 12 \rho_{12} ρ 12 ρ 38 \rho_{38} ρ 38
In this paper, that weighted average (for a particular bin) is called the “optimal estimator.” In several places, we also consider a special case, which we call the “narrow-bin limit.” This is the (mathematical) limit in which the width (in γ \gamma γ
Conditions (1) and (2) lead immediately to a simple formula for the weights. The weights depend upon the sky directions to all of the pulsars which contribute to a particular bin, and are easily computed for any specific set of directions. In the limit of low pulsar noise, the resulting optimal estimator tells an observer (in a universe described by the Gaussian ensemble) the best way to combine measured pulsar-pair correlations to estimate the Hellings and Downs correlation in a particular bin.
As a byproduct, we also obtain simple formulas for (a) the variance of the optimal estimator for a given angular separation bin, and (b) the covariance of the optimal estimators for any pair of bins.
By definition these are the variance and covariance of the Hellings and Downs correlation. In particular, the variance, which by condition (2) is as small as possible, quantifies the uncertainty in the Hellings and Downs correlation for any particular bin. The variance depends upon the sky directions to all of the pulsars that contribute to that bin.
To illustrate this, Figure 3 shows the variance obtained by using the 88 pulsars currently employed by three active PTAs, after dividing the 88 × 87 / 2 = 3828 88 \times87/2=3828 88 × 87/2 = 3828 6 ∘ 6^\circ 6 ∘
Figure 3. The dots (lines are to guide the eye) show the predicted GW-induced variance σ o p t 2 \sigma_{\mathrm{opt}}^2 σ opt 2 × \times × 6 ∘ 6^\circ 6 ∘ σ t o t 2 \sigma_{\mathrm{tot}}^2 σ tot 2 σ c o s 2 \sigma_{\mathrm{cos}}^2 σ cos 2 Figure 9 , i.e., α = 1 \alpha=1 α = 1 h 2 2 ≈ 0.5622 h_2^2 \approx0.5622 h 2 2 ≈ 0.5622 § 24 for more details.
(ii) Second, we extend [11 ] by showing how the variance and covariance of the expected Hellings and Downs correlation decrease as pulsar pairs are added. For example, the variance of the optimal estimator for narrow bins drops from the total variance σ t o t 2 ( γ ) \sigma_{\mathrm{tot}}^2(\gamma) σ tot 2 ( γ ) σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^2(\gamma) σ cos 2 ( γ ) Figure 6 .
Figure 6. Left panel: the variance of the optimal estimator as the number of pulsar pairs at each angular separation γ \gamma γ σ o p t ( γ ) \sigma_{\mathrm{opt}}(\gamma) σ opt ( γ ) κ 4 = 1 / 2 \kappa^{4}=1/2 κ 4 = 1/2 σ o p t 2 ( γ ) \sigma_{\mathrm{opt}}^{2}(\gamma) σ opt 2 ( γ ) 1 / e ≈ 37 % 1/e\approx37\% 1/ e ≈ 37% σ o p t 2 ( γ ) ≈ ( 1 + 1 / e ) σ c o s 2 ( γ ) \sigma_{\mathrm{opt}}^{2}(\gamma)\approx(1+1/e)\sigma_{\mathrm{cos}}^{2}(\gamma) σ opt 2 ( γ ) ≈ ( 1 + 1/ e ) σ cos 2 ( γ )
Our approach to deriving the cosmic variance is different than that given in Sec. IV of [11 ] , and provides a useful
alternative interpretation. Because different pulsar-pair correlations are correlated with one another, as pulsar pairs are added to a bin, the variance does not approach zero. The reason is straightforward: once there are a sufficient number of pulsar pairs in the bin, adding more pairs to that bin provides negligible additional information about the GW-induced correlations. (Section III E demonstrates this behavior with a simple example.)
We also derive an elegant analytic expression for the covariance of the optimal estimators in narrow angular separation bins labeled by j j j k k k [11 ] , nor have we found it elsewhere in the literature. We give it both in “position-space” form and in harmonic space, where it is a diagonal sum of products of Legendre polynomials, dominated by the quadrupole plus a few additional multipoles. The cosmic covariance has a surprisingly simple structure, showing that the expected fluctuations away from the Hellings and Downs mean correlation are strongly correlated/anticorrelated in the three γ \gamma γ
(iii) Third, we address the question: “What is the strength of the GW stochastic background in the PTA band?” For simplicity, we assume that the frequency-dependent shape of the GW spectrum is known (e.g., ∝ ∣ f ∣ − 7 / 3 \propto|f|^{-7/3} ∝ ∣ f ∣ − 7/3 h 2 h^2 h 2
We construct optimal estimators of h 2 h^2 h 2 ρ a b \rho_{ab} ρ ab h 2 h^2 h 2
We show that if cross-correlations are included in the squared-strain estimator, then the variance tends to zero as N p u l → 0 N_{\rm pul} \to0 N pul → 0 h μ ν h μ ν h_{\mu\nu}h^{\mu\nu} h μν h μν
(iv) Fourth, to enable straightforward tests for consistency with the predictions of the Gaussian ensemble, we construct a set of χ 2 \chi^2 χ 2 h 2 μ u ( γ ) h^2\mu_u(\gamma) h 2 μ u ( γ ) χ 2 \chi^2 χ 2 h 2 h^2 h 2
To set the scale of the GW strain, there are two options: (1) assume that h 2 h^2 h 2 h 2 h^2 h 2 χ 2 \chi^2 χ 2 h 2 h^2 h 2
The χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2
(v) Our final extension to [11 ] , which assumes noise-free data, is to also consider the effects of pulsar and measurement noise. While most of our paper considers ideal data, noise is easily included. We do this starting in Sec. IX, assuming that the noise is uncorrelated between different pulsars. The noise modifies the pulsar-pair correlations and thus adds additional terms to the pulsar-pair expectation value, covariance matrix, and higher-order moments. This means that our entire approach and methodology can still be used in the presence of noise.
The noise contributions to the pulsar-pair correlations are fundamentally different than those which are induced by the GWs. For the GW-induced correlations, as the number of pulsar pairs increases, the variance in the Hellings and Downs correlation approaches the (nonzero) cosmic variance. In contrast, since we assume that the different pulsars have independent noise, the contribution of their noise to the variance of the Hellings and Downs correlation vanishes as the number of pulsar pairs increases. Thus, there are reasons to hope that during the coming decades, as PTAs add additional pulsars and improve their measurements, the limit of the cosmic variance can be achieved in practice.
Outline
Here, we give a brief outline of the paper, and introduce the most important notation. Note that our methods and results apply both to pulsar redshifts (set α = 0 \alpha= 0 α = 0 α = 1 \alpha= 1 α = 1
We begin in § 2 by defining the time-averaged correlation ρ a b \rho_{ab} ρ ab a a a b b b a b ab ab § 2 A we calculate the first moment ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ a ≠ b a \ne b a = b μ u ( γ a b ) \mu_u(\gamma_{ab}) μ u ( γ ab ) (1) , where γ a b \gamma_{ab} γ ab a = b a=b a = b μ a b \mu_{ab} μ ab (5) . In § 2 B, we use this to compute the “second moment” ⟨ ρ a b ρ c d ⟩ \langle\rho_{ab}\rho_{cd}\rangle ⟨ ρ ab ρ c d ⟩ C a b ; c d ≡ ⟨ ρ a b ρ c d ⟩ − ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ C_{ab;cd}\equiv\langle\rho_{ab}\rho_{cd}\rangle-\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle C ab ; c d ≡ ⟨ ρ ab ρ c d ⟩ − ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ c d cd c d c c c d d d ρ a b \rho_{ab} ρ ab § 2 C and Appendix § 12.6 .
In this paper, angle brackets ⟨ Q ⟩ \langle Q\rangle ⟨ Q ⟩ Q Q Q ⟨ 1 ⟩ = 1 \langle1\rangle=1 ⟨ 1 ⟩ = 1 [12 , 13 , 14 ] . However, in some places we (state that we) use it differently. For example, in (185) it denotes an average over a Gaussian ensemble of noise sources and GW sources. In (50) it denotes an average over pulsar pairs at fixed separation angles (indicated by the subscript c d ∈ γ k cd\in\gamma_k c d ∈ γ k
The covariance matrix C \mathbf{C} C C \mathbf{C} C C a b ; c d C_{ab;cd} C ab ; c d § 2 D, we derive two identities satisfied by the entries C a b ; c d C_{ab;cd} C ab ; c d
The way that C \mathbf{C} C § 2 E. Ordering pulsar pairs a b ab ab γ a b \gamma_{ab} γ ab C \mathbf{C} C (15) . The different blocks correspond to different angular bins, which we label with integers j j j k k k C j k C_{jk} C j k a b ab ab c d cd c d j j j k k k
Pulsar pairs a b ab ab c d cd c d j j j C \mathbf{C} C C = C j j C=C_{jj} C = C j j
In Sec. III, we turn our attention to defining the Hellings and Downs correlation for an arbitrary collection of pulsars. This is defined as the optimal estimator of correlation in a particular angular-separation bin, which is a weighted average of pulsar-pair correlations in that bin: ρ o p t = ∑ a b w a b ρ a b \rho_{\mathrm{opt}}=\sum_{ab}w_{ab}\rho_{ab} ρ opt = ∑ ab w ab ρ ab w a b w_{ab} w ab B j k B_{jk} B j k j j j k k k § 11 , where we define χ 2 \chi^2 χ 2 1 \mathbf{1} 1 C C C C \mathbf{C} C
Using this framework, Sec. IV investigates the behavior of the binned variance and covariance matrix B j k B_{jk} B j k [11 ] , the variances at the two extremes are called the “total variance” (one pair) and the “cosmic variance” (many pairs). We begin in Sec. IV A by averaging the entries C a b , c d C_{ab,cd} C ab , c d C \mathbf{C} C a b ab ab c d cd c d j j j k k k (57) , for the cosmic covariance matrix s j k s_{jk} s j k s j k s_{jk} s j k (61) , and calculate its inverse, s j k − 1 s_{jk}^{-1} s j k − 1 § 11 C, where we characterize the (fractional) variance of cross-correlation estimators h ^ 2 \hat{h}^{2} h ^ 2 (98) . Finally, in Sec. IV D, we obtain the cosmic variance computed in [11 ] by simply setting the bins’ indices, j j j k k k 1 \mathbf{1} 1 C j j C_{jj} C j j C \mathbf{C} C γ j \gamma_j γ j s j j s_{jj} s j j [11 ] , but the results are in exact agreement.
Current PTAs do not have enough pulsar pairs to be well described by the narrow-bin limit. In § 6 , we study the variance for this case. Since the bins have nonzero width in γ \gamma γ § 6 A we show that there is a compensating benefit: the variance of the optimal estimator can be smaller than the cosmic variance. In § 6 B, we apply our formalism to the sets of pulsars currently monitored by three active PTAs. With plots, we show how these sets reduce pulsar variance, illustrating (for the ideal noiseless case) how closely the variance approaches the cosmic variance.
In Sec. VI, for pedagogic purposes, we introduce hypothetical “non-PTA” detectors, whose response to GWs differ from that of PTAs. (Among the infinite set of possible detector responses, standard PTAs are simply a particular special case.) This is helpful in understanding the statistical properties of estimators of the squared GW strain h 2 h^2 h 2
In Sec. VII, we study the statistical properties of estimators h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 H ( f ) H(f) H ( f ) h 2 h^2 h 2
In all cases, the variance σ h ^ 2 2 \sigma^2_{\hat{h}^2} σ h ^ 2 2 N p u l N_{\mathrm{pul}} N pul Figure 10 , the rate of falloff depends upon which types of correlation measurements are included in the squared strain estimator. Section VII A presents the (mathematically) simplest case, where both auto- and cross-correlations are included in the estimator. A direct calculation shows that as the number of pulsars grows, the variance vanishes as 1 / N p u l 1/N_{\mathrm{pul}} 1/ N pul N p u l → ∞ N_{\mathrm{pul}} \to\infty N pul → ∞ Figure 13 , the variance vanishes as N p u l − 1 / 2 N_{\mathrm{pul}}^{-1/2} N pul − 1/2
Figure 10. The inverse variance of three different estimators of the squared GW strain, plotted as a function of the number of uniformly distributed pulsars. For large numbers of pulsars, the estimator based only on auto-correlations has a finite limit corresponding to 48 degrees of freedom. The estimator which employs only cross-correlations, and the estimator which employs both auto- and cross-correlations, have variances that vanish as N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞
Figure 13. The black curve shows the variance σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 h ^ 2 \hat{h}^2 h ^ 2 (100) for N p u l N_{\rm pul} N pul (124) : for large numbers of pulsars σ h ^ 2 2 ∝ N p u l − 1 / 2 \sigma_{\hat{h}^2}^2 \propto N_{\rm pul}^{-1/2} σ h ^ 2 2 ∝ N pul − 1/2 N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞
Since the Hellings and Downs correlation of the Gaussian ensemble has nonzero cosmic variance, we found it surprising that (if pulsar-pair cross-correlations are employed) there is no cosmic variance for the h 2 h^2 h 2 σ h ^ 2 2 → 0 \sigma^2_{\hat{h}^2} \to0 σ h ^ 2 2 → 0 s j k − 1 s_{jk}^{-1} s j k − 1 s j k s_{jk} s j k σ h ^ 2 2 \sigma^2_{\hat{h}^2} σ h ^ 2 2 (82) ] has contributions from an infinite number of (harmonic decomposition) modes. This contrasts with conventional detectors, such as the LIGO interferometers. Because the arms of LIGO-like detectors are small compared to the GW wavelength [16 ] they only respond to the five l = 2 l = 2 l = 2 (82) ].
In Sec. VII D, we calculate the fractional uncertainty σ h ^ 2 2 / ⟨ h ^ 2 ⟩ \sigma_{\hat{h}^2}^2/\langle\hat{h}^2\rangle σ h ^ 2 2 / ⟨ h ^ 2 ⟩ Table I . We also demonstrate a graphical method for constructing a “self-consistent interval” for h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 h 2 h^2 h 2 1 σ 1\sigma 1 σ
Table I. Fractional uncertainties for the different squared-strain estimators discussed in this section, assuming noise-free measurements for each PTA’s pulsars. The values in row (1) are obtained directly from (109) , whereas those in rows (2) and (3) are obtained by numerically evaluating (100) for the PTA’s specific pulsar sky directions. While the order of increasing uncertainty is (1), (2), (3), if there were more than ≈ 1500 \approx1500 ≈ 1500 Figure 10 . The values in row (4) are the fractional error σ s / ⟨ s ⟩ = 2 / 5 h ^ 2 / h 2 \sigma_s/\langle s\rangle=\sqrt{2/5}\,\hat{h}^2/h^2 σ s / ⟨ s ⟩ = 2/5 h ^ 2 / h 2 (112) if all pulsars are replaced by one-arm LIGO-like detectors sensitive to only the l = 2 l=2 l = 2 h ^ 2 / h 2 \hat{h}^2/h^2 h ^ 2 / h 2 α = 0 \alpha=0 α = 0 h ^ 2 / h 2 ≈ 0.3905 \hat{h}^2/h^2\approx0.3905 h ^ 2 / h 2 ≈ 0.3905 α = 1 \alpha=1 α = 1 h ^ 2 / h 2 ≈ 0.5622 \hat{h}^2/h^2\approx0.5622 h ^ 2 / h 2 ≈ 0.5622
In § 11 , we turn our attention to χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 § 11 A and VIII B, can either use “external” estimates of the squared GW strain h 2 h^2 h 2 χ 2 \chi^2 χ 2 h ^ 2 \hat{h}^2 h ^ 2
The χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 N b i n s = N p a i r s N_{\mathrm{bins}} = N_{\mathrm{pairs}} N bins = N pairs
In § 11 C, we investigate the variance of the χ 2 \chi^2 χ 2 N b i n s = N p a i r s N_{\mathrm{bins}} = N_{\mathrm{pairs}} N bins = N pairs σ χ 2 2 \sigma^2_{\chi^2} σ χ 2 2 ρ a b \rho_{ab} ρ ab § 12.6 . In all cases, the variance is dominated by a non-Gaussian term E \mathbb{E} E ρ a b \rho_{ab} ρ ab § 11 C 1) and numerical investigations (§ 11 C 3) show that as N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞ σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ ∝ N b i n s − 1 / 4 \propto N_{\rm bins}^{-1/4} ∝ N bins − 1/4 ∝ N b i n s − 1 / 2 \propto N_{\rm bins}^{-1/2} ∝ N bins − 1/2 ρ a b \rho_{ab} ρ ab § 11 C 2) it appears that the limit is positive, meaning that there is a (nonzero) cosmic variance for χ 2 \chi^2 χ 2 § 23 .
In § 11 C 4, we evaluate the (best-case, noise-free) fractional uncertainty in χ 2 \chi^2 χ 2 Table II gives σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 h 2 h^2 h 2
Table II. Fractional uncertainties for the different (unprojected) χ 2 \chi^2 χ 2 [57 ] . The total number of bins for each analysis is shown in parentheses. The values in row (1) are obtained directly from (8.47) for Ξ \Xi Ξ (162) (for specific pulsar sky directions) to determine Ξ \Xi Ξ
Section IX examines the effects of instrumental and pulsar timing noise, assuming that the noise in different pulsars is uncorrelated. We show how to modify quantities previously computed assuming noise-free measurements. These are (1) the optimal estimator of the Hellings and Downs correlation, (2) the optimal estimator of the squared GW strain h 2 h^2 h 2 χ 2 \chi^2 χ 2 ρ a b \rho_{ab} ρ ab ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ C a b ; c d C_{ab;cd} C ab ; c d D a b ; c d ; e f \mathcal{D}_{ab;cd;ef} D ab ; c d ; e f E a b ; c d ; e f ; g h \mathcal{E}_{ab;cd;ef;gh} E ab ; c d ; e f ; g h
In Sec. IX D, we examine how (2) is affected. In contrast to the noise-free case, the optimal estimator h ^ 2 \hat{h}^2 h ^ 2 ρ a b \rho_{ab} ρ ab h ^ 2 \hat{h}^2 h ^ 2 h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2
Lastly, in Sec. IX F, we show that for large numbers of pulsar pairs, the noise has no effect on the variance of the Hellings and Downs correlation. This relies on a detailed argument presented in Appendix C.
Our definition of the Hellings and Downs correlation, and our approach for computing the weights and corresponding variance, could be applied to any source of GWs. However, for much of the paper, we assume that the GW background arises from a Gaussian ensemble, which implies a special form for the covariance matrix. In § 13 , we explain the consequence: our predictions for the cosmic variance create a consistency test. If the Hellings and Downs correlation (estimated according to the recipe we present) matches the Hellings and Downs prediction much more (or much less) closely than the variance we compute, then it is very unlikely that our Universe has a GW background which is described by the Gaussian ensemble. As also argued in [111], this shows that the Hellings and Downs variance is an observable quantity: it could (in principle or practice) be used to falsify the widely accepted “Gaussian ensemble” model of the PTA GW background.
This is followed by a short conclusion in Sec. XI, and a number of technical appendices, Appendix A–Appendix § 24 . A summary of their contents may be found prior to the appendices.
Throughout the paper we use units in which the speed of light c = 1 c=1 c = 1 L \mathcal{L} L
Statistics of Pulsar-Pair Redshift Correlations
Consider a PTA built from a collection of N p u l N_{\rm pul} N pul a a a b b b c c c d d d 1 , 2 , … , N p u l 1,2,\ldots,N_{\rm pul} 1 , 2 , … , N pul Z a ( t ) Z_a(t) Z a ( t )
The main objects of our analysis are the correlations between two pulsars. The corresponding pulsar pair is labeled with two letters a b ab ab c d cd c d e f ef e f … \ldots … N p a i r s = N p u l ( N p u l − 1 ) / 2 N_{\rm pairs}=N_{\rm pul}(N_{\rm pul}-1)/2 N pairs = N pul ( N pul − 1 ) /2 a a a b b b γ a b \gamma_{ab} γ ab
The time-averaged redshift correlation between the pulses arriving from pulsars a a a b b b a b ab ab
ρ a b ≡ Z a Z b ‾ . (2) \rho_{ab} \equiv\overline{Z_a Z_b}. \tag*{(2)} ρ ab ≡ Z a Z b . (2) Here, the overline denotes an average over the time interval − T / 2 < t < T / 2 -T/2 < t < T/2 − T /2 < t < T /2 T T T [17 ] . For simplicity we take T T T
Note that (a) our calculations, formalism, results, and plots also apply to pulsar timing-residual correlations (as well as to redshifts) as explained in Appendix A. (b) Here, and in most of the paper, we assume noise-free pulsar redshift measurements and thus noise-free pulsar-pair correlations ρ a b \rho_{ab} ρ ab
In § 2 A and II B, we compute the first moment ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab
C a b , c d ≡ ⟨ ρ a b ρ c d ⟩ − ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ . (3) C_{ab,cd} \equiv\langle\rho_{ab}\rho_{cd}\rangle-\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle. \tag*{(3)} C ab , c d ≡ ⟨ ρ ab ρ c d ⟩ − ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ . (3) Note that the pairs a b ab ab c d cd c d ⟨ ρ a b ρ c d ⟩ \langle\rho_{ab}\rho_{cd}\rangle ⟨ ρ ab ρ c d ⟩ correlation of the correlations . To avoid such awkward language, we write “second moment” even when a b ab ab c d cd c d
The angle brackets denote the average over a Gaussian ensemble. As shown in Appendix C of [11 ] , this ensemble corresponds to an infinite collection of weak, unpolarized, time-stationary GW sources, distributed uniformly in space [12 , 13 , 14 ] . In any narrow frequency band, there are an infinite number of sources radiating at indistinguishably close GW frequencies, but with different (random) phases. This generates “confusion noise,” giving rise via the central-limit theorem to a stationary and Gaussian stochastic process.
As explained in § 0.1 B, our definition of the Hellings and Downs correlation is based on angular bins. For this reason, it is helpful to arrange the entries of C a b , c d C_{ab,cd} C ab , c d C C C § 2 E. First, the pulsar pairs a b ab ab γ a b \gamma_{ab} γ ab C C C
First moment
For the unpolarized isotropic stationary Gaussian ensemble, Eq. (C15) in [11 ] gives the first moment of the correlation as
⟨ ρ a b ⟩ = h 2 μ a b , (4) \langle\rho_{ab}\rangle= h^2\mu_{ab}, \tag*{(4)} ⟨ ρ ab ⟩ = h 2 μ ab , (4) where
μ a b ≡ μ u ( γ a b ) + δ a b μ u ( 0 ) . (5) \mu_{ab} \equiv\mu_u(\gamma_{ab})+\delta_{ab}\mu_u(0). \tag*{(5)} μ ab ≡ μ u ( γ ab ) + δ ab μ u ( 0 ) . (5) Here, δ a b \delta_{ab} δ ab a a a b b b μ u ( γ a b ) \mu_u(\gamma_{ab}) μ u ( γ ab ) (1) . Its fundamental underlying definition is [3 ]
μ u ( γ a b ) ≡ 1 4 π ∫ d Ω ^ ∑ A F a A ( Ω ^ ) F b A ( Ω ^ ) . (6) \mu_u(\gamma_{ab}) \equiv\frac{1}{4\pi}\int\mathrm{d}\widehat{\Omega}\sum_A F_a^A(\widehat{\Omega})F_b^A(\widehat{\Omega}). \tag*{(6)} μ u ( γ ab ) ≡ 4 π 1 ∫ d Ω A ∑ F a A ( Ω ) F b A ( Ω ) . (6) This is an average of the products of antenna pattern functions F a A ( Ω ^ ) F_a^A(\widehat{\Omega}) F a A ( Ω ) F b A ( Ω ^ ) F_b^A(\widehat{\Omega}) F b A ( Ω ) Ω ^ \widehat{\Omega} Ω Ω ^ \widehat{\Omega} Ω A = + , × A=+,\times A = + , × (1) starting from (6) are given in Appendix D in [11 ] .
The function μ u ( γ a b ) \mu_u(\gamma_{ab}) μ u ( γ ab ) distinct pulsars a a a b b b γ a b \gamma_{ab} γ ab (5) for time-stationary GW signals is discussed in detail in Appendix C.§ 11.5 of [11 ] . It arises because the expected correlation for a pair of identical pulsars a = b a=b a = b distinct pulsars a a a b b b [18 ] from Earth. If pulsars a a a b b b
The overall scale of the first moment ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩
h 2 ≡ 4 π ∫ − ∞ ∞ d f ( 2 π f ) − 2 α H ( f ) . (7) h^2 \equiv4\pi\int_{-\infty}^{\infty}\mathrm{d}f\,(2\pi f)^{-2\alpha}H(f). \tag*{(7)} h 2 ≡ 4 π ∫ − ∞ ∞ d f ( 2 π f ) − 2 α H ( f ) . (7) Set α = 0 \alpha=0 α = 0 α = 1 \alpha=1 α = 1 h h h h h h × \times ×
The real two-sided squared-strain spectral density H ( f ) H(f) H ( f ) Ω g w ( f ) \Omega_{\rm gw}(f) Ω gw ( f ) h c 2 ( f ) h_c^2(f) h c 2 ( f )
H ( f ) = 3 H 0 2 32 π 3 1 ∣ f ∣ 3 Ω g w ( ∣ f ∣ ) = 1 16 π 2 1 ∣ f ∣ h c 2 ( ∣ f ∣ ) . (8) H(f)=\frac{3H_0^2}{32\pi^3}\frac{1}{|f|^3}\Omega_{\rm gw}(|f|)=\frac{1}{16\pi^2}\frac{1}{|f|}h_c^2(|f|). \tag*{(8)} H ( f ) = 32 π 3 3 H 0 2 ∣ f ∣ 3 1 Ω gw ( ∣ f ∣ ) = 16 π 2 1 ∣ f ∣ 1 h c 2 ( ∣ f ∣ ) . (8) with H ( f ) ≥ 0 H(f)\geq0 H ( f ) ≥ 0 H ( f ) = H ( − f ) H(f)=H(-f) H ( f ) = H ( − f ) [11 , 12 , 13 ] .
Note that the derivation of (4) assumes that the spectral function H ( f ) H(f) H ( f )
Second moment and covariance matrix
For a Gaussian ensemble, the second moment of the pulsar-pair correlation can be found following the methods in Appendix C.§ 11.6 in [11 ] . There, the ensemble average ⟨ ρ 1 ρ 2 ⟩ \langle\rho_1\rho_2\rangle ⟨ ρ 1 ρ 2 ⟩ a a a b b b c c c d d d [19 ] , which expresses the four-point function for zero-mean Gaussian variables as a sum of three products of two-point functions, giving
⟨ ρ a b ρ c d ⟩ = ( h 2 μ a b ) ( h 2 μ c d ) + h 4 μ a c μ b d + h 4 μ a d μ b c = ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ + h 4 ( μ a c μ b d + μ a d μ b c ) . (9) \begin{aligned}
\langle\rho_{ab}\rho_{cd}\rangle
&=(h^2\mu_{ab})(h^2\mu_{cd})+h^4\mu_{ac}\mu_{bd}+h^4\mu_{ad}\mu_{bc}\\
&=\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle+h^4(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}). \tag*{(9)}
\end{aligned} ⟨ ρ ab ρ c d ⟩ = ( h 2 μ ab ) ( h 2 μ c d ) + h 4 μ a c μ b d + h 4 μ a d μ b c = ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ + h 4 ( μ a c μ b d + μ a d μ b c ) . (9) The second line of (9) is obtained from the first line by using (4) .
The factor which appears on the rhs of (9) sets the scale of the covariance, and is
h ^ 4 ≡ ( 4 π ) 2 ∫ − ∞ ∞ d f ∫ − ∞ ∞ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α H ( f ) H ( f ′ ) (10) \hat{h}^{4} \equiv(4\pi)^{2} \int_{-\infty}^{\infty} \mathrm{d}f \int_{-\infty}^{\infty} \mathrm{d}f' \, \operatorname{sinc}^{2}\left(\pi(f-f')T\right)(4\pi^{2}ff')^{-2\alpha}H(f)H(f') \tag*{(10)} h ^ 4 ≡ ( 4 π ) 2 ∫ − ∞ ∞ d f ∫ − ∞ ∞ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α H ( f ) H ( f ′ ) (10) where sinc x ≡ sin ( x ) / x \operatorname{sinc}x \equiv\sin(x)/x sinc x ≡ sin ( x ) / x [20 ] and T T T α \alpha α α = 0 \alpha=0 α = 0 h ^ \hat{h} h ^ α = 1 \alpha=1 α = 1 h ^ \hat{h} h ^ × \times ×
To compare the variance (proportional to h ^ 4 \hat{h}^{4} h ^ 4 h 4 h^{4} h 4 h ^ 4 / h 4 \hat{h}^{4}/h^{4} h ^ 4 / h 4 [11 ] . Note that since H ( f ) ≥ 0 H(f)\geq0 H ( f ) ≥ 0 0 ≤ sinc 2 x ≤ 1 0\leq\operatorname{sinc}^{2}x\leq1 0 ≤ sinc 2 x ≤ 1 (7) and (10) that h ^ 4 ≤ h 4 \hat{h}^{4}\leq h^{4} h ^ 4 ≤ h 4 h ^ 2 / h 2 \hat{h}^{2}/h^{2} h ^ 2 / h 2 Figure 21 .
Figure 21. The quantity R 2 / h 2 \mathcal{R}^2/h^2 R 2 / h 2 σ \sigma σ ⟨ ρ ⟩ \langle\rho\rangle ⟨ ρ ⟩ H ( f ) H(f) H ( f ) f 0 f_0 f 0 1 / T 1/T 1/ T α = 0 \alpha=0 α = 0 α = 1 \alpha=1 α = 1
The covariance matrix is defined by (3) . For the Gaussian ensemble, making use of (9) we obtain
C a b , c d = h ^ 4 ( μ a c μ b d + μ a d μ b c ) . (11) C_{ab,cd}=\hat{h}^{4}\left(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}\right). \tag*{(11)} C ab , c d = h ^ 4 ( μ a c μ b d + μ a d μ b c ) . (11) In this expression, the pulsars a a a b b b c c c d d d a b ab ab c d cd c d
Expression (11) for the entries of the covariance matrix has an interesting structure. Although the lhs is (at least conceptually) a function of the two pulsar pairs a b ab ab c d cd c d a c ac a c a d ad a d b c bc b c b d bd b d a b ab ab c d cd c d
Expression (11) also shows that the covariance matrix is not diagonal. A diagonal covariance matrix could be obtained by replacing the μ \mu μ μ e f \mu_{ef} μ e f e e e f f f
As a sanity check, consider a diagonal element of the covariance matrix C a b , a b C_{ab,ab} C ab , ab a b ab ab a ≠ b a\ne b a = b c = a c=a c = a d = b d=b d = b (11) can be simplified using (5) . This gives the total (pulsar plus cosmic) variance of ρ a b \rho_{ab} ρ ab
C a b , a b = σ t o t 2 = h ^ 4 ( μ u 2 ( γ a b ) + 4 μ u 2 ( 0 ) ) . (12) C_{ab,ab}=\sigma_{\mathrm{tot}}^{2}=\hat{h}^{4}\left(\mu_{u}^{2}(\gamma_{ab})+4\mu_{u}^{2}(0)\right). \tag*{(12)} C ab , ab = σ tot 2 = h ^ 4 ( μ u 2 ( γ ab ) + 4 μ u 2 ( 0 ) ) . (12) This diagonal element agrees exactly with the result found in Eq. (C28) of [11 ] , which corresponds to (11) with a = 1 a=1 a = 1 b = 2 b=2 b = 2 c = 1 c=1 c = 1 d = 2 d=2 d = 2
Third and fourth moments
The third- and fourth-order moments of the pulsar-pair correlation ρ a b \rho_{ab} ρ ab C a b , c d C_{ab,cd} C ab , c d Z a , Z b , … , Z h Z_a,Z_b,\ldots,Z_h Z a , Z b , … , Z h χ 2 \chi^{2} χ 2 § 11 , we put these calculations in Appendix § 12.6 . For the remaining analyses of this paper, the central object of interest is the covariance matrix C a b , c d C_{ab,cd} C ab , c d
Useful identities
For later use, we provide two useful identities involving the covariance matrix C a b , c d C_{ab,cd} C ab , c d (5) to expand the rhs of (11) for C a b , c d C_{ab,cd} C ab , c d
μ a c μ b d + μ a d μ b c = μ u ( γ a c ) μ u ( γ b d ) + μ u ( γ a d ) μ u ( γ b c ) + μ u 2 ( 0 ) [ δ a c δ b d + δ a d δ b c ] + μ u ( 0 ) [ μ ( γ b d ) δ a c + μ ( γ a c ) δ b d + μ ( γ b c ) δ a d + μ ( γ a d ) δ b c ] . (13) \mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}
=\mu_{u}(\gamma_{ac})\mu_{u}(\gamma_{bd})+\mu_{u}(\gamma_{ad})\mu_{u}(\gamma_{bc})+\mu_{u}^{2}(0)\left[\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right]
+\mu_{u}(0)\left[\mu(\gamma_{bd})\delta_{ac}+\mu(\gamma_{ac})\delta_{bd}+\mu(\gamma_{bc})\delta_{ad}+\mu(\gamma_{ad})\delta_{bc}\right]. \tag*{(13)} μ a c μ b d + μ a d μ b c = μ u ( γ a c ) μ u ( γ b d ) + μ u ( γ a d ) μ u ( γ b c ) + μ u 2 ( 0 ) [ δ a c δ b d + δ a d δ b c ] + μ u ( 0 ) [ μ ( γ b d ) δ a c + μ ( γ a c ) δ b d + μ ( γ b c ) δ a d + μ ( γ a d ) δ b c ] . (13) The second identity provides an alternative form for the first two terms on the rhs of (13) . Making use of (6) , the products of Hellings and Downs curves may be written in terms of antenna pattern functions as
μ u ( γ a c ) μ u ( γ b d ) + μ u ( γ a d ) μ u ( γ b c ) = ∑ A ∑ A ′ ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π ( F a A ( Ω ^ ) F c A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F d A ′ ( Ω ^ ′ ) + F a A ( Ω ^ ) F d A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F c A ′ ( Ω ^ ′ ) ) . (14) \begin{aligned}
\mu_{u}(\gamma_{ac})\mu_{u}(\gamma_{bd})+\mu_{u}(\gamma_{ad})\mu_{u}(\gamma_{bc})
={}&\sum_{A}\sum_{A'}\int\frac{\mathrm{d}\hat{\Omega}}{4\pi}\int\frac{\mathrm{d}\hat{\Omega}'}{4\pi}
\left(F_{a}^{A}(\hat{\Omega})F_{c}^{A}(\hat{\Omega})F_{b}^{A'}(\hat{\Omega}')F_{d}^{A'}(\hat{\Omega}')
+F_{a}^{A}(\hat{\Omega})F_{d}^{A}(\hat{\Omega})F_{b}^{A'}(\hat{\Omega}')F_{c}^{A'}(\hat{\Omega}')\right). \tag*{(14)}
\end{aligned} μ u ( γ a c ) μ u ( γ b d ) + μ u ( γ a d ) μ u ( γ b c ) = A ∑ A ′ ∑ ∫ 4 π d Ω ^ ∫ 4 π d Ω ^ ′ ( F a A ( Ω ^ ) F c A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F d A ′ ( Ω ^ ′ ) + F a A ( Ω ^ ) F d A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F c A ′ ( Ω ^ ′ ) ) . (14) We use these identities in Sec. IV to calculate the average value of a row of C j k C_{jk} C j k (15) , j j j k k k [11 ] .
To form the matrix C \mathcal{C} C C a b , c d \mathcal{C}_{ab,cd} C ab , c d γ a b \gamma_{ab} γ ab (3) , each angular bin provides a lower and upper bound for this index, making it easy to restrict attention to pulsar pairs within the bin around angular separation γ \gamma γ
To explicitly illustrate this indexing scheme, suppose that we have N p u l = 101 N_{\mathrm{pul}}=101 N pul = 101 N p a i r s = 5050 N_{\mathrm{pairs}}=5050 N pairs = 5050 cos γ a b ∈ [ − 1 , 1 ] \cos\gamma_{ab}\in[-1,1] cos γ ab ∈ [ − 1 , 1 ] κ = 50 \kappa=50 κ = 50 d ( cos γ ) = 0.04 \mathrm{d}(\cos\gamma)=0.04 d ( cos γ ) = 0.04 n p a i r s = 101 n_{\mathrm{pairs}}=101 n pairs = 101 cos γ \cos\gamma cos γ ± 0.04 \pm0.04 ± 0.04 n p a i r s n_{\mathrm{pairs}} n pairs N p a i r s N_{\mathrm{pairs}} N pairs n p a i r s , j n_{\mathrm{pairs},j} n pairs , j j = 1 , 2 , … , 50 j=1,2,\ldots,50 j = 1 , 2 , … , 50 γ j \gamma_j γ j γ a b \gamma_{ab} γ ab j j j [21 ] .
With this angular-separation binning scheme, the covariance matrix C \mathcal{C} C (11) , takes the following “block-matrix form”:
C = ( C 11 C 12 ⋯ C 1 κ C 21 C 22 ⋯ C 2 κ ⋮ ⋮ ⋱ ⋮ C κ 1 C κ 2 ⋯ C κ κ ) (15) \mathcal{C}=\begin{pmatrix}
C_{11} & C_{12} & \cdots& C_{1\kappa}\\
C_{21} & C_{22} & \cdots& C_{2\kappa}\\
\vdots& \vdots& \ddots& \vdots\\
C_{\kappa1} & C_{\kappa2} & \cdots& C_{\kappa\kappa}
\end{pmatrix} \tag*{(15)} C = C 11 C 21 ⋮ C κ 1 C 12 C 22 ⋮ C κ 2 ⋯ ⋯ ⋱ ⋯ C 1 κ C 2 κ ⋮ C κκ (15) with κ = 50 \kappa=50 κ = 50 C j k C_{jk} C j k j , k ∈ 1 , 2 , … , κ j,k\in1,2,\ldots,\kappa j , k ∈ 1 , 2 , … , κ C \mathcal{C} C C C C
In this example, each block C j k C_{jk} C j k 101 × 101 101\times101 101 × 101 j j j k k k a b ab ab γ a b ≈ γ j \gamma_{ab}\approx\gamma_j γ ab ≈ γ j c d cd c d γ c d ≈ γ k \gamma_{cd}\approx\gamma_k γ c d ≈ γ k j j j C C C C j j C_{jj} C j j (15) . These diagonal blocks inherit the most important properties of C \mathcal{C} C [22 ] .
If j j j k k k C j k C_{jk} C j k C j k C_{jk} C j k independent of the row. In this case, the only thing we will need for our analysis is the average value s j k s_{jk} s j k C j k C_{jk} C j k s j j s_{jj} s j j σ c o s 2 ( γ j ) \sigma_{\mathrm{cos}}^2(\gamma_j) σ cos 2 ( γ j ) s j k s_{jk} s j k
Optimal estimation of the Hellings and Downs correlation
In this section, we define the Hellings and Downs correlation for a set of pulsars, as described in Sec. I B. For a given angular bin (in γ \gamma γ unbiased and have minimum variance .
Derivation of the optimal estimator and its variance
We now derive the optimal way to combine pulsar-pair correlation measurements ρ a b \rho_{ab} ρ ab γ a b \gamma_{ab} γ ab j j j n p a i r s n_{\mathrm{pairs}} n pairs ρ ≡ ρ a b \boldsymbol{\rho}\equiv\rho_{ab} ρ ≡ ρ ab j j j C ≡ C j j C\equiv C_{jj} C ≡ C j j ρ a b \rho_{ab} ρ ab ρ c d \rho_{cd} ρ c d
The optimal estimator is a linear combination
ρ o p t = w T ρ = ∑ a b = 1 n p a i r s w a b ρ a b (16) \rho_{\mathrm{opt}}=\mathbf{w}^{\mathsf{T}}\boldsymbol{\rho}=\sum_{ab=1}^{n_{\mathrm{pairs}}}w_{ab}\rho_{ab} \tag*{(16)} ρ opt = w T ρ = ab = 1 ∑ n pairs w ab ρ ab (16) of the correlations in the bin, where w \mathbf{w} w w a b w_{ab} w ab w T \mathbf{w}^{\mathsf{T}} w T w \mathbf{w} w ρ o p t \rho_{\mathrm{opt}} ρ opt n p a i r s n_{\mathrm{pairs}} n pairs a b ab ab j j j
Note that (16) is proportional to ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab a ≠ b a\ne b a = b (16) is sufficient to construct an optimal (unbiased and minimum-variance ) estimator of the Hellings and Downs correlation. Likewise, if the sum (16) includes auto-correlations (a = b a=b a = b (16) is not general enough to form an unbiased estimator. For that, an additional constant term must be subtracted from the rhs of (16) . In what follows, we concentrate on the ideal noise-free case, where the constant term is zero. Later, when we consider the effects of noise in Sec. IX, this constant term must be nonzero to obtain an unbiased minimum-variance estimator.
We start by calculating the variance of the optimal estimator:
σ o p t 2 ≡ ⟨ ρ o p t 2 ⟩ − ⟨ ρ o p t ⟩ 2 , (17) \sigma_{\mathrm{opt}}^2 \equiv\langle\rho_{\mathrm{opt}}^2 \rangle- \langle\rho_{\mathrm{opt}} \rangle^2, \tag*{(17)} σ opt 2 ≡ ⟨ ρ opt 2 ⟩ − ⟨ ρ opt ⟩ 2 , (17) where angle brackets denote the expectation value for the Gaussian ensemble used in, e.g., (4) and (9) . Substituting (16) into (17) , we obtain
σ o p t 2 = w T ⟨ ρ ρ T ⟩ − ⟨ ρ ⟩ ⟨ ρ T ⟩ w = w T C w = ( C w , C w ) , (18) \sigma_{\mathrm{opt}}^2 = \mathbf{w}^{\mathsf{T}}\left\langle\rho\rho^{\mathsf{T}}\right\rangle- \langle\rho\rangle\left\langle\rho^{\mathsf{T}}\right\rangle\mathbf{w} = \mathbf{w}^{\mathsf{T}}C\mathbf{w} = (C\mathbf{w},C\mathbf{w}), \tag*{(18)} σ opt 2 = w T ⟨ ρ ρ T ⟩ − ⟨ ρ ⟩ ⟨ ρ T ⟩ w = w T C w = ( C w , C w ) , (18) where the second equality follows from the definition (3) of the covariance matrix (restricted to block C ≡ C j j C \equiv C_{jj} C ≡ C j j j j j A A A B B B
( A , B ) = A T C − 1 B , (19) (A,B) = A^{\mathsf{T}}C^{-1}B, \tag*{(19)} ( A , B ) = A T C − 1 B , (19) and use the property that the diagonal block C C C C T = C C^{\mathsf{T}} = C C T = C
To completely determine the weights, we need a normalization constraint. The expected value of the estimator follows immediately from (4) and (16) , as
⟨ ρ o p t ⟩ = h 2 w T μ = h 2 μ T w = h 2 ( μ , C w ) , (20) \langle\rho_{\mathrm{opt}}\rangle= h^2\mathbf{w}^{\mathsf{T}}\mu= h^2\mu^{\mathsf{T}}\mathbf{w} = h^2(\mu,C\mathbf{w}), \tag*{(20)} ⟨ ρ opt ⟩ = h 2 w T μ = h 2 μ T w = h 2 ( μ , C w ) , (20) where μ \mu μ μ a b = μ u ( γ a b ) \mu_{ab}=\mu_u(\gamma_{ab}) μ ab = μ u ( γ ab ) a ≠ b a\ne b a = b ⟨ ρ o p t ⟩ \langle\rho_{\mathrm{opt}}\rangle ⟨ ρ opt ⟩ unbiased .
Using the subscript “bin” to denote this somewhat arbitrary choice for the normalization constraint, the mean value of our estimator should correspond to the expected Hellings and Downs correlation
⟨ ρ o p t ⟩ ≡ ρ b i n ≡ h 2 μ b i n (21) \langle\rho_{\mathrm{opt}}\rangle\equiv\rho_{\mathrm{bin}} \equiv h^2\mu_{\mathrm{bin}} \tag*{(21)} ⟨ ρ opt ⟩ ≡ ρ bin ≡ h 2 μ bin (21) for some choice of μ b i n \mu_{\mathrm{bin}} μ bin j j j
(a) μ b i n = μ u ( γ b i n ) \mu_{\mathrm{bin}}=\mu_u(\gamma_{\mathrm{bin}}) μ bin = μ u ( γ bin ) γ b i n \gamma_{\mathrm{bin}} γ bin
(b) μ b i n = μ T 1 / n p a i r s \mu_{\mathrm{bin}}=\mu^{\mathsf{T}}\mathbf{1}/n_{\mathrm{pairs}} μ bin = μ T 1 / n pairs 1 \mathbf{1} 1 n p a i r s n_{\mathrm{pairs}} n pairs
1 ≡ ( 1 , 1 , … , 1 ) T . (22) \mathbf{1}\equiv\left(1,1,\ldots,1\right)^{\mathsf{T}}. \tag*{(22)} 1 ≡ ( 1 , 1 , … , 1 ) T . (22) (c) μ b i n = μ u ( γ b i n ) \mu_{\mathrm{bin}}=\mu_u(\gamma_{\mathrm{bin}}) μ bin = μ u ( γ bin ) γ b i n = γ T 1 / n p a i r s \gamma_{\mathrm{bin}}=\gamma^{\mathsf{T}}\mathbf{1}/n_{\mathrm{pairs}} γ bin = γ T 1 / n pairs γ \gamma γ γ a b \gamma_{ab} γ ab
We will see that this arbitrary choice only enters the mean and the (square root of the) variance as an overall scale. So, while there may be other justifiable choices which are not listed above, we will see that the choice has no effect on the most important quantity: the fractional uncertainty σ o p t / ρ o p t \sigma_{\mathrm{opt}}/\rho_{\mathrm{opt}} σ opt / ρ opt
For any of these arbitrary normalization choices, we can now solve for the weights that define the optimal (minimum-variance) estimator. Since the expected value ⟨ ρ o p t ⟩ ≡ ρ b i n ≡ h 2 μ b i n \langle\rho_{\mathrm{opt}}\rangle\equiv\rho_{\mathrm{bin}}\equiv h^2\mu_{\mathrm{bin}} ⟨ ρ opt ⟩ ≡ ρ bin ≡ h 2 μ bin w \mathbf{w} w σ o p t 2 \sigma_{\mathrm{opt}}^2 σ opt 2 w \mathbf{w} w w \mathbf{w} w
σ o p t 2 ρ b i n 2 = σ o p t 2 ⟨ ρ o p t ⟩ 2 = ( C w , C w ) h 4 ( μ , C w ) 2 , (23) \frac{\sigma_{\mathrm{opt}}^2}{\rho_{\mathrm{bin}}^2}=\frac{\sigma_{\mathrm{opt}}^2}{\langle\rho_{\mathrm{opt}}\rangle^2}=\frac{(C\mathbf{w},C\mathbf{w})}{h^4(\mu,C\mathbf{w})^2}, \tag*{(23)} ρ bin 2 σ opt 2 = ⟨ ρ opt ⟩ 2 σ opt 2 = h 4 ( μ , C w ) 2 ( C w , C w ) , (23) where we have used (18) and (20) .
The inner product defined by (19) is positive definite and obeys the Schwarz inequality. Hence, the denominator of (23) is maximum when the vectors μ \mu μ C w C\mathbf{w} C w C w = q μ C\mathbf{w}=q\mu C w = q μ q q q w = q C − 1 μ \mathbf{w}=qC^{-1}\mu w = q C − 1 μ (20) then lets us solve for q q q
h 2 ( μ , q μ ) = h 2 μ b i n ⇒ q = μ b i n μ T C − 1 μ ⇒ w = μ b i n μ T C − 1 μ C − 1 μ . (24) h^2(\mu,q\mu)=h^2\mu_{\mathrm{bin}}\ \Rightarrow\ q=\frac{\mu_{\mathrm{bin}}}{\mu^{\mathsf{T}}C^{-1}\mu}\ \Rightarrow\ \mathbf{w}=\frac{\mu_{\mathrm{bin}}}{\mu^{\mathsf{T}}C^{-1}\mu}C^{-1}\mu. \tag*{(24)} h 2 ( μ , q μ ) = h 2 μ bin ⇒ q = μ T C − 1 μ μ bin ⇒ w = μ T C − 1 μ μ bin C − 1 μ . (24) From the optimal weights given in (24) , the optimal estimator and its variance are then given by (16) and (18) as
ρ o p t = μ b i n μ T C − 1 ρ μ T C − 1 μ (25) \rho_{\mathrm{opt}}=\mu_{\mathrm{bin}}\frac{\mu^{\mathsf{T}}C^{-1}\rho}{\mu^{\mathsf{T}}C^{-1}\mu} \tag*{(25)} ρ opt = μ bin μ T C − 1 μ μ T C − 1 ρ (25) and
σ o p t 2 = μ b i n 2 μ T C − 1 μ . (26) \sigma_{\mathrm{opt}}^2=\frac{\mu_{\mathrm{bin}}^2}{\mu^{\mathsf{T}}C^{-1}\mu}. \tag*{(26)} σ opt 2 = μ T C − 1 μ μ bin 2 . (26) As discussed after (22) , the ratio ρ o p t / σ o p t \rho_{\mathrm{opt}}/\sigma_{\mathrm{opt}} ρ opt / σ opt μ b i n \mu_{\mathrm{bin}} μ bin
While this paper is specifically focused on GW backgrounds that are described by a Gaussian ensemble, the results of this section apply to any GW ensemble. This is because any ensemble has a covariance matrix, from which one can define the (best estimate of the) pulsar-pair correlation (25) and the corresponding variance (26) . So, this way of defining the Hellings and Downs correlation for some set of pulsar pairs, and computing its variance, is quite general.
Covariance between optimal estimates
In Sec. I B and Figure 2 , we explained the importance of correlations (in the Hellings and Downs correlation) between different pairs of pulsars. This correlation extends to the optimal estimators which we have constructed for different angular bins. For example, if the optimal estimator in one angular bin lies somewhat above the Hellings and Downs curve, then it is very likely that the optimal estimator for an adjacent angular bin also lies above the curve. This is because pulsar pairs are correlated, even if they do not lie in the same (arbitrarily defined) angular bin.
The correlation between deviations from the Hellings and Downs curve in the j j j k k k (16) as
ρ o p t , j = w j T ρ j and ρ o p t , k = w k T ρ k , (27) \rho_{\mathrm{opt},j} = w_j^{\mathsf{T}}\rho_j \quad\text{and}\quad\rho_{\mathrm{opt},k} = w_k^{\mathsf{T}}\rho_k, \tag*{(27)} ρ opt , j = w j T ρ j and ρ opt , k = w k T ρ k , (27) with optimal weights given by (24) :
w j = μ b i n , j μ j T C j j − 1 μ j C j j − 1 μ j and w k = μ b i n , k μ k T C k k − 1 μ k C k k − 1 μ k . (28) w_j = \frac{\mu_{\mathrm{bin},j}}{\mu_j^{\mathsf{T}}C_{jj}^{-1}\mu_j}C_{jj}^{-1}\mu_j \quad\text{and}\quad w_k = \frac{\mu_{\mathrm{bin},k}}{\mu_k^{\mathsf{T}}C_{kk}^{-1}\mu_k}C_{kk}^{-1}\mu_k. \tag*{(28)} w j = μ j T C j j − 1 μ j μ bin , j C j j − 1 μ j and w k = μ k T C k k − 1 μ k μ bin , k C k k − 1 μ k . (28) The covariance between ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j ρ o p t , k \rho_{\mathrm{opt},k} ρ opt , k
B j k ≡ ⟨ ( ρ o p t , j − ⟨ ρ o p t , j ⟩ ) ( ρ o p t , k − ⟨ ρ o p t , k ⟩ ) ⟩ = ⟨ ρ o p t , j ρ o p t , k ⟩ − ⟨ ρ o p t , j ⟩ ⟨ ρ o p t , k ⟩ . (29) B_{jk} \equiv\left\langle\left(\rho_{\mathrm{opt},j}-\left\langle\rho_{\mathrm{opt},j}\right\rangle\right)\left(\rho_{\mathrm{opt},k}-\left\langle\rho_{\mathrm{opt},k}\right\rangle\right)\right\rangle= \left\langle\rho_{\mathrm{opt},j}\rho_{\mathrm{opt},k}\right\rangle-\left\langle\rho_{\mathrm{opt},j}\right\rangle\left\langle\rho_{\mathrm{opt},k}\right\rangle. \tag*{(29)} B j k ≡ ⟨ ( ρ opt , j − ⟨ ρ opt , j ⟩ ) ( ρ opt , k − ⟨ ρ opt , k ⟩ ) ⟩ = ⟨ ρ opt , j ρ opt , k ⟩ − ⟨ ρ opt , j ⟩ ⟨ ρ opt , k ⟩ . (29) Expanding the rhs of (29) using (27) , this becomes
B j k = w j T ( ⟨ ρ j ρ k T ⟩ − ⟨ ρ j ⟩ ⟨ ρ k T ⟩ ) w k = w j T C j k w k = μ b i n , j μ b i n , k ( μ j T C j j − 1 μ j ) ( μ k T C k k − 1 μ k ) μ j T C j j − 1 C j k C k k − 1 μ k . (30) \begin{aligned}
B_{jk} &= w_j^{\mathsf{T}}\left(\left\langle\rho_j\rho_k^{\mathsf{T}}\right\rangle-\left\langle\rho_j\right\rangle\left\langle\rho_k^{\mathsf{T}}\right\rangle\right)w_k \\
&= w_j^{\mathsf{T}}C_{jk}w_k \\
&= \frac{\mu_{\mathrm{bin},j}\mu_{\mathrm{bin},k}}{\left(\mu_j^{\mathsf{T}}C_{jj}^{-1}\mu_j\right)\left(\mu_k^{\mathsf{T}}C_{kk}^{-1}\mu_k\right)}\mu_j^{\mathsf{T}}C_{jj}^{-1}C_{jk}C_{kk}^{-1}\mu_k. \tag*{(30)}
\end{aligned} B j k = w j T ( ⟨ ρ j ρ k T ⟩ − ⟨ ρ j ⟩ ⟨ ρ k T ⟩ ) w k = w j T C j k w k = ( μ j T C j j − 1 μ j ) ( μ k T C k k − 1 μ k ) μ bin , j μ bin , k μ j T C j j − 1 C j k C k k − 1 μ k . (30) where on the second line we use the notation of (15) for the j k jk j k (28) .
Note that for the case j = k j=k j = k B j j B_{jj} B j j (26) . The lack of the index j j j
We will return to this topic in Sec. IV B, where we will compute this interbin covariance for the special case where both bins are narrow and contain enough pulsar pairs to faithfully cover the sky.
Simplifications for narrow-bin correlation measurements
The formulas above simplify for narrow angular bins. For these, the pulsar pairs a b ab ab γ \gamma γ γ = γ a b \gamma=\gamma_{ab} γ = γ ab ρ a b \rho_{ab} ρ ab ⟨ ρ a b ⟩ = h 2 μ a b = h 2 μ ( γ a b ) \left\langle\rho_{ab}\right\rangle=h^2\mu_{ab}=h^2\mu(\gamma_{ab}) ⟨ ρ ab ⟩ = h 2 μ ab = h 2 μ ( γ ab )
⟨ ρ ⟩ = h 2 μ = h 2 μ u ( γ ) 1 , (31) \langle\rho\rangle= h^2\mu= h^2\mu_u(\gamma)\mathbf{1}, \tag*{(31)} ⟨ ρ ⟩ = h 2 μ = h 2 μ u ( γ ) 1 , (31) where 1 \mathbf{1} 1 (22) . This implies that the three “different” reasonable choices for the normalization condition described in Sec. III A all correspond to the same condition:
μ b i n = μ u ( γ ) ⟺ μ = μ b i n 1 ⟺ w T 1 = 1. (32) \mu_{\mathrm{bin}}=\mu_u(\gamma)\ \Longleftrightarrow\ \mu=\mu_{\mathrm{bin}}\mathbf{1}\ \Longleftrightarrow\ w^{\mathsf{T}}\mathbf{1}=1. \tag*{(32)} μ bin = μ u ( γ ) ⟺ μ = μ bin 1 ⟺ w T 1 = 1. (32) For this narrow-bin case, the optimal weights (24) , optimal estimator (25) , and variance (26) are
w = C − 1 1 1 T C − 1 1 , (33) w=\frac{C^{-1}\mathbf{1}}{\mathbf{1}^{\mathsf{T}}C^{-1}\mathbf{1}}, \tag*{(33)} w = 1 T C − 1 1 C − 1 1 , (33) ρ o p t = 1 T C − 1 ρ 1 T C − 1 1 , (34) \rho_{\mathrm{opt}}=\frac{\mathbf{1}^{\mathsf{T}}C^{-1}\rho}{\mathbf{1}^{\mathsf{T}}C^{-1}\mathbf{1}}, \tag*{(34)} ρ opt = 1 T C − 1 1 1 T C − 1 ρ , (34) σ o p t 2 = 1 1 T C − 1 1 . (35) \sigma_{\mathrm{opt}}^2=\frac{1}{\mathbf{1}^{\mathsf{T}}C^{-1}\mathbf{1}}. \tag*{(35)} σ opt 2 = 1 T C − 1 1 1 . (35) The denominators of (33) –(35) are the inverse of the covariance matrix block, summed over all of its elements. This is called the “grand sum” of C − 1 C^{-1} C − 1 (33) –(35) are standard expressions for the optimal combination of correlated measurements that have the same expectation value, and may be found in, e.g., [10 , 23 , 24 ] .
An important special case is that of a single pulsar pair a b ab ab ρ a b \rho_{ab} ρ ab σ o p t 2 = C a b ; a b \sigma_{\mathrm{opt}}^2 = C_{ab;ab} σ opt 2 = C ab ; ab (12) . As anticipated, this is precisely the (one-pulsar-pair) total variance σ 2 ( γ ) \sigma^2(\gamma) σ 2 ( γ ) [11 ] for the Gaussian ensemble.
Simplifications for narrow-bin correlations when 1 \mathbf{1} 1 C C C
We obtain a further simplification for the narrow-bin case if the sum of every row of the corresponding covariance matrix block has the same value λ \lambda λ
C 1 = λ 1 , (36) C\mathbf{1}=\lambda\mathbf{1}, \tag*{(36)} C 1 = λ 1 , (36) meaning that 1 \mathbf{1} 1 C C C λ \lambda λ (35) , can be expressed in terms of the average value of the entries of C C C
To see this, note that since C C C λ \lambda λ (36) from the left by λ − 1 C − 1 \lambda^{-1}C^{-1} λ − 1 C − 1
C − 1 1 = λ − 1 1. (37) C^{-1}\mathbf{1}=\lambda^{-1}\mathbf{1}. \tag*{(37)} C − 1 1 = λ − 1 1 . (37) This means that 1 \mathbf{1} 1 C − 1 C^{-1} C − 1 λ − 1 \lambda^{-1} λ − 1 (37) on the left with 1 T \mathbf{1}^{\mathsf{T}} 1 T
1 T C − 1 1 = 1 T λ − 1 1 = n p a i r s / λ , (38) \mathbf{1}^{\mathsf{T}}C^{-1}\mathbf{1}=\mathbf{1}^{\mathsf{T}}\lambda^{-1}\mathbf{1}=n_{\mathrm{pairs}}/\lambda, \tag*{(38)} 1 T C − 1 1 = 1 T λ − 1 1 = n pairs / λ , (38) where the number of pairs has appeared because it is the dimension of the vector 1 \mathbf{1} 1
So, with these assumptions (35) simplifies to
σ o p t 2 = λ / n p a i r s ≡ s , (39) \sigma_{\mathrm{opt}}^2=\lambda/n_{\mathrm{pairs}}\equiv s, \tag*{(39)} σ opt 2 = λ / n pairs ≡ s , (39) where s s s C C C (33) are then given by
w = 1 / n p a i r s . (40) w=1/n_{\mathrm{pairs}}. \tag*{(40)} w = 1/ n pairs . (40) In this case, the optimal weights are all equal, and (34) becomes
ρ o p t = 1 T ρ / n p a i r s . (41) \rho_{\mathrm{opt}}=\mathbf{1}^{\mathsf{T}}\rho/n_{\mathrm{pairs}}. \tag*{(41)} ρ opt = 1 T ρ / n pairs . (41) This is the uniform average of the correlation measurements ρ a b \rho_{ab} ρ ab
To summarize, if the sum of each row of the covariance matrix C C C λ \lambda λ s ≡ λ / n p a i r s s\equiv\lambda/n_{\mathrm{pairs}} s ≡ λ / n pairs C C C
We will show in Sec. IV that, in the limit of an infinite number of pulsar pairs distributed uniformly on the sky, each row of a given narrow-bin covariance matrix block C C C C C C γ \gamma γ Figure 2 do not apply to this limiting case.
Simple example: An infinite set of measurements with nonzero variance
We would normally expect that as more and more noise-free measurements are included in some estimate, the uncertainty in the estimate decreases to zero. However, as we have explained, the Hellings and Downs correlation does not behave this way. In the limit of many measurements, the uncertainty decreases to a nonzero value, which is the cosmic variance.
Here, we provide a simple example to illustrate how correlations between the different measurements (described by off-diagonal terms in the covariance matrix) are responsible for this behavior. Consider the following n p a i r s × n p a i r s n_{\mathrm{pairs}}\times n_{\mathrm{pairs}} n pairs × n pairs § 2 E, where the terms proportional to a a a
C = ( 1 a ⋯ a a 1 ⋯ a ⋮ ⋮ ⋱ ⋮ a a ⋯ 1 ) . (42) C=\begin{pmatrix}
1 & a & \cdots& a\\
a & 1 & \cdots& a\\
\vdots& \vdots& \ddots& \vdots\\
a & a & \cdots& 1
\end{pmatrix}. \tag*{(42)} C = 1 a ⋮ a a 1 ⋮ a ⋯ ⋯ ⋱ ⋯ a a ⋮ 1 . (42) The parameter a a a − 1 / ( n p a i r s − 1 ) < a < 1 -1/(n_{\mathrm{pairs}}-1)<a<1 − 1/ ( n pairs − 1 ) < a < 1 C C C [25 ] . We find the optimal estimator and its variance using the results of Sec. III D. It is easy to verify that 1 \mathbf{1} 1 C C C
λ = 1 + ( n p a i r s − 1 ) a . (43) \lambda=1+(n_{\mathrm{pairs}}-1)a. \tag*{(43)} λ = 1 + ( n pairs − 1 ) a . (43) Thus, the optimal estimator is
ρ o p t = 1 T ρ / n p a i r s , (44) \rho_{\mathrm{opt}}=\mathbf{1}^{\mathsf{T}}\rho/n_{\mathrm{pairs}}, \tag*{(44)} ρ opt = 1 T ρ / n pairs , (44) and its variance is
σ o p t 2 = λ n p a i r s = 1 + ( n p a i r s − 1 ) a n p a i r s . (45) \sigma_{\mathrm{opt}}^2=\frac{\lambda}{n_{\mathrm{pairs}}}=\frac{1+(n_{\mathrm{pairs}}-1)a}{n_{\mathrm{pairs}}}. \tag*{(45)} σ opt 2 = n pairs λ = n pairs 1 + ( n pairs − 1 ) a . (45) Now, one might think that since ρ o p t \rho_{\mathrm{opt}} ρ opt ρ a b \rho_{ab} ρ ab 1 / n p a i r s 1/n_{\mathrm{pairs}} 1/ n pairs n p a i r s → ∞ n_{\mathrm{pairs}}\to\infty n pairs → ∞ (45) that in the limit of large numbers of measurements the variance approaches
lim n p a i r s → ∞ σ o p t 2 = a , (46) \lim_{n_{\mathrm{pairs}}\to\infty}\sigma_{\mathrm{opt}}^2=a, \tag*{(46)} n pairs → ∞ lim σ opt 2 = a , (46) which is nonzero for a > 0 a>0 a > 0
In the next section, we will see that the cosmic variance exhibits the same behavior. From this example, we conclude that it is the nonzero off-diagonal elements of the covariance matrix that are responsible for this. (It does not arise from unequal weighting of the correlation measurements because in the limit of many pulsar pairs, uniformly distributed on the sky, all of the weights are equal.)
Cosmic Covariance and Cosmic Variance
This section considers a pair of narrow Hellings and Downs correlation bins at two different angles, each containing a large number of pulsar pairs, uniformly distributed on the sky. For each bin, we construct the optimal estimator of the Hellings and Downs correlation in that bin. We then compute the covariance of those two optimal estimators, as defined in Sec. III B, obtaining a simple analytic expression for the cosmic covariance. If the two bins are the same, we prove that this reduces to the cosmic variance found in [11 ] .
At the end of the section, we consider the more realistic case, where the number of pulsars is finite, and show how the cosmic variance is approached as the number of pulsars grows.
Narrow-bin pulsar averaging
Our approach to deriving the cosmic covariance follows the formulation of Secs. III B, III C, and III D. Since each angular bin is narrow, the quantities of interest are the average values of the entries in the inverse covariance matrix for that bin, as shown in Sec. III C. The averages are computed along each row of the matrix. If each angular bin contains a large number of pulsar pairs, uniformly distributed on the sky, then we can prove that every row of the covariance matrix block has the same average value. This allows further simplifications, as shown in Sec. III D.
Throughout this section we consider a pair of narrow angular-separation bins j j j k k k γ j \gamma_j γ j γ k \gamma_k γ k a b ab ab c d cd c d (11) and identities (13) and (14) provide the following expression for the entries of the covariance matrix C j k C_{jk} C j k
C a b , c d = K 4 ( μ a c μ b d + μ a d μ b c ) = K 4 ∑ A ∑ A ′ ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π ( F a A ( Ω ^ ) F c A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F d A ′ ( Ω ^ ′ ) + F a A ( Ω ^ ) F d A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) F c A ′ ( Ω ^ ′ ) ) + K 4 μ u ( 0 ) [ μ u ( 0 ) ( δ a c δ b d + δ a d δ b c ) + μ u ( γ b d ) δ a c + μ u ( γ a c ) δ b d + μ u ( γ b c ) δ a d + μ u ( γ a d ) δ b c ] . (47) \begin{aligned}
\mathcal{C}_{ab,cd}=\mathcal{K}^4(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}) \\
&=\mathcal{K}^4\sum_A\sum_{A'}\int\frac{\mathrm{d}\widehat{\Omega}}{4\pi}\int\frac{\mathrm{d}\widehat{\Omega}'}{4\pi}\left(F_a^A(\widehat{\Omega})F_c^A(\widehat{\Omega})F_b^{A'}(\widehat{\Omega}')F_d^{A'}(\widehat{\Omega}')+F_a^A(\widehat{\Omega})F_d^A(\widehat{\Omega})F_b^{A'}(\widehat{\Omega}')F_c^{A'}(\widehat{\Omega}')\right) \\
&+\mathcal{K}^4\mu_u(0)\left[\mu_u(0)(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})+\mu_u(\gamma_{bd})\delta_{ac}+\mu_u(\gamma_{ac})\delta_{bd}+\mu_u(\gamma_{bc})\delta_{ad}+\mu_u(\gamma_{ad})\delta_{bc}\right]. \tag*{(47)}
\end{aligned} C ab , c d = K 4 ( μ a c μ b d + μ a d μ b c ) = K 4 A ∑ A ′ ∑ ∫ 4 π d Ω ∫ 4 π d Ω ′ ( F a A ( Ω ) F c A ( Ω ) F b A ′ ( Ω ′ ) F d A ′ ( Ω ′ ) + F a A ( Ω ) F d A ( Ω ) F b A ′ ( Ω ′ ) F c A ′ ( Ω ′ ) ) + K 4 μ u ( 0 ) [ μ u ( 0 ) ( δ a c δ b d + δ a d δ b c ) + μ u ( γ b d ) δ a c + μ u ( γ a c ) δ b d + μ u ( γ b c ) δ a d + μ u ( γ a d ) δ b c ] . (47) The second equality can be written more compactly using “symmetrization notation” [26 ] for indices,
Q ( c d ) = 1 2 ( Q c d + Q d c ) . (48) \mathcal{Q}_{(cd)}=\frac{1}{2}\left(\mathcal{Q}_{cd}+\mathcal{Q}_{dc}\right). \tag*{(48)} Q ( c d ) = 2 1 ( Q c d + Q d c ) . (48) This yields
C a b , c d = 2 K 4 [ ∑ A ∑ A ′ ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π F a A ( Ω ^ ) F c A ( Ω ^ ) F d A ′ ( Ω ^ ′ ) F b A ′ ( Ω ^ ′ ) \mathcal{C}_{ab,cd}=2\mathcal{K}^4\left[\sum_A\sum_{A'}\int\frac{\mathrm{d}\widehat{\Omega}}{4\pi}\int\frac{\mathrm{d}\widehat{\Omega}'}{4\pi}F_a^A(\widehat{\Omega})F_c^A(\widehat{\Omega})F_d^{A'}(\widehat{\Omega}')F_b^{A'}(\widehat{\Omega}')\right. C ab , c d = 2 K 4 [ A ∑ A ′ ∑ ∫ 4 π d Ω ∫ 4 π d Ω ′ F a A ( Ω ) F c A ( Ω ) F d A ′ ( Ω ′ ) F b A ′ ( Ω ′ ) + μ u 2 ( 0 ) δ a c δ b d + μ u ( 0 ) ( μ u ( γ a c ) δ b d + μ u ( γ b c ) δ a d ) ] . (49) \left.+\mu_u^2(0)\delta_{ac}\delta_{bd}+\mu_u(0)\left(\mu_u(\gamma_{ac})\delta_{bd}+\mu_u(\gamma_{bc})\delta_{ad}\right)\right]. \tag*{(49)} + μ u 2 ( 0 ) δ a c δ b d + μ u ( 0 ) ( μ u ( γ a c ) δ b d + μ u ( γ b c ) δ a d ) ] . (49) Recall [see the discussion after (11) ] that the above expression for C a b , c d \mathcal{C}_{ab,cd} C ab , c d a a a b b b c c c d d d a < b a<b a < b c < d c<d c < d
We now compute the average value of the entries C a b , c d \mathcal{C}_{ab,cd} C ab , c d (49) , where the average is computed along the a b ab ab C j k C_{jk} C j k a b ab ab C j k C_{jk} C j k 1 k \mathbf{1}_k 1 k n p a i r s , k n_{\mathrm{pairs},k} n pairs , k 1 j \mathbf{1}_j 1 j n p a i r s , j n_{\mathrm{pairs},j} n pairs , j
Figure 4 provides a pictorial guide to clarify our notation. The top shows the full covariance matrix C \mathsf{C} C C j k \mathsf{C}_{jk} C j k j j j k k k
Figure 4. Four sub-blocks of the full covariance matrix C \mathsf{C} C a b ab ab c d cd c d C j k \mathsf{C}_{jk} C j k s j k s_{jk} s j k λ \lambda λ 1 \mathbf{1} 1
The bottom left of Figure 4 shows the rectangular submatrix C 73 \mathsf{C}_{73} C 73 a b ab ab γ 7 \gamma_7 γ 7 c d cd c d γ 3 \gamma_3 γ 3
The bottom right illustrates the implications of the statement “the average values of the submatrix entries, averaged along a row, are row-independent” [27 ] . This is true if and only if each of the four submatrix blocks satisfies an equation of the form C j k 1 = λ j k 1 \mathsf{C}_{jk}\mathbf{1}=\lambda_{jk}\mathbf{1} C j k 1 = λ j k 1 Figure 4 , keeping in mind that the 1 \mathbf{1} 1 s j k s_{jk} s j k s 33 s_{33} s 33 s 37 = s 73 s_{37}=s_{73} s 37 = s 73 s 77 s_{77} s 77
In what follows, we derive a formula for s j k s_{jk} s j k j j j k k k Figure 4 in mind.
To compute the average value s j k s_{jk} s j k a b ab ab c d cd c d c d cd c d γ k \gamma_k γ k ( s j k ) a b (s_{jk})_{ab} ( s j k ) ab
( s j k ) a b ≡ ⟨ C a b , c d ⟩ c d ∈ γ k , (50) (s_{jk})_{ab} \equiv\langle C_{ab,cd}\rangle_{cd\in\gamma_k}, \tag*{(50)} ( s j k ) ab ≡ ⟨ C ab , c d ⟩ c d ∈ γ k , (50) where ⟨ ⟩ c d ∈ γ k \langle\rangle_{cd\in\gamma_k} ⟨ ⟩ c d ∈ γ k c d cd c d γ k \gamma_k γ k [11 ] , this is called a “pulsar average” and is explicitly defined in Appendix A of [11 ] as a normalized integral over three variables.
From (49) , one can see that there are three terms that contribute to the average (50) . The first explicitly involves the antenna pattern functions, and the second and third involve either a single Kronecker delta function or a product of two Kronecker deltas. We show in Appendix C that, in the limit of many pulsar pairs, the latter two terms make negligible contributions to the average: they correspond to a set of measure zero.
Thus, for large numbers of pulsar pairs, the only term that contributes to the average value of the entries along the row labeled by a b ab ab
( s j k ) a b = 2 h 2 β 4 ∑ A ∑ A ′ ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π F a A ( Ω ^ ) F b A ′ ( Ω ^ ′ ) ⟨ F c A ( Ω ^ ) F d A ′ ( Ω ^ ′ ) ⟩ c d ∈ γ k . (51) (s_{jk})_{ab}=2h^2\beta^4\sum_A\sum_{A'}\int\frac{\mathrm{d}\widehat{\Omega}}{4\pi}\int\frac{\mathrm{d}\widehat{\Omega}'}{4\pi}F_a^A(\widehat{\Omega})F_b^{A'}(\widehat{\Omega}')\left\langle F_c^A(\widehat{\Omega})F_d^{A'}(\widehat{\Omega}')\right\rangle_{cd\in\gamma_k}. \tag*{(51)} ( s j k ) ab = 2 h 2 β 4 A ∑ A ′ ∑ ∫ 4 π d Ω ∫ 4 π d Ω ′ F a A ( Ω ) F b A ′ ( Ω ′ ) ⟨ F c A ( Ω ) F d A ′ ( Ω ′ ) ⟩ c d ∈ γ k . (51) (The symmetrization on c c c d d d γ k \gamma_k γ k § 23 in [11 ] , where they are called two-point functions, and the specific average that appears in (51) is evaluated. The uniform sky average over all pulsar pairs c d cd c d γ k \gamma_k γ k
⟨ F c A ( Ω ^ ) F d A ′ ( Ω ^ ′ ) ⟩ c d ∈ γ k = μ A A ′ ( γ k , β ) , (52) \left\langle F_{\mathrm{c}}^{A}(\hat{\Omega})F_{\mathrm{d}}^{A'}(\hat{\Omega}')\right\rangle_{cd\in\gamma_k}=\mu_{AA'}(\gamma_k,\beta), \tag*{(52)} ⟨ F c A ( Ω ^ ) F d A ′ ( Ω ^ ′ ) ⟩ c d ∈ γ k = μ A A ′ ( γ k , β ) , (52) where the functions μ A A ′ ( γ , β ) \mu_{AA'}(\gamma,\beta) μ A A ′ ( γ , β ) [11 ] ; these functions vanish if A ≠ A ′ A\ne A' A = A ′ β \beta β Ω ^ \hat{\Omega} Ω ^ Ω ^ ′ \hat{\Omega}' Ω ^ ′ cos β = Ω ^ ⋅ Ω ^ ′ \cos\beta=\hat{\Omega}\cdot\hat{\Omega}' cos β = Ω ^ ⋅ Ω ^ ′ (52) into (51) and summing over polarizations A ′ A' A ′
( s j k ) a b = 2 h 4 ∑ A ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π F a A ( Ω ^ ) F b A ( Ω ^ ′ ) μ A A ( γ k , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) . (53) (s_{jk})_{ab}=2h^4\sum_A\int\frac{\mathrm{d}\hat{\Omega}}{4\pi}\int\frac{\mathrm{d}\hat{\Omega}'}{4\pi}F_a^A(\hat{\Omega})F_b^A(\hat{\Omega}')\mu_{AA}\left(\gamma_k,\cos^{-1}(\hat{\Omega}\cdot\hat{\Omega}')\right). \tag*{(53)} ( s j k ) ab = 2 h 4 A ∑ ∫ 4 π d Ω ^ ∫ 4 π d Ω ^ ′ F a A ( Ω ^ ) F b A ( Ω ^ ′ ) μ AA ( γ k , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) . (53) It is easy to show that this expression is independent of the pulsar pair (row index) a b ab ab a a a F a A ( Ω ^ ) F_a^A(\hat{\Omega}) F a A ( Ω ^ ) b b b F b A ( Ω ^ ′ ) F_b^A(\hat{\Omega}') F b A ( Ω ^ ′ )
To see this, pick coordinates for the Ω ^ \hat{\Omega} Ω ^ a a a z z z Ω ^ ′ \hat{\Omega}' Ω ^ ′ b b b x , z x,z x , z a b ab ab γ a b = γ j \gamma_{ab}=\gamma_j γ ab = γ j a b ab ab
Since the value of ( s j k ) a b (s_{jk})_{ab} ( s j k ) ab a b ab ab ( s j k ) a b = s j k (s_{jk})_{ab}=s_{jk} ( s j k ) ab = s j k (52) again (with pulsar labels c d cd c d a b ab ab γ k \gamma_k γ k γ j \gamma_j γ j (53) over a b ab ab γ j \gamma_j γ j
s j k = 2 h 4 ∑ A ∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π μ A A ( γ j , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) μ A A ( γ k , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) = h 4 ∫ 0 π d β sin β ( μ + + ( γ j , β ) μ + + ( γ k , β ) + μ × × ( γ j , β ) μ × × ( γ k , β ) ) . (54) \begin{aligned}
s_{jk}&=2h^4\sum_A\int\frac{\mathrm{d}\hat{\Omega}}{4\pi}\int\frac{\mathrm{d}\hat{\Omega}'}{4\pi}\mu_{AA}\left(\gamma_j,\cos^{-1}(\hat{\Omega}\cdot\hat{\Omega}')\right)\mu_{AA}\left(\gamma_k,\cos^{-1}(\hat{\Omega}\cdot\hat{\Omega}')\right)\\
&=h^4\int_0^\pi\mathrm{d}\beta\sin\beta\left(\mu_{++}(\gamma_j,\beta)\mu_{++}(\gamma_k,\beta)+\mu_{\times\times}(\gamma_j,\beta)\mu_{\times\times}(\gamma_k,\beta)\right). \tag*{(54)}
\end{aligned} s j k = 2 h 4 A ∑ ∫ 4 π d Ω ^ ∫ 4 π d Ω ^ ′ μ AA ( γ j , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) μ AA ( γ k , cos − 1 ( Ω ^ ⋅ Ω ^ ′ ) ) = h 4 ∫ 0 π d β sin β ( μ ++ ( γ j , β ) μ ++ ( γ k , β ) + μ ×× ( γ j , β ) μ ×× ( γ k , β ) ) . (54) To obtain the final line of (54) , we use
∫ d Ω ^ 4 π ∫ d Ω ^ ′ 4 π Q ( Ω ^ ⋅ Ω ^ ′ ) = 1 2 ∫ 0 π d β sin β Q ( cos β ) (55) \int\frac{\mathrm{d}\hat{\Omega}}{4\pi}\int\frac{\mathrm{d}\hat{\Omega}'}{4\pi}Q(\hat{\Omega}\cdot\hat{\Omega}')=\frac{1}{2}\int_0^\pi\mathrm{d}\beta\sin\beta Q(\cos\beta) \tag*{(55)} ∫ 4 π d Ω ^ ∫ 4 π d Ω ^ ′ Q ( Ω ^ ⋅ Ω ^ ′ ) = 2 1 ∫ 0 π d β sin β Q ( cos β ) (55) for the spherical average of any function Q ( x ) Q(x) Q ( x ) Q ( x ) = 1 Q(x)=1 Q ( x ) = 1
The quantity s j k s_{jk} s j k γ j \gamma_j γ j γ k \gamma_k γ k s j k s_{jk} s j k C j k C_{jk} C j k 1 k \mathbf{1}_k 1 k C j k C_{jk} C j k
Cosmic covariance and correlation between optimal estimators for two narrow bins
In Sec. IV A, we obtained a simple formula for the average value s j k s_{jk} s j k j j j k k k
The covariance between the optimal estimators of the Hellings and Downs correlation for the two bins is given in (30) . Since the bins are narrow, the quantities μ T C − 1 μ \boldsymbol{\mu}^{\mathsf{T}}C^{-1}\boldsymbol{\mu} μ T C − 1 μ Figure 4 because the column vector μ \boldsymbol{\mu} μ 1 \mathbf{1} 1 (30) simplifies to
B j k = w j T C j k w k = 1 n p a i r s , j n p a i r s , k 1 j T C j k 1 k = ⟨ C a b , c d ⟩ a b ∈ γ j ; c d ∈ γ k = s j k . (56) B_{jk}=\mathbf{w}_j^{\mathsf{T}}C_{jk}\mathbf{w}_k=\frac{1}{n_{\mathrm{pairs},j}n_{\mathrm{pairs},k}}\mathbf{1}_j^{\mathsf{T}}C_{jk}\mathbf{1}_k=\langle C_{ab,cd}\rangle_{ab\in\gamma_j;cd\in\gamma_k}=s_{jk}. \tag*{(56)} B j k = w j T C j k w k = n pairs , j n pairs , k 1 1 j T C j k 1 k = ⟨ C ab , c d ⟩ ab ∈ γ j ; c d ∈ γ k = s j k . (56) where s j k s_{jk} s j k (54) . This cosmic covariance s j k s_{jk} s j k γ j \gamma_j γ j γ k \gamma_k γ k j j j k k k (54) , using the functions μ A A ′ ( γ , β ) \mu_{AA'}(\gamma,\beta) μ A A ′ ( γ , β ) (261) and (265) in [11 ] . To evaluate the integrals, it is convenient to set γ < = min ( γ j , γ k ) \gamma_< = \min(\gamma_j,\gamma_k) γ < = min ( γ j , γ k ) γ > = max ( γ j , γ k ) \gamma_> = \max(\gamma_j,\gamma_k) γ > = max ( γ j , γ k ) s j k s_{jk} s j k γ < \gamma_< γ < γ > \gamma_> γ > β \beta β β ∈ [ 0 , γ < ] \beta\in[0,\gamma_<] β ∈ [ 0 , γ < ] β ∈ [ γ < , γ > ] \beta\in[\gamma_<,\gamma_>] β ∈ [ γ < , γ > ] β ∈ [ γ > , π ] \beta\in[\gamma_>,\pi] β ∈ [ γ > , π ]
s j k 2 h 4 = 1 12 ( cos γ < cos γ > − cos γ < − cos γ > − 3 ) ln 1 + cos γ < 2 + 49 432 cos γ < cos γ > + 1 12 ( cos γ < cos γ > + cos γ < + cos γ > − 3 ) ln 1 − cos γ > 2 + 1 4 ( cos γ < − cos γ > ) − 5 48 + 1 12 ( cos γ < cos γ > + 3 ) ( Li 2 1 − cos γ < 2 − Li 2 1 − cos γ > 2 − ln 1 − cos γ > 2 ln ( 1 + cos γ < ) ( 1 + cos γ > ) 4 ) . (57) \frac{s_{jk}}{2\mathcal{h}^4} = \frac{1}{12}\left(\cos\gamma_<\cos\gamma_>-\cos\gamma_<-\cos\gamma_>-3\right)\ln\frac{1+\cos\gamma_<}{2}+\frac{49}{432}\cos\gamma_<\cos\gamma_>
+\frac{1}{12}\left(\cos\gamma_<\cos\gamma_>+\cos\gamma_<+\cos\gamma_>-3\right)\ln\frac{1-\cos\gamma_>}{2}+\frac{1}{4}\left(\cos\gamma_<-\cos\gamma_>\right)-\frac{5}{48}
+\frac{1}{12}\left(\cos\gamma_<\cos\gamma_>+3\right)\left(\operatorname{Li}_2\frac{1-\cos\gamma_<}{2}-\operatorname{Li}_2\frac{1-\cos\gamma_>}{2}-\ln\frac{1-\cos\gamma_>}{2}\ln\frac{(1+\cos\gamma_<)(1+\cos\gamma_>)}{4}\right). \tag*{(57)} 2 h 4 s j k = 12 1 ( cos γ < cos γ > − cos γ < − cos γ > − 3 ) ln 2 1 + cos γ < + 432 49 cos γ < cos γ > + 12 1 ( cos γ < cos γ > + cos γ < + cos γ > − 3 ) ln 2 1 − cos γ > + 4 1 ( cos γ < − cos γ > ) − 48 5 + 12 1 ( cos γ < cos γ > + 3 ) ( Li 2 2 1 − cos γ < − Li 2 2 1 − cos γ > − ln 2 1 − cos γ > ln 4 ( 1 + cos γ < ) ( 1 + cos γ > ) ) . (57) This is explicitly symmetric s j k = s k j s_{jk}=s_{kj} s j k = s k j γ j \gamma_j γ j γ k \gamma_k γ k γ < \gamma_< γ < γ > \gamma_> γ >
The special function Li 2 \operatorname{Li}_2 Li 2 [28 , 29 , 30 ] . This remarkable function [31 ] is defined by the sum
Li 2 ( z ) = ∑ n = 1 ∞ z n n 2 , (58) \operatorname{Li}_2(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^2}, \tag*{(58)} Li 2 ( z ) = n = 1 ∑ ∞ n 2 z n , (58) which converges on and inside the unit circle ∣ z ∣ 2 ≤ 1 |z|^2\le1 ∣ z ∣ 2 ≤ 1 (58) is sufficient for our purposes, because the arguments in (57) lie on the real axis between − 1 -1 − 1 1 1 1
The structure of the cosmic covariance is displayed in the left panel of Figure 5 . This shows that the angular range between 0 0 0 180 ∘ 180^\circ 18 0 ∘ 0 0 0 ≈ 50 ∘ \approx50^\circ ≈ 5 0 ∘ ≈ 50 ∘ \approx50^\circ ≈ 5 0 ∘ ≈ 130 ∘ \approx130^\circ ≈ 13 0 ∘ ≈ 130 ∘ \approx130^\circ ≈ 13 0 ∘ 180 ∘ 180^\circ 18 0 ∘ coherence function s j k / s j j s k k s_{jk}/\sqrt{s_{jj}s_{kk}} s j k / s j j s k k Figure 5 , which is normalized to take values between − 1 -1 − 1 + 1 +1 + 1
Figure 5. Left panel: the cosmic covariance s j k s_{jk} s j k γ j \gamma_j γ j γ k \gamma_k γ k (57) scaled by a factor of one thousand, which is equivalent to setting 2 h 4 = 1000 2\mathcal{h}^4=1000 2 h 4 = 1000
This is also called the Pearson correlation coefficient or “Pearson’s r r r + 1 +1 + 1 − 1 -1 − 1
Diagonalizing and inverting the cosmic covariance
The inverse of the covariance matrix appears in (the weights and variances associated with) the optimal estimator of the Hellings and Downs correlation, for example in (3.18)–(3.20). We want to characterize these in the cosmic variance limit, when there are many pulsar pairs uniformly distributed on the sky. In this limit, the average value of the covariance matrix approaches the cosmic covariance s j k s_{jk} s j k s j k s_{jk} s j k s j k s_{jk} s j k
To carry out this diagonalization, we expand the Hellings and Downs correlation in harmonic functions. This approach was pioneered in [32 , 33 ] and then further developed in [34 , 35 ] . We follow the treatment given in Appendix C 6 in [11 ] . For example, the Hellings and Downs curve (1) may be expressed ([11 ] , Eq. (C49)) as
μ u ( γ ) = ∑ l = 0 ∞ ( 2 l + 1 ) C l P l ( x ) , (59) \mu_u(\gamma) = \sum_{l=0}^{\infty} (2l+1) C_l P_l(x), \tag*{(59)} μ u ( γ ) = l = 0 ∑ ∞ ( 2 l + 1 ) C l P l ( x ) , (59) where P l P_l P l l l l x = cos γ x=\cos\gamma x = cos γ
C l = { 0 if l < 2 ( l − 2 ) ! / ( l + 2 ) ! = 1 / [ ( l + 2 ) ( l + 1 ) l ( l − 1 ) ] if l ≥ 2. (60) C_l =
\begin{cases}
0 & \text{if } l<2 \\
(l-2)!/(l+2)! = 1/[(l+2)(l+1)l(l-1)] & \text{if } l\ge2.
\end{cases} \tag*{(60)} C l = { 0 ( l − 2 )! / ( l + 2 )! = 1/ [( l + 2 ) ( l + 1 ) l ( l − 1 )] if l < 2 if l ≥ 2. (60) The C l C_l C l
The cosmic covariance s j k s_{jk} s j k [11 ] to obtain
s j k = 4 ℏ 4 ∑ l = 2 ∞ 2 l + 1 2 C l 2 P l ( cos γ j ) P l ( cos γ k ) . (61) s_{jk}=4\hbar^4\sum_{l=2}^{\infty}\frac{2l+1}{2}C_l^2P_l(\cos\gamma_j)P_l(\cos\gamma_k). \tag*{(61)} s j k = 4 ℏ 4 l = 2 ∑ ∞ 2 2 l + 1 C l 2 P l ( cos γ j ) P l ( cos γ k ) . (61) As a simple check, we can set γ j = γ k \gamma_j=\gamma_k γ j = γ k [11 ] . A numerical check (it is sufficient to sum l = 2 , 3 , … , 10 l=2,3,\ldots,10 l = 2 , 3 , … , 10 (57) and (61) .
To invert s j k s_{jk} s j k M − 1 M^{-1} M − 1 M M M M M − 1 = I MM^{-1}=I M M − 1 = I I I I
∑ k M j k M k m − 1 = δ j m . (62) \sum_k M_{jk}M^{-1}_{km}=\delta_{jm}. \tag*{(62)} k ∑ M j k M k m − 1 = δ j m . (62) For the case of interest, the matrix M M M s j k s_{jk} s j k
s ( x , x ′ ) = 4 ℏ 4 ∑ l = 2 ∞ 2 l + 1 2 C l 2 P l ( x ) P l ( x ′ ) , (63) s(x,x')=4\hbar^4\sum_{l=2}^{\infty}\frac{2l+1}{2}C_l^2P_l(x)P_l(x'), \tag*{(63)} s ( x , x ′ ) = 4 ℏ 4 l = 2 ∑ ∞ 2 2 l + 1 C l 2 P l ( x ) P l ( x ′ ) , (63) where x = cos γ x=\cos\gamma x = cos γ x ′ = cos γ ′ x'=\cos\gamma' x ′ = cos γ ′ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] (62) , if M ( x , x ′ ) M(x,x') M ( x , x ′ ) M − 1 ( x , x ′ ) M^{-1}(x,x') M − 1 ( x , x ′ ) (71) ] to
∫ − 1 1 d x ′ M ( x , x ′ ) M − 1 ( x ′ , x ′ ′ ) = δ ( x − x ′ ′ ) , (64) \int_{-1}^{1}\mathrm{d}x'M(x,x')M^{-1}(x',x'')=\delta(x-x''), \tag*{(64)} ∫ − 1 1 d x ′ M ( x , x ′ ) M − 1 ( x ′ , x ′′ ) = δ ( x − x ′′ ) , (64) where δ ( x ) \delta(x) δ ( x ) d x ′ = sin γ ′ d γ ′ \mathrm{d}x'=\sin\gamma'\mathrm{d}\gamma' d x ′ = sin γ ′ d γ ′
The Dirac delta function on the rhs of (64) may be expressed as a sum of Legendre polynomials. The orthogonality relation for Legendre polynomials,
∫ − 1 1 d x P l ( x ) P l ′ ( x ) = 2 2 l + 1 δ l l ′ , (65) \int_{-1}^{1}\mathrm{d}x\,P_l(x)P_{l'}(x)=\frac{2}{2l+1}\delta_{ll'}, \tag*{(65)} ∫ − 1 1 d x P l ( x ) P l ′ ( x ) = 2 l + 1 2 δ l l ′ , (65) implies that for x x x x ′ x' x ′ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
δ ( x − x ′ ) = ∑ l 2 l + 1 2 P l ( x ) P l ( x ′ ) . (66) \delta(x-x')=\sum_l\frac{2l+1}{2}P_l(x)P_l(x'). \tag*{(66)} δ ( x − x ′ ) = l ∑ 2 2 l + 1 P l ( x ) P l ( x ′ ) . (66) This allows us to rewrite the rhs of (64) . We now demonstrate a convenient way to invert M M M M M M
M ( x , x ′ ) = ∑ l 2 l + 1 2 Q l P l ( x ) P l ( x ′ ) , and M(x,x')=\sum_l\frac{2l+1}{2}Q_lP_l(x)P_l(x'), \qquad\text{and} M ( x , x ′ ) = l ∑ 2 2 l + 1 Q l P l ( x ) P l ( x ′ ) , and M − 1 ( x , x ′ ) = ∑ l 2 l + 1 2 q l P l ( x ) P l ( x ′ ) . (67) M^{-1}(x,x')=\sum_l\frac{2l+1}{2}q_lP_l(x)P_l(x'). \tag*{(67)} M − 1 ( x , x ′ ) = l ∑ 2 2 l + 1 q l P l ( x ) P l ( x ′ ) . (67) M − 1 ( x , x ′ ) = ∑ l 2 l + 1 2 q l P l ( x ) P l ( x ′ ) . (68) M^{-1}(x,x')=\sum_l\frac{2l+1}{2}q_lP_l(x)P_l(x'). \tag*{(68)} M − 1 ( x , x ′ ) = l ∑ 2 2 l + 1 q l P l ( x ) P l ( x ′ ) . (68) Substituting these into the lhs of (64) , using (65) to carry out the integral over x ′ x' x ′ (64) with (66) , we obtain
∑ l 2 l + 1 2 Q l q l P l ( x ) P l ( x ′ ′ ) = ∑ l 2 l + 1 2 P l ( x ) P l ( x ′ ′ ) . (69) \sum_l\frac{2l+1}{2}Q_lq_lP_l(x)P_l(x'')=\sum_l\frac{2l+1}{2}P_l(x)P_l(x''). \tag*{(69)} l ∑ 2 2 l + 1 Q l q l P l ( x ) P l ( x ′′ ) = l ∑ 2 2 l + 1 P l ( x ) P l ( x ′′ ) . (69) Identifying the Legendre expansion coefficients on both sides implies
Q l q l = 1. (70) Q_l q_l = 1. \tag*{(70)} Q l q l = 1. (70) Thus, to invert M M M q l = 1 / Q l q_l = 1/Q_l q l = 1/ Q l
We now use (70) to invert the continuous matrix s ( x , x ′ ) s(x,x') s ( x , x ′ ) M ≡ s M \equiv s M ≡ s (67) gives Q l = 4 ℏ 4 C l 2 Q_l = 4\hbar^4 C_l^2 Q l = 4 ℏ 4 C l 2 (70) implies q l = 1 / 4 ℏ 4 C l 2 q_l = 1/4\hbar^4 C_l^2 q l = 1/4 ℏ 4 C l 2 (68) as
s − 1 ( x , x ′ ) = ( 4 ℏ 4 ) − 1 ∑ l = 2 ∞ 2 l + 1 2 C l 2 P l ( x ) P l ( x ′ ) . (71) s^{-1}(x,x') = (4\hbar^4)^{-1} \sum_{l=2}^{\infty} \frac{2l+1}{2C_l^2}P_l(x)P_l(x'). \tag*{(71)} s − 1 ( x , x ′ ) = ( 4 ℏ 4 ) − 1 l = 2 ∑ ∞ 2 C l 2 2 l + 1 P l ( x ) P l ( x ′ ) . (71) Note that the l = 0 l=0 l = 0 l = 1 l=1 l = 1 (63) and (71) cannot satisfy (64) . Rather, they satisfy this equation if the l = 0 l=0 l = 0 l = 1 l=1 l = 1 (66) . This is because s ( x , x ′ ) s(x,x') s ( x , x ′ ) P 0 ( x ) P_0(x) P 0 ( x ) P 1 ( x ) P_1(x) P 1 ( x ) s − 1 ( x , x ′ ) s^{-1}(x,x') s − 1 ( x , x ′ )
Later, we will need to evaluate sums over uniformly distributed pulsars, when the number of pulsar pairs goes to infinity. These sums, which typically involve the vector μ a b ≡ μ j \mu_{ab} \equiv\mu_j μ ab ≡ μ j s j k s_{jk} s j k s j k − 1 s^{-1}_{jk} s j k − 1
To construct vectors of dimension N b i n s N_{\rm bins} N bins N b i n s × N b i n s N_{\rm bins} \times N_{\rm bins} N bins × N bins N b i n s N_{\rm bins} N bins x = cos γ x=\cos\gamma x = cos γ ( − 1 , 1 ) (-1,1) ( − 1 , 1 )
x j = ( 2 j + 1 − N b i n s ) / N b i n s for j = 0 , 1 , … N b i n s − 1. (72) x_j = (2j+1-N_{\rm bins})/N_{\rm bins}\quad\text{for }j=0,1,\ldots N_{\rm bins}-1. \tag*{(72)} x j = ( 2 j + 1 − N bins ) / N bins for j = 0 , 1 , … N bins − 1. (72) The vector μ j \mu_j μ j μ u \mu_u μ u (1) or (59) , at these discrete values: μ j = μ u ( x j ) \mu_j=\mu_u(x_j) μ j = μ u ( x j ) x x x γ \gamma γ s j k s_{jk} s j k (61) , and (63) shows that its entries are obtained by evaluating the continuous function at the grid of points, hence
s j k = s ( x j , x k ) . (73) s_{jk} = s(x_j,x_k). \tag*{(73)} s j k = s ( x j , x k ) . (73) While the continuous matrix inverse s − 1 ( x , x ′ ) s^{-1}(x,x') s − 1 ( x , x ′ ) (71) may also be evaluated on the discrete grid of points, this would not yield the matrix inverse of (73) : it has the wrong normalization to satisfy
∑ k s j k s k m − 1 = δ j m , (74) \sum_k s_{jk}s^{-1}_{km}=\delta_{jm}, \tag*{(74)} k ∑ s j k s k m − 1 = δ j m , (74) even in the limit of large N b i n s N_{\rm bins} N bins
To obtain an inverse matrix s j k − 1 s^{-1}_{jk} s j k − 1 (74) , in the limit of many pairs, one must multiply s − 1 ( x j , x k ) s^{-1}(x_j,x_k) s − 1 ( x j , x k ) Δ x ≡ 2 / N b i n s \Delta x \equiv2/N_{\rm bins} Δ x ≡ 2/ N bins (71) this gives
s j k − 1 = ( 2 N b i n s ) 2 ( 4 ℏ 4 ) − 1 ∑ l = 2 ∞ 2 l + 1 2 C l 2 P l ( cos γ j ) P l ( cos γ k ) . (75) s^{-1}_{jk}=\left(\frac{2}{N_{\rm bins}}\right)^2(4\hbar^4)^{-1}\sum_{l=2}^{\infty}\frac{2l+1}{2C_l^2}P_l(\cos\gamma_j)P_l(\cos\gamma_k). \tag*{(75)} s j k − 1 = ( N bins 2 ) 2 ( 4 ℏ 4 ) − 1 l = 2 ∑ ∞ 2 C l 2 2 l + 1 P l ( cos γ j ) P l ( cos γ k ) . (75) Note that the equality in (75) holds only in the limit of large N b i n s N_{\rm bins} N bins (74) may be demonstrated by converting the sum to a Riemann sum and then to an integral [36 ] . We make use of s j k − 1 s^{-1}_{jk} s j k − 1 s − 1 ( x , x ′ ) s^{-1}(x,x') s − 1 ( x , x ′ )
The failure of s j k − 1 s^{-1}_{jk} s j k − 1 s − 1 ( x j , x k ) s^{-1}(x_j,x_k) s − 1 ( x j , x k ) y ( t ) y(t) y ( t ) y ~ ( f ) \tilde y(f) y ~ ( f ) y ( t ) y(t) y ( t ) t = k Δ t t=k\Delta t t = k Δ t Δ t \Delta t Δ t k = 0 , 1 , … , N − 1 k=0,1,\ldots,N-1 k = 0 , 1 , … , N − 1 f f f y ~ ( f ) \tilde y(f) y ~ ( f ) N N N N Δ t N\Delta t N Δ t [14 ] .
Relating the variance of the narrow-bin optimal estimator to the cosmic variance
Using (54) and the results of Sec. III D, an expression for the variance of the optimal estimator in the narrow-bin limit follows immediately from setting γ j = γ k \gamma_j=\gamma_k γ j = γ k C \mathcal{C} C j = k j=k j = k γ j = γ k = γ \gamma_j=\gamma_k=\gamma γ j = γ k = γ (54) , we see that l l l C j j \mathcal{C}_{jj} C j j
s = 2 ℏ 4 1 2 ∫ 0 π d β sin β ( μ + + 2 ( γ , β ) + μ × × 2 ( γ , β ) ) = 2 ℏ 4 μ ˉ 2 ( γ ) , (76) s=2\hbar^4\frac{1}{2}\int_0^\pi d\beta\sin\beta\left(\mu_{++}^2(\gamma,\beta)+\mu_{\times\times}^2(\gamma,\beta)\right)=2\hbar^4\bar{\mu}^2(\gamma), \tag*{(76)} s = 2 ℏ 4 2 1 ∫ 0 π d β sin β ( μ ++ 2 ( γ , β ) + μ ×× 2 ( γ , β ) ) = 2 ℏ 4 μ ˉ 2 ( γ ) , (76) where we dropped the indices from s j k = s j j s_{jk}=s_{jj} s j k = s j j
The (cosmic variance) function μ ~ 2 ( γ ) \tilde{\mu}^{2}(\gamma) μ ~ 2 ( γ ) § 23 in [11 ] and may also be obtained directly from the diagonal of the cosmic covariance (57) by setting γ j = γ k = γ \gamma_j=\gamma_k=\gamma γ j = γ k = γ
μ ~ 2 ( γ ) μ u 2 ( 0 ) = 49 48 cos 2 γ − 15 16 − 3 2 ( cos 2 γ + 3 ) ln ( 1 − cos γ 2 ) ln ( 1 + cos γ 2 ) + 3 4 ( cos γ − 1 ) ( cos γ + 3 ) ln ( 1 − cos γ 2 ) + 3 4 ( cos γ + 1 ) ( cos γ − 3 ) ln ( 1 + cos γ 2 ) , (77) \frac{\tilde{\mu}^{2}(\gamma)}{\mu_u^{2}(0)}=\frac{49}{48}\cos^{2}\gamma-\frac{15}{16}-\frac{3}{2}(\cos^{2}\gamma+3)\ln\left(\frac{1-\cos\gamma}{2}\right)\ln\left(\frac{1+\cos\gamma}{2}\right)
+\frac{3}{4}(\cos\gamma-1)(\cos\gamma+3)\ln\left(\frac{1-\cos\gamma}{2}\right)+\frac{3}{4}(\cos\gamma+1)(\cos\gamma-3)\ln\left(\frac{1+\cos\gamma}{2}\right), \tag*{(77)} μ u 2 ( 0 ) μ ~ 2 ( γ ) = 48 49 cos 2 γ − 16 15 − 2 3 ( cos 2 γ + 3 ) ln ( 2 1 − cos γ ) ln ( 2 1 + cos γ ) + 4 3 ( cos γ − 1 ) ( cos γ + 3 ) ln ( 2 1 − cos γ ) + 4 3 ( cos γ + 1 ) ( cos γ − 3 ) ln ( 2 1 + cos γ ) , (77) which agrees with Eq. (G11) in [11 ] . We have divided μ ~ 2 ( γ ) \tilde{\mu}^{2}(\gamma) μ ~ 2 ( γ ) μ u 2 ( 0 ) \mu_u^{2}(0) μ u 2 ( 0 ) (77) independent of the choice of normalization of the Hellings and Downs correlation.
The variance of the optimal estimator of the Hellings and Downs correlation is given by (39) as the average value s s s (76) . Hence,
σ o p t 2 ( γ ) = s = 2 κ 4 μ ~ 2 ( γ ) = σ c o s 2 ( γ ) , (78) \sigma_{\mathrm{opt}}^{2}(\gamma)=s=2\kappa^{4}\tilde{\mu}^{2}(\gamma)=\sigma_{\mathrm{cos}}^{2}(\gamma), \tag*{(78)} σ opt 2 ( γ ) = s = 2 κ 4 μ ~ 2 ( γ ) = σ cos 2 ( γ ) , (78) where σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^{2}(\gamma) σ cos 2 ( γ ) [11 ] . This provides a simple and alternative way to derive the cosmic variance.
Numerical illustration showing the approach to the cosmic variance
There are two limiting cases for which we can give analytic expressions for the variance. If there is only a single pulsar pair at each angular separation, then the variance is the total variance, given by σ t o t 2 ( γ ) \sigma_{\mathrm{tot}}^{2}(\gamma) σ tot 2 ( γ ) (12) . If there are a large number of uniformly distributed pulsar pairs, then the variance is the cosmic variance σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^{2}(\gamma) σ cos 2 ( γ ) (78) . Here, we study the transition between these two limits, when the number of pulsar pairs is finite.
In the left panel of Figure 6 , we show the transition from σ t o t 2 ( γ ) \sigma_{\mathrm{tot}}^{2}(\gamma) σ tot 2 ( γ ) σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^{2}(\gamma) σ cos 2 ( γ ) γ \gamma γ n p a i r s = 1 n_{\mathrm{pairs}}=1 n pairs = 1 γ \gamma γ σ o p t 2 ( γ ) \sigma_{\mathrm{opt}}^{2}(\gamma) σ opt 2 ( γ ) n p a i r s n_{\mathrm{pairs}} n pairs γ \gamma γ
In the right panel of Figure 6 , we plot the minimum number of pulsar pairs needed for the optimal variance σ o p t 2 ( γ ) \sigma_{\mathrm{opt}}^{2}(\gamma) σ opt 2 ( γ ) 1 / e ≈ 0.37 1/e\approx0.37 1/ e ≈ 0.37 σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^{2}(\gamma) σ cos 2 ( γ ) σ o p t ( γ ) \sigma_{\mathrm{opt}}(\gamma) σ opt ( γ ) 1.17 σ c o s ( γ ) 1.17\sigma_{\mathrm{cos}}(\gamma) 1.17 σ cos ( γ ) γ \gamma γ σ c o s ( γ ) \sigma_{\mathrm{cos}}(\gamma) σ cos ( γ ) γ ≈ 54 ∘ \gamma\approx54^\circ γ ≈ 5 4 ∘ γ ≈ 126 ∘ \gamma\approx126^\circ γ ≈ 12 6 ∘
Variance of the optimal binned estimator
Beating the cosmic variance
One interesting consequence of doing the binned estimation is that for some ranges of angular separation γ \gamma γ
the loss of angular resolution allows one to “beat” the cosmic variance for a sufficiently large number of pulsar pairs in the bin. This is illustrated numerically in Figure 7 for the case of a simulation involving 400 pulsars distributed uniformly on the sky and 6° angular-separation bins. For this calculation, we used normalization condition (b) in Sec. III A, which corresponds to μ b i n ≡ μ T 1 / n p a i r s \mu_{\mathrm{bin}}\equiv\mu^{\mathsf{T}}\mathbf{1}/n_{\mathrm{pairs}} μ bin ≡ μ T 1 / n pairs
Figure 7. Left panel: the variance of the optimal estimator for a simulation with 400 pulsars distributed uniformly on the sky, with 6° angular bins. This is compared to the (narrow-bin) cosmic variance. The variance of the optimal estimator dips below the cosmic variance for sufficiently large numbers of pulsar pairs per bin. This is possible because angular resolution has been sacrificed, see Appendix E. (For this plot we have set h ˉ 4 = 1 / 2 \bar{h}^{4}=1/2 h ˉ 4 = 1/2 sin γ \sin\gamma sin γ
Examples
Here, we apply the results of Sec. III A to make plots showing the total variance σ t o t 2 ( γ ) \sigma_{\mathrm{tot}}^{2}(\gamma) σ tot 2 ( γ ) σ c o s 2 ( γ ) \sigma_{\mathrm{cos}}^{2}(\gamma) σ cos 2 ( γ ) σ o p t 2 ( γ ) \sigma_{\mathrm{opt}}^{2}(\gamma) σ opt 2 ( γ ) Table IV in Appendix § 24 for the names and angular coordinates of the pulsars.
Table IV. The 88 pulsars currently employed by the European Pulsar Timing Array, the North American Nanohertz Observatory for Gravitational Waves, and the Parkes Pulsar Timing Array collaborations. The final column identifies which PTAs employ this pulsar. The sky location is given by right ascension and declination in degrees. The International Pulsar Timing Array (IPTA) pulsars are the full list.
The individual PTA collaborations are currently monitoring 42, 66, and 26 pulsars respectively, with a total of 88 distinct pulsars for the IPTA [37 ] . A skymap of the pulsars is shown in Figure 8 , which is a Mollweide projection in equatorial coordinates. Note that the pulsars are clustered in the direction of the galactic center, which has equatorial coordinates ( r a , d e c ) = ( 17 h 46 m , − 29 ∘ ) (\mathrm{ra},\mathrm{dec})=(17\mathrm{h}46\mathrm{m},-29^\circ) ( ra , dec ) = ( 17 h 46 m , − 2 9 ∘ ) Figure 8 . For reference, the center of the sky maps is ( r a , d e c ) = ( 12 h , 0 ∘ ) (\mathrm{ra},\mathrm{dec})=(12\mathrm{h},0^\circ) ( ra , dec ) = ( 12 h , 0 ∘ )
Figure 8. Sky locations of the pulsars employed by the EPTA, NANOGrav, PPTA, and IPTA collaborations. The black dot indicates the direction to the Galactic center. Table IV in Appendix § 24 lists the pulsar names and sky locations.
Plots showing the expected Hellings and Downs correlation plus/minus the uncertainties associated with the total variance, cosmic variance, and variance of the optimal binned estimator are given in Figure 9 .
Figure 9. The mean Hellings and Downs timing-residual correlations (solid black curve) and the ± σ o p t \pm\sigma_{\mathrm{opt}} ± σ opt 30 × 6 ∘ 30 \times6^\circ 30 × 6 ∘ Figure 3 for a closeup of the IPTA variances.) We assume the binary-inspiral GW background described in Appendix B, which has h ˉ 2 / h 2 ≈ 0.5622 \bar{h}^{2}/h^{2} \approx0.5622 h ˉ 2 / h 2 ≈ 0.5622 α = 1 \alpha=1 α = 1 h 2 = 1 h^{2}=1 h 2 = 1 ± σ t o t \pm\sigma_{\mathrm{tot}} ± σ tot ± σ c o s \pm\sigma_{\mathrm{cos}} ± σ cos 3 ∘ 3^\circ 3 ∘ 147 ∘ 147^\circ 14 7 ∘ 159 ∘ 159^\circ 15 9 ∘ 177 ∘ 177^\circ 17 7 ∘ n p a i r s = 0 n_{\mathrm{pairs}}=0 n pairs = 0
For these plots, we use thirty 6°-wide angular-separation bins, equally spaced between 0 and 180°. To model the relative amplitude of the expected correlations and their uncertainties, we set h ˉ 2 = 0.5622 h 2 \bar{h}^{2}=0.5622h^{2} h ˉ 2 = 0.5622 h 2 h 2 = 1 h^{2}=1 h 2 = 1
(c) in Sec. III A, that is, μ b i n ≡ μ u ( γ b i n ) \mu_{\mathrm{bin}}\equiv\mu_u(\gamma_{\mathrm{bin}}) μ bin ≡ μ u ( γ bin ) γ b i n ≡ γ T 1 / n p a i r s \gamma_{\mathrm{bin}}\equiv\gamma^{\mathsf{T}}1/n_{\mathrm{pairs}} γ bin ≡ γ T 1/ n pairs
These plots do not include any contributions from measurement noise or intrinsic pulsar noise. As such, they show the fundamental limits on the recovery of the Hellings and Downs correlation, based on using optimal estimators for the given set of pulsars and a specific choice of binning.
Since the PTAs are improving their results by adding additional pulsars, these plots will improve with time. They may be easily constructed as follows:
starting with a set of pulsar sky directions, form all pulsar pairs;
divide the pulsar pairs among a set of angular-separation bins;
for each bin, compute the average angular separation γ b i n \gamma_{\mathrm{bin}} γ bin n p a i r s n_{\mathrm{pairs}} n pairs
for each bin, choose a normalization μ b i n \mu_{\mathrm{bin}} μ bin μ b i n ≡ μ u ( γ b i n ) \mu_{\mathrm{bin}}\equiv\mu_u(\gamma_{\mathrm{bin}}) μ bin ≡ μ u ( γ bin )
for each bin, calculate the n p a i r s n_{\mathrm{pairs}} n pairs μ a b \mu_{ab} μ ab C a b , c d C_{ab,cd} C ab , c d a b ab ab c d cd c d (5) and (11) ; the n p a i r s × n p a i r s n_{\mathrm{pairs}}\times n_{\mathrm{pairs}} n pairs × n pairs C C C C C C
for each bin, calculate the variance σ o p t 2 \sigma_{\mathrm{opt}}^{2} σ opt 2 (26) for each bin.
Following this simple procedure, interested readers can produce similar plots for different arrays of pulsars and different choices of angular bins.
NON-PTA DETECTORS
The focus of this paper is the Hellings and Downs correlation (1) induced by GWs in PTAs. However, to aid in our understanding, it is useful to consider “non-PTA” detectors, which have a different response to GWs than a PTA. This is particularly helpful in understanding the properties of estimators of the squared GW strain h 2 h^{2} h 2 [38 ] . These estimators, which are linear combinations of pulsar-pair correlation measurements ρ a b \rho_{ab} ρ ab
The most familiar non-PTA detector type is the “short-arm interferometer” GW detector, described in Appendix D. 4 of [32 ] . To construct this, place one arm of a LIGO-like detector along the line of sight p ^ \hat{p} p ^ [16 ] ) is small compared to the GW wavelength. In practice, at typical PTA frequencies this detector would be overwhelmed by low-frequency noise, but in our idealized thought experiment we replace pulsar redshift with the fractional change in length between the mirrors.
Let us contrast the response (“antenna pattern”) of this LIGO-like detector with one arm along direction p ^ \hat{p} p ^ p ^ \hat{p} p ^ Ω ^ \hat{\Omega} Ω ^
one-arm LIGO: F ( Ω ^ ) = 1 2 ( 2 π i f L ) p ^ μ p ^ ν e μ ν ( Ω ^ ) and (79) \text{one-arm LIGO:}\quad F(\hat{\Omega}) = \frac{1}{2}(2\pi if\mathcal{L})\hat{p}^{\mu}\hat{p}^{\nu}e_{\mu\nu}(\hat{\Omega})\quad\text{and} \tag*{(79)} one-arm LIGO: F ( Ω ^ ) = 2 1 ( 2 π i f L ) p ^ μ p ^ ν e μν ( Ω ^ ) and (79) PTA pulsar: F ( Ω ^ ) = 1 2 p ^ μ p ^ ν 1 + Ω ^ ⋅ p ^ e μ ν ( Ω ^ ) [ 1 − η e − 2 π i f L ( 1 + Ω ^ ⋅ p ^ ) ] . (80) \text{PTA pulsar:}\quad F(\hat{\Omega}) = \frac{1}{2}\frac{\hat{p}^{\mu}\hat{p}^{\nu}}{1+\hat{\Omega}\cdot\hat{p}}e_{\mu\nu}(\hat{\Omega})\left[1-\eta e^{-2\pi if\mathcal{L}(1+\hat{\Omega}\cdot\hat{p})}\right]. \tag*{(80)} PTA pulsar: F ( Ω ^ ) = 2 1 1 + Ω ^ ⋅ p ^ p ^ μ p ^ ν e μν ( Ω ^ ) [ 1 − η e − 2 π i f L ( 1 + Ω ^ ⋅ p ^ ) ] . (80) Here, the Einstein summation convention is in effect for the spatial indices μ \mu μ ν \nu ν e μ ν ≡ e μ ν + − i e μ ν × e_{\mu\nu}\equiv e^{+}_{\mu\nu}-ie^{\times}_{\mu\nu} e μν ≡ e μν + − i e μν × (221) in [11 ] ), and f L f\mathcal{L} f L
The factor of 2 π i f 2\pi if 2 π i f (79) arises because F ( Ω ^ ) F(\hat{\Omega}) F ( Ω ^ ) [14 ] , which is the fractional variation h = Δ L / L = 1 2 p ^ μ p ^ ν e μ ν ( Ω ^ ) h=\Delta\mathcal{L}/\mathcal{L}=\frac{1}{2}\hat{p}^{\mu}\hat{p}^{\nu}e_{\mu\nu}(\hat{\Omega}) h = Δ L / L = 2 1 p ^ μ p ^ ν e μν ( Ω ^ ) L \mathcal{L} L Δ L \Delta\mathcal{L} Δ L 2 π i f 2\pi if 2 π i f (79) follows.
In (80) , the unity term in square brackets gives the Earth term, and the second term in square brackets gives the pulsar term. If desired, a similar pulsar term could be incorporated into the one-arm LIGO-like detector [39 ] . The real constant η \eta η η = 1 \eta=1 η = 1 [40 ] , or to turn off the pulsar term by setting η = 0 \eta=0 η = 0
The response (79) of a one-arm LIGO-like detector may be obtained as the low-frequency (≡ \equiv ≡ (80) [41 , 42 ] . This is no coincidence. If our fictional one-arm LIGO-like detector could also operate at wavelengths much smaller than its arm length, then its response would be described by the same expression (80) with η = 1 \eta=1 η = 1 [43 ] . If we start with (80) , and take the limit f L → 0 f\mathcal{L}\to0 f L → 0 (79) . This is because the pulsar term in (80) may be approximated using e x ≈ 1 + x e^{x}\approx1+x e x ≈ 1 + x 1 + Ω ^ ⋅ p ^ 1+\hat{\Omega}\cdot\hat{p} 1 + Ω ^ ⋅ p ^
This short-arm limit is valid for the currently operating advanced LIGO detectors in their most sensitive frequency band around f ≈ 100 H z f\approx100\ \mathrm{Hz} f ≈ 100 Hz ≈ 3000 k m \approx3000\ \mathrm{km} ≈ 3000 km 4 k m 4\ \mathrm{km} 4 km [16 ] . By assumption/construction, the short-arm limit is valid for our hypothetical one-arm LIGO-like detectors.
Equations (79) and (80) for the detector response can be recast in harmonic space by decomposing the GW perturbations into a sum of (gradients and curls of) spherical harmonic functions Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi) Y l m ( θ , ϕ ) [32 ] . Any GW field may be expressed in this form, whose value is determined by a countably infinite set of amplitudes, labeled by l = 2 , 3 , … l=2,3,\ldots l = 2 , 3 , … m = − l , − l + 1 , … , l m=-l,-l+1,\ldots,l m = − l , − l + 1 , … , l l l l l , m l,m l , m (219) from [32 ] , and by eq:92 (including the pulsar term) from that same reference. These are
one-arm LIGO: F l m ( p ^ ) ∝ δ l 2 Y l m ( p ^ ) and (81) \text{one-arm LIGO:}\quad F_{lm}(\hat{p})\propto\delta_{l2}Y_{lm}(\hat{p})\quad\text{and} \tag*{(81)} one-arm LIGO: F l m ( p ^ ) ∝ δ l 2 Y l m ( p ^ ) and (81) PTA pulsar: F l m ( p ^ ) ∝ C l 1 / 2 [ 1 − η e − 2 π i f L ( 1 + Ω ^ ⋅ p ^ ) ] Y l m ( p ^ ) , (82) \text{PTA pulsar:}\quad F_{lm}(\hat{p})\propto C_l^{1/2}\left[1-\eta e^{-2\pi if\mathcal{L}(1+\hat{\Omega}\cdot\hat{p})}\right]Y_{lm}(\hat{p}), \tag*{(82)} PTA pulsar: F l m ( p ^ ) ∝ C l 1/2 [ 1 − η e − 2 π i f L ( 1 + Ω ^ ⋅ p ^ ) ] Y l m ( p ^ ) , (82) where the C l C_l C l (60) , and l l l only the l = 2 l=2 l = 2 modes , whereas the PTA pulsar detector responds to all modes with l ≥ 2 l\ge2 l ≥ 2 Y l m ( p ^ ) Y_{lm}(\hat{p}) Y l m ( p ^ ) Y l m ( θ p , ϕ p ) Y_{lm}(\theta_p,\phi_p) Y l m ( θ p , ϕ p ) θ p , ϕ p \theta_p,\phi_p θ p , ϕ p p ^ \hat{p} p ^
Our claim about the distinct patterns of response may appear surprising. Fundamentally, it is a statement about the differing antenna patterns of the detectors, and can be understood by comparing (79) and (80) . Said succinctly, a LIGO-like detector has a purely quadrupolar antenna pattern and response, whereas a PTA pulsar detector responds to all modes.
To determine the output at time t t t h μ ν ( t ) h_{\mu\nu}(t) h μν ( t ) [44 ] at time t t t 1 + Ω ^ ⋅ p ^ 1+\hat{\Omega}\cdot\hat{p} 1 + Ω ^ ⋅ p ^ (80) . Because the denominator depends upon Ω ^ \hat{\Omega} Ω ^ h μ ν ( t ) h_{\mu\nu}(t) h μν ( t )
As shown in (79) , a LIGO detector with one arm in direction p ^ μ \hat{p}^{\mu} p ^ μ p ^ μ p ^ ν \hat{p}^{\mu}\hat{p}^{\nu} p ^ μ p ^ ν p ^ a , p ^ b \hat{p}_{a},\hat{p}_{b} p ^ a , p ^ b p ^ a μ p ^ a ν − p ^ b μ p ^ b ν \hat{p}_{a}^{\mu}\hat{p}_{a}^{\nu}-\hat{p}_{b}^{\mu}\hat{p}_{b}^{\nu} p ^ a μ p ^ a ν − p ^ b μ p ^ b ν p ^ x , p ^ y , p ^ z \hat{p}^{x},\hat{p}^{y},\hat{p}^{z} p ^ x , p ^ y , p ^ z l = 2 l=2 l = 2 l l l 1 + Ω ^ ⋅ p ^ 1+\hat{\Omega}\cdot\hat{p} 1 + Ω ^ ⋅ p ^ (79) gives the PTA pulsar detector an antenna pattern that also responds to modes with l > 2 l>2 l > 2
Note that even for a (one- or two-arm) LIGO-like detector, the five components of h μ ν h_{\mu\nu} h μν l = 2 l=2 l = 2 [32 ] . Thus, there are five of these per frequency bin; the restriction to l = 2 l=2 l = 2
Returning back to (81) and (82) , we can use these response functions to calculate the expected value of the correlation between detector responses Z a ( t ) Z_a(t) Z a ( t ) Z b ( t ) Z_b(t) Z b ( t ) l l l m m m [12 , 13 , 32 ] . Let p ^ a \hat{p}_a p ^ a p ^ b \hat{p}_b p ^ b a a a b b b
⟨ ρ a b ⟩ = h 2 μ a b = h 2 ∑ l ∑ m = − l l F l m ( p ^ a ) F l m ∗ ( p ^ b ) , (83) \langle\rho_{ab} \rangle= h^2\mu_{ab} = h^2 \sum_l \sum_{m=-l}^{l} F_{lm}(\hat{p}_a)F^*_{lm}(\hat{p}_b), \tag*{(83)} ⟨ ρ ab ⟩ = h 2 μ ab = h 2 l ∑ m = − l ∑ l F l m ( p ^ a ) F l m ∗ ( p ^ b ) , (83) where μ a b \mu_{ab} μ ab
To evaluate the expected correlation for either of the two different GW detector types, substitute (81) or (82) into (83) and use the addition theorem for spherical harmonics,
∑ m = − l l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) = 2 l + 1 4 π P l ( cos γ a b ) , (84) \sum_{m=-l}^{l}Y_{lm}(\hat{p}_a)Y^*_{lm}(\hat{p}_b)=\frac{2l+1}{4\pi}P_l(\cos\gamma_{ab}), \tag*{(84)} m = − l ∑ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) = 4 π 2 l + 1 P l ( cos γ ab ) , (84) to evaluate the sum over m m m cos γ a b ≡ p ^ a ⋅ p ^ b \cos\gamma_{ab}\equiv\hat{p}_a\cdot\hat{p}_b cos γ ab ≡ p ^ a ⋅ p ^ b
one-arm LIGO: μ a b = P 2 ( cos γ a b ) \mu_{ab}=P_2(\cos\gamma_{ab}) μ ab = P 2 ( cos γ ab )
μ a b = P 2 ( cos γ a b ) (85) \mu_{ab}=P_2(\cos\gamma_{ab}) \tag*{(85)} μ ab = P 2 ( cos γ ab ) (85) PTA pulsar: μ a b = ∑ l ( 2 l + 1 ) C l P l ( cos γ a b ) [ 1 + η 2 δ a b ] \mu_{ab}=\sum_l(2l+1)C_lP_l(\cos\gamma_{ab})[1+\eta^2\delta_{ab}] μ ab = ∑ l ( 2 l + 1 ) C l P l ( cos γ ab ) [ 1 + η 2 δ ab ]
μ a b = ∑ l ( 2 l + 1 ) C l P l ( cos γ a b ) [ 1 + η 2 δ a b ] , (86) \mu_{ab}=\sum_l(2l+1)C_lP_l(\cos\gamma_{ab})[1+\eta^2\delta_{ab}], \tag*{(86)} μ ab = l ∑ ( 2 l + 1 ) C l P l ( cos γ ab ) [ 1 + η 2 δ ab ] , (86) where for (86) we assume (i) that the Earth-to-pulsar distances are many GW wavelengths l ≪ f L l\ll f\mathcal{L} l ≪ f L L a \mathcal{L}_a L a L b \mathcal{L}_b L b l ≪ f ∣ L a − L b ∣ l\ll f\lvert\mathcal{L}_a-\mathcal{L}_b\rvert l ≪ f ∣ L a − L b ∣ [11 ] . Thus, (86) contains only Earth-Earth and pulsar-pulsar terms. For pairs of distinct pulsars a ≠ b a\ne b a = b μ u ( γ a b ) \mu_u(\gamma_{ab}) μ u ( γ ab ) (59) and (60) .
It should now be clear to the reader that the two examples we have just given are members of an infinite collection of possible detector response functions. To define these, we replace ( 2 l + 1 ) C l (2l+1)C_l ( 2 l + 1 ) C l (86) by an arbitrary set of (positive) coefficients Q l Q_l Q l Q l = 0 Q_l=0 Q l = 0 l ≠ 2 l\ne2 l = 2 l ≥ 2 l\ge2 l ≥ 2 μ a b \mu_{ab} μ ab
The generalized detector response functions are defined in analogy with (86) . They have a correlation matrix
μ a b = ( 1 + η 2 δ a b ) U a b , (87) \mu_{ab}=(1+\eta^2\delta_{ab})U_{ab}, \tag*{(87)} μ ab = ( 1 + η 2 δ ab ) U ab , (87) where U U U N p u l × N p u l N_{\mathrm{pul}}\times N_{\mathrm{pul}} N pul × N pul
U a b = ∑ l Q l P l ( cos γ a b ) . (88) U_{ab}=\sum_l Q_lP_l(\cos\gamma_{ab}). \tag*{(88)} U ab = l ∑ Q l P l ( cos γ ab ) . (88) The non-negative quantities Q l ≥ 0 Q_l\ge0 Q l ≥ 0 U a b U_{ab} U ab η 2 > 0 \eta^2>0 η 2 > 0 (87) allows us to control the pulsar term. Depending upon the context, N p u l N_{\mathrm{pul}} N pul
A PTA has η = 1 \eta=1 η = 1 Q l = ( 2 l + 1 ) C l Q_l=(2l+1)C_l Q l = ( 2 l + 1 ) C l C l C_l C l (60) . Alternative detectors have different values for the Q l Q_l Q l Q l Q_l Q l l = L l=L l = L 2 ≤ l ≤ L 2 \le l \le L 2 ≤ l ≤ L Q l = δ l 2 Q_l=\delta_{l2} Q l = δ l 2 η = 0 \eta=0 η = 0
Eigenvectors and eigenvalues of the correlation matrix μ a b \mu_{ab} μ ab
We now investigate the properties of the correlation matrix μ a b \mu_{ab} μ ab (87) and (88) . We start by finding its eigenvalues and eigenvectors. For this, it is helpful to first examine the matrix U a b U_{ab} U ab
The eigenvalues of U a b U_{ab} U ab Q l ≥ 0 Q_l \ge0 Q l ≥ 0 (88) . To see this, use the addition theorem (84) to write U a b U_{ab} U ab
U a b = ∑ l 4 π 2 l + 1 Q l ∑ m = − l l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) , (89) U_{ab}=\sum_l\frac{4\pi}{2l+1}Q_l\sum_{m=-l}^{l}Y_{lm}(\hat p_a)Y^*_{lm}(\hat p_b), \tag*{(89)} U ab = l ∑ 2 l + 1 4 π Q l m = − l ∑ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) , (89) where p ^ a \hat p_a p ^ a p ^ b \hat p_b p ^ b v T U v = ∑ a , b v a U a b v b v^{\mathsf T}Uv=\sum_{a,b}v^aU_{ab}v^b v T U v = ∑ a , b v a U ab v b v a v^a v a N p u l N_{\mathrm{pul}} N pul U U U v T U v > 0 v^{\mathsf T}Uv>0 v T U v > 0 v v v v T U v ≥ 0 v^{\mathsf T}Uv\ge0 v T U v ≥ 0 v v v
∑ a , b v a U a b v b = ∑ l 4 π 2 l + 1 Q l ∑ m = − l l ∣ y l m ∣ 2 , (90) \sum_{a,b}v^aU_{ab}v^b=\sum_l\frac{4\pi}{2l+1}Q_l\sum_{m=-l}^{l}\lvert y_{lm}\rvert^2, \tag*{(90)} a , b ∑ v a U ab v b = l ∑ 2 l + 1 4 π Q l m = − l ∑ l ∣ y l m ∣ 2 , (90) where y l m ≡ ∑ a v a Y l m ( Ω ^ a ) y_{lm}\equiv\sum_a v^aY_{lm}(\hat\Omega_a) y l m ≡ ∑ a v a Y l m ( Ω ^ a ) Q l ≥ 0 Q_l\ge0 Q l ≥ 0 (90) is non-negative. This establishes that U U U
To count the vanishing eigenvalues of U a b U_{ab} U ab Q l Q_l Q l l = L l=L l = L Q L Q_L Q L (90) an eigenvector v a v^a v a
∑ a v a Y L m ( p ^ a ) = 0 , m = − L , … , L . \sum_a v^aY_{Lm}(\hat p_a)=0,\qquad m=-L,\ldots,L. a ∑ v a Y L m ( p ^ a ) = 0 , m = − L , … , L . This is a set of 2 L + 1 2L+1 2 L + 1 N p u l N_{\mathrm{pul}} N pul v a v^a v a N p u l ≤ 2 L + 1 N_{\mathrm{pul}}\le2L+1 N pul ≤ 2 L + 1 v a = 0 v^a=0 v a = 0 N p u l ≤ 2 L + 1 N_{\mathrm{pul}}\le2L+1 N pul ≤ 2 L + 1 N p u l N_{\mathrm{pul}} N pul N p u l > 2 L + 1 N_{\mathrm{pul}}>2L+1 N pul > 2 L + 1 ∑ a v a Y L m ( p ^ a ) = 0 \sum_a v^aY_{Lm}(\hat p_a)=0 ∑ a v a Y L m ( p ^ a ) = 0 N p u l − ( 2 L + 1 ) N_{\mathrm{pul}}-(2L+1) N pul − ( 2 L + 1 ) v a v^a v a N p u l − ( 2 L + 1 ) N_{\mathrm{pul}}-(2L+1) N pul − ( 2 L + 1 ) 2 L + 1 2L+1 2 L + 1
The result generalizes immediately to the case where more than one Q l Q_l Q l
N D O F ≡ ∑ { l ∣ Q l ≠ 0 } ( 2 l + 1 ) , (91) N_{\mathrm{DOF}}\equiv\sum_{\{l\mid Q_l\ne0\}}(2l+1), \tag*{(91)} N DOF ≡ { l ∣ Q l = 0 } ∑ ( 2 l + 1 ) , (91) which is the number of real degrees of freedom (DOF) associated with the nonzero Q l Q_l Q l N p u l ≤ N D O F N_{\mathrm{pul}}\le N_{\mathrm{DOF}} N pul ≤ N DOF U U U U U U N D O F N_{\mathrm{DOF}} N DOF N p u l − N D O F N_{\mathrm{pul}}-N_{\mathrm{DOF}} N pul − N DOF
For notational clarity, for any square matrix M M M
N M + ≡ number of nonzero eigenvalues of M . (92) N_M^+\equiv\text{number of nonzero eigenvalues of }M. \tag*{(92)} N M + ≡ number of nonzero eigenvalues of M . (92) For example, N U + N_U^+ N U + U a b U_{ab} U ab M M M N M + N_M^+ N M + M M M
With this notation, the number of positive and zero eigenvalues of U a b U_{ab} U ab
number of positive eigenvalues of U = N U + = min ( N D O F , N p u l ) U=N_U^+=\min(N_{\mathrm{DOF}},N_{\mathrm{pul}}) U = N U + = min ( N DOF , N pul )
number of zero eigenvalues of U = N p u l − N U + = max ( N p u l − N D O F , 0 ) U=N_{\mathrm{pul}}-N_U^+=\max(N_{\mathrm{pul}}-N_{\mathrm{DOF}},0) U = N pul − N U + = max ( N pul − N DOF , 0 )
number of positive eigenvalues of U = N U + = min ( N D O F , N p u l ) , and number of zero eigenvalues of U = N p u l − N U + = max ( N p u l − N D O F , 0 ) . (93) \begin{aligned}
\text{number of positive eigenvalues of }U&=N_U^+=\min(N_{\mathrm{DOF}},N_{\mathrm{pul}}), \qquad\text{and}\\
\text{number of zero eigenvalues of }U&=N_{\mathrm{pul}}-N_U^+=\max(N_{\mathrm{pul}}-N_{\mathrm{DOF}},0). \tag*{(93)}
\end{aligned} number of positive eigenvalues of U number of zero eigenvalues of U = N U + = min ( N DOF , N pul ) , and = N pul − N U + = max ( N pul − N DOF , 0 ) . (93) Note that a real PTA has Q l ≠ 0 Q_l\ne0 Q l = 0 l ≥ 2 l\ge2 l ≥ 2 N U + = N p u l N_U^+=N_{\mathrm{pul}} N U + = N pul U a b U_{ab} U ab
It is now straightforward to count the number of vanishing eigenvalues of the correlation matrix μ a b \mu_{ab} μ ab η = 0 \eta=0 η = 0 (87) one has μ a b = U a b \mu_{ab}=U_{ab} μ ab = U ab (93) also describes the eigenvalues of μ a b \mu_{ab} μ ab η ≠ 0 \eta\ne0 η = 0 μ a b \mu_{ab} μ ab
To see this, note from (88) that the diagonal terms of U a b U_{ab} U ab P l ( cos γ a a ) = P l ( cos 0 ) = P l ( 1 ) = 1 P_l(\cos\gamma_{aa})=P_l(\cos0)=P_l(1)=1 P l ( cos γ aa ) = P l ( cos 0 ) = P l ( 1 ) = 1 Q l ≥ 0 Q_l\ge0 Q l ≥ 0
U ( 0 ) = U a a = ∑ l Q l > 0. (94) U(0)=U_{aa}=\sum_l Q_l>0. \tag*{(94)} U ( 0 ) = U aa = l ∑ Q l > 0. (94) Since they are equal, from (87) we can write
μ a b = U a b + η 2 U ( 0 ) δ a b . (95) \mu_{ab}=U_{ab}+\eta^2U(0)\delta_{ab}. \tag*{(95)} μ ab = U ab + η 2 U ( 0 ) δ ab . (95) where δ a b \delta_{ab} δ ab v a v^{a} v a λ \lambda λ U U U v a v^{a} v a λ + η 2 U ( 0 ) \lambda+\eta^{2}U(0) λ + η 2 U ( 0 ) μ a b \mu_{ab} μ ab λ ≥ 0 \lambda\geq0 λ ≥ 0 η 2 \eta^{2} η 2 U ( 0 ) U(0) U ( 0 ) μ a b \mu_{ab} μ ab η ≠ 0 \eta\ne0 η = 0 μ a b \mu_{ab} μ ab
Inverse of the correlation matrix $\mu_{ab}`
Having understood the properties of the eigenvalues of the correlation matrix μ a b \mu_{ab} μ ab
For a PTA, or for any GW detector response which includes a pulsar term, the matrix μ a b \mu_{ab} μ ab μ a b − 1 \mu^{-1}_{ab} μ ab − 1 η = 0 \eta=0 η = 0 N D O F N_{\mathrm{DOF}} N DOF N D O F N_{\mathrm{DOF}} N DOF μ a b \mu_{ab} μ ab
In such cases, it is helpful to define a modified inverse, which we denote by μ a b + \mu^{+}_{ab} μ ab +
μ a b + = ∑ { n ∣ λ n > 0 } λ n − 1 v n a v n b , (96) \mu^{+}_{ab}=\sum_{\{n\mid\lambda_n>0\}}\lambda_n^{-1}v_n^{a}v_n^{b}, \tag*{(96)} μ ab + = { n ∣ λ n > 0 } ∑ λ n − 1 v n a v n b , (96) where λ n \lambda_n λ n v n a v_n^{a} v n a μ a b \mu_{ab} μ ab Moore-Penrose pseudoinverse of the matrix: it was found by Moore [45 , 46 ] and rediscovered three decades later by Penrose [47 ] ; see [48 ] for an account. If all of the eigenvalues of μ a b \mu_{ab} μ ab μ a b + \mu^{+}_{ab} μ ab + μ a b − 1 \mu^{-1}_{ab} μ ab − 1
∑ b μ a b + μ b c = ∑ b μ a b μ b c + = δ a c . \sum_b\mu^{+}_{ab}\mu_{bc}=\sum_b\mu_{ab}\mu^{+}_{bc}=\delta_{ac}. b ∑ μ ab + μ b c = b ∑ μ ab μ b c + = δ a c . If some of the eigenvalues of μ a b \mu_{ab} μ ab μ a b + \mu^{+}_{ab} μ ab +
∑ b μ a b + μ b c = ∑ b μ a b μ b c + = π a c , \sum_b\mu^{+}_{ab}\mu_{bc}=\sum_b\mu_{ab}\mu^{+}_{bc}=\pi_{ac}, b ∑ μ ab + μ b c = b ∑ μ ab μ b c + = π a c , where π a b \pi_{ab} π ab
π a b ≡ ∑ { n ∣ λ n > 0 } v n a v n b . (97) \pi_{ab}\equiv\sum_{\{n\mid\lambda_n>0\}}v_n^{a}v_n^{b}. \tag*{(97)} π ab ≡ { n ∣ λ n > 0 } ∑ v n a v n b . (97) In a basis formed from the eigenvectors v n a v_n^{a} v n a (97) shows that the matrix π a b \pi_{ab} π ab N μ + N_{\mu}^{+} N μ + N p u l − N μ + N_{\mathrm{pul}}-N_{\mu}^{+} N pul − N μ + π a b \pi_{ab} π ab ∑ a π a a = N μ + \sum_a\pi_{aa}=N_{\mu}^{+} ∑ a π aa = N μ + (97) , which is the number of nonzero eigenvalues of μ a b \mu_{ab} μ ab
Estimating the squared GW strain $h^2`
PTAs are constructed to detect low-frequency GWs; the most important quantity that they should determine is the squared amplitude h 2 h^2 h 2 h 2 h^2 h 2 h 2 ^ \widehat{h^2} h 2
If the pulsar measurements are free of noise, or if their noise is uncorrelated between different pulsars, then our expectation is that including more pulsars in the PTA will reduce the uncertainty in the strain estimate, because it adds more information. This is indeed the case. However, we will see that the rate at which that uncertainty decreases, and whether it tends to zero for large numbers of pulsars, depends upon the details.
In this section, we consider three cases, starting with data from N p u l N_{\mathrm{pul}} N pul N p u l ( N p u l + 1 ) / 2 N_{\mathrm{pul}}(N_{\mathrm{pul}}+1)/2 N pul ( N pul + 1 ) /2 N p u l N_{\mathrm{pul}} N pul N p u l ( N p u l − 1 ) / 2 N_{\mathrm{pul}}(N_{\mathrm{pul}}-1)/2 N pul ( N pul − 1 ) /2 h 2 h^2 h 2 Figure 10 , which shows the (inverse) variance of the estimators. Later in this section, we derive these results, and discuss and compare them in detail.
In addition, to gain some insight, we will also consider GW responses which differ from those of PTA pulsars. As described in Sec. VI, these could correspond, for example, to fictitious LIGO-like detectors, floating in space near
Earth, but sensitive to the same low frequencies as current PTAs.
In all three cases (i)–(iii) above, we use h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 ρ a b \rho_{ab} ρ ab w a b w_{ab} w ab
h ^ 2 ≡ ∑ a b ∈ S w a b ρ a b . (98) \hat{h}^2 \equiv\sum_{ab\in\mathcal{S}} w_{ab}\rho_{ab}. \tag*{(98)} h ^ 2 ≡ ab ∈ S ∑ w ab ρ ab . (98) The only difference between cases (i)–(iii) is the choice of the set S \mathcal{S} S a b ab ab S = { a b ∣ a ≤ b } \mathcal{S}=\{ab|a\le b\} S = { ab ∣ a ≤ b } S = { a b ∣ a = b } \mathcal{S}=\{ab|a=b\} S = { ab ∣ a = b } S = { a b ∣ a < b } \mathcal{S}=\{ab|a<b\} S = { ab ∣ a < b } a ≤ b a\le b a ≤ b a = b a=b a = b a < b a<b a < b
Note that (98) is proportional to ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab (16) , which gives the optimal estimator of the Hellings and Downs correlation for a particular angular-separation bin. The situation here is the same. A constant term in (98) is only needed if (i) auto-correlations (ρ a a \rho_{aa} ρ aa (16) also applies here. If both (i) and (ii) hold, then (98) is not general enough to form an unbiased estimator for h 2 h^2 h 2 (98) . Since we primarily focus on the ideal noise-free case, we do not include a constant term in (98) or in what follows. We will return to this issue later, in Sec. IX, when we consider the effects of noise.
To find the optimal (unbiased, minimum-variance) estimator for the squared strain h 2 h^2 h 2 (98) with exactly the same technique that was used in Sec. III A. The resulting estimator h ^ 2 \hat{h}^2 h ^ 2 ρ a b \rho_{ab} ρ ab
h ^ 2 = ( ∑ a b , c d ∈ S μ a b C a b , c d − 1 μ c d ) − 1 ∑ e f , g h ∈ S μ e f C e f , g h − 1 ρ g h . (99) \hat{h}^2=
\left(
\sum_{ab,cd\in\mathcal{S}}\mu_{ab}\mathcal{C}^{-1}_{ab,cd}\mu_{cd}
\right)^{-1}
\sum_{ef,gh\in\mathcal{S}}\mu_{ef}\mathcal{C}^{-1}_{ef,gh}\rho_{gh}. \tag*{(99)} h ^ 2 = ab , c d ∈ S ∑ μ ab C ab , c d − 1 μ c d − 1 e f , g h ∈ S ∑ μ e f C e f , g h − 1 ρ g h . (99) The expected value of h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2
σ h ^ 2 2 = ( ∑ a b , c d ∈ S μ a b C a b , c d − 1 μ c d ) − 1 . (100) \sigma_{\hat{h}^2}^2=
\left(
\sum_{ab,cd\in\mathcal{S}}\mu_{ab}\mathcal{C}^{-1}_{ab,cd}\mu_{cd}
\right)^{-1}. \tag*{(100)} σ h ^ 2 2 = ab , c d ∈ S ∑ μ ab C ab , c d − 1 μ c d − 1 . (100) Note that h ^ 2 \hat{h}^2 h ^ 2 generalized chi-squared distribution [49 ] .
The pairs a b ab ab c d cd c d (99) and (100) differ for cases (i)–(iii), exactly as described after (98) .
In the following three subsections, we investigate the behavior of the variance for those three different cases. Expressions (99) and (100) for h ^ 2 \hat{h}^2 h ^ 2 σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 [42 , 50 , 51 ] ; in the literature this is called the “optimal statistic.” Our expression differs in two ways. First, the standard cross-correlation estimator corresponds to choice (iii) above, and only includes cross-correlations. With choices (i) and (ii) our definition can also include auto-correlations. Second, the standard optimal statistic assumes that the covariance matrix is dominated by instrumental and pulsar noise. Our expressions are general enough to cover the entire range from noise-free to noise-dominated, although we primarily focus on the noise-free case.
In the absence of noise, the optimal estimator h ^ 2 \hat{h}^2 h ^ 2 ρ a b \rho_{ab} ρ ab R 4 \mathcal{R}^4 R 4 C \mathcal{C} C (11) ] cancel out from the numerator and denominator of (99) . In contrast, the variance σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 R 4 \mathcal{R}^4 R 4 h 2 h^2 h 2 unknown value of the squared GW strain that we are trying to estimate.
If these estimators are applied to observational data, one cannot avoid the question: “what value of h 2 h^2 h 2 σ h ^ 2 2 = σ h ^ 2 2 ( h 2 ) \sigma_{\hat{h}^2}^2=\sigma_{\hat{h}^2}^2(h^2) σ h ^ 2 2 = σ h ^ 2 2 ( h 2 ) h 2 h^2 h 2 h 2 h^2 h 2 h 2 ± σ h ^ 2 ( h 2 ) h^2\pm\sigma_{\hat{h}^2}(h^2) h 2 ± σ h ^ 2 ( h 2 ) h 2 h^2 h 2 h 2 h^2 h 2 ± σ h ^ 2 ( h 2 ) \pm\sigma_{\hat{h}^2}(h^2) ± σ h ^ 2 ( h 2 ) h ^ 2 \hat{h}^2 h ^ 2
Case (i): Using auto-correlations and cross-correlations
We first consider the optimal estimator (98) for the squared strain h 2 h^2 h 2 both the cross-correlations and the auto-correlations. While this is defined by the index set a ≤ b a\le b a ≤ b a a a b b b 1 , … , N p u l 1,\ldots,N_{\rm pul} 1 , … , N pul (98) includes N p u l 2 N_{\rm pul}^2 N pul 2 a = b a=b a = b ρ 37 \rho_{37} ρ 37 ρ 73 \rho_{73} ρ 73 w 37 = w 73 w_{37}=w_{73} w 37 = w 73 N p u l 2 N_{\rm pul}^2 N pul 2 a ≤ b a \le b a ≤ b N p u l 2 N_{\mathrm{pul}}^2 N pul 2 (98) , so that
h ^ 2 ≡ ∑ a = 1 N p u l ∑ b = 1 N p u l w a b ρ a b ≡ ∑ a , b w a b ρ a b , (101) \hat{h}^2 \equiv\sum_{a=1}^{N_{\mathrm{pul}}}\sum_{b=1}^{N_{\mathrm{pul}}} w_{ab}\rho_{ab} \equiv\sum_{a,b} w_{ab}\rho_{ab}, \tag*{(101)} h ^ 2 ≡ a = 1 ∑ N pul b = 1 ∑ N pul w ab ρ ab ≡ a , b ∑ w ab ρ ab , (101) where the final summation is shorthand notation for the previous two sums. The same extension of the summations apply to (99) and (100) .
Here, and elsewhere in this paper, we use the following convention for commas that occur under summation signs. A comma between indices under a summation sign, for example the “a , b a, b a , b (101) , means that within the sum, each index ranges separately over 1 , … , N p u l 1,\ldots,N_{\mathrm{pul}} 1 , … , N pul a b ab ab a b ab ab S S S C a b , c d \mathcal{C}_{ab,cd} C ab , c d D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h
The variance of h ^ 2 \hat{h}^2 h ^ 2 (100) , where each index a a a b b b c c c d d d 1 1 1 N p u l N_{\mathrm{pul}} N pul (100) in closed form, by explicitly inverting the covariance matrix. [With these index conventions, the inverse C − 1 \mathcal{C}^{-1} C − 1 μ a b \mu_{ab} μ ab (100) is also well defined.]
With the index labeling conventions described in the previous two paragraphs, the inverse of C a b , c d \mathcal{C}_{ab,cd} C ab , c d
C a b , c d − 1 = 1 4 K − 4 ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) . (102) \mathcal{C}^{-1}_{ab,cd} = \frac{1}{4}\mathcal{K}^{-4}\left(\mu^{-1}_{ac}\mu^{-1}_{bd}+\mu^{-1}_{ad}\mu^{-1}_{bc}\right). \tag*{(102)} C ab , c d − 1 = 4 1 K − 4 ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) . (102) To understand the restriction, consider the original definition of C \mathcal{C} C (11) , but with the indexing conventions of this subsection: each of the four indices runs from 1 1 1 N p u l N_{\mathrm{pul}} N pul C \mathcal{C} C N p u l × N p u l N_{\mathrm{pul}}\times N_{\mathrm{pul}} N pul × N pul Λ n \Lambda_n Λ n W n a b W_n^{ab} W n ab C \mathcal{C} C
∑ c , d C a b , c d W n c d = Λ n W n a b and ∑ a , b W n a b W m a b = δ n m . (103) \sum_{c,d}\mathcal{C}_{ab,cd}W_n^{cd}=\Lambda_n W_n^{ab}\quad\text{and}\quad\sum_{a,b}W_n^{ab}W_m^{ab}=\delta_{nm}. \tag*{(103)} c , d ∑ C ab , c d W n c d = Λ n W n ab and a , b ∑ W n ab W m ab = δ nm . (103) In terms of these, C \mathcal{C} C
C a b , c d = ∑ n Λ n W n a b W n c d , (104) \mathcal{C}_{ab,cd}=\sum_n\Lambda_n W_n^{ab}W_n^{cd}, \tag*{(104)} C ab , c d = n ∑ Λ n W n ab W n c d , (104) which can be checked by confirming the action of C \mathcal{C} C (103) .
Since the number of eigenvectors is the dimension of the space of linearly independent N p u l × N p u l N_{\mathrm{pul}}\times N_{\mathrm{pul}} N pul × N pul n n n n = 1 , 2 , … , N p u l 2 n=1,2,\ldots,N_{\mathrm{pul}}^2 n = 1 , 2 , … , N pul 2 C \mathcal{C} C (11) , one can see that it is symmetric in the indices c c c d d d N p u l ( N p u l − 1 ) / 2 N_{\mathrm{pul}}(N_{\mathrm{pul}}-1)/2 N pul ( N pul − 1 ) /2 N p u l ( N p u l + 1 ) / 2 N_{\mathrm{pul}}(N_{\mathrm{pul}}+1)/2 N pul ( N pul + 1 ) /2 μ a b \mu_{ab} μ ab
By the inverse of C \mathcal{C} C
C a b , c d − 1 = ∑ { n ∣ Λ n > 0 } Λ n − 1 W n a b W n c d , (105) \mathcal{C}^{-1}_{ab,cd}=\sum_{\{n\mid\Lambda_n>0\}}\Lambda_n^{-1}W_n^{ab}W_n^{cd}, \tag*{(105)} C ab , c d − 1 = { n ∣ Λ n > 0 } ∑ Λ n − 1 W n ab W n c d , (105) which excludes the eigenvectors corresponding to vanishing eigenvalues. Strictly speaking, C − 1 \mathcal{C}^{-1} C − 1 (105) is the Moore-Penrose pseudoinverse C + \mathcal{C}^{+} C + C − 1 \mathcal{C}^{-1} C − 1 C \mathcal{C} C
∑ c , d C a b , c d C c d , e f − 1 = ∑ { n ∣ Λ n ≠ 0 } W n a b W n e f . (106) \sum_{c,d}\mathcal{C}_{ab,cd}\mathcal{C}^{-1}_{cd,ef}=\sum_{\{n\mid\Lambda_n\ne0\}}W_n^{ab}W_n^{ef}. \tag*{(106)} c , d ∑ C ab , c d C c d , e f − 1 = { n ∣ Λ n = 0 } ∑ W n ab W n e f . (106) Because the zero eigenvectors are missing from the sum, the rhs of (106) is not the identity matrix δ a e δ b f \delta_{ae}\delta_{bf} δ a e δ b f
Making use of (11) and (102) we obtain
∑ c , d C a b , c d C c d , e f − 1 = 1 4 ∑ c , d ( μ a c μ b d + μ a d μ b c ) ( μ c e − 1 μ d f − 1 + μ c f − 1 μ d e − 1 ) = 1 2 ( δ a e δ b f + δ a f δ b e ) . (107) \begin{aligned}
\sum_{c,d}\mathcal{C}_{ab,cd}\mathcal{C}^{-1}_{cd,ef} \\
&=\frac{1}{4}\sum_{c,d}\left(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}\right)\left(\mu^{-1}_{ce}\mu^{-1}_{df}+\mu^{-1}_{cf}\mu^{-1}_{de}\right) \\
&=\frac{1}{2}\left(\delta_{ae}\delta_{bf}+\delta_{af}\delta_{be}\right). \tag*{(107)}
\end{aligned} c , d ∑ C ab , c d C c d , e f − 1 = 4 1 c , d ∑ ( μ a c μ b d + μ a d μ b c ) ( μ ce − 1 μ df − 1 + μ c f − 1 μ d e − 1 ) = 2 1 ( δ a e δ b f + δ a f δ b e ) . (107) The rhs is precisely (106) . It is easy to show that this is the linear map of rank N p u l ( N p u l + 1 ) / 2 N_{\mathrm{pul}}(N_{\mathrm{pul}}+1)/2 N pul ( N pul + 1 ) /2 M M M ( M + M T ) / 2 (M+M^{\mathsf T})/2 ( M + M T ) /2
Please be warned that (102) does not give the inverse of C \mathcal{C} C C \mathcal{C} C
We now evaluate the (inverse) variance σ ^ h ^ 2 2 \hat{\sigma}_{\hat{h}^2}^2 σ ^ h ^ 2 2 (100) . Making use of (102) we find
( σ ^ h ^ 2 2 ) − 1 = ∑ a , b , c , d μ a b C a b , c d − 1 μ c d = 1 4 K − 4 ∑ a , b , c , d μ a b ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) μ c d = 1 2 K − 4 ∑ b δ b b = 1 2 K − 4 N p u l . (108) \begin{aligned}
\left(\hat{\sigma}_{\hat{h}^2}^2\right)^{-1}=\sum_{a,b,c,d}\mu_{ab}\mathcal{C}^{-1}_{ab,cd}\mu_{cd} \\
&=\frac{1}{4}\mathcal{K}^{-4}\sum_{a,b,c,d}\mu_{ab}\left(\mu^{-1}_{ac}\mu^{-1}_{bd}+\mu^{-1}_{ad}\mu^{-1}_{bc}\right)\mu_{cd} \\
&=\frac{1}{2}\mathcal{K}^{-4}\sum_b\delta_{bb} \\
&=\frac{1}{2}\mathcal{K}^{-4}N_{\mathrm{pul}}. \tag*{(108)}
\end{aligned} ( σ ^ h ^ 2 2 ) − 1 = a , b , c , d ∑ μ ab C ab , c d − 1 μ c d = 4 1 K − 4 a , b , c , d ∑ μ ab ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) μ c d = 2 1 K − 4 b ∑ δ bb = 2 1 K − 4 N pul . (108) Thus, the fractional variance for the squared GW strain is
σ h ^ 2 2 ⟨ h ^ 2 ⟩ 2 = 2 N p u l h 4 h 4 , (109) \frac{\sigma_{\hat{h}^{2}}^{2}}{\langle\hat{h}^{2}\rangle^{2}} = \frac{2}{N_{\mathrm{pul}}}\frac{\mathcal{h}^{4}}{h^{4}}, \tag*{(109)} ⟨ h ^ 2 ⟩ 2 σ h ^ 2 2 = N pul 2 h 4 h 4 , (109) where we have inverted (108) and used ⟨ h ^ 2 ⟩ = h 2 \langle\hat{h}^{2}\rangle=h^{2} ⟨ h ^ 2 ⟩ = h 2 Figure 11 .
Figure 11. The inverse variance (111) of the squared-strain estimator which optimally exploits both auto- and cross-correlations, as a function of the number of pulsars employed. Diagonal line: for any type of detector with a pulsar term, such as PTAs, σ h ^ 2 − 2 \sigma_{\hat{h}^{2}}^{-2} σ h ^ 2 − 2 h 2 h^{2} h 2 σ h ^ 2 2 \sigma_{\hat{h}^{2}}^{2} σ h ^ 2 2
Initially, we found this result surprising for two reasons. First, in the many-pulsar limit, the optimal estimator of the Hellings and Downs correlation has a nonzero (cosmic) variance. Why does the variance of the squared-strain estimate vanish in that same limit? Second, previous work (e.g., Eq. (C40) in Sec. C 4 of Ref. [11 ] ) shows that an apparently similar measure of the squared strain (the time average of s ≡ h μ ν h μ ν s \equiv h_{\mu\nu}h^{\mu\nu} s ≡ h μν h μν 2 / 5 2/5 2/5 h ^ 2 \hat{h}^{2} h ^ 2 (101) provide more precise information?
To answer these two questions, consider GW detector responses which differ than those of a PTA, as examined in Sec. VI A.
Sec. VI A. If there are a sufficient number of pulsars, then for some responses, the correlation matrix μ a b \mu_{ab} μ ab μ a b \mu_{ab} μ ab μ a b − 1 \mu_{ab}^{-1} μ ab − 1 (102) change to μ a b + \mu_{ab}^{+} μ ab + μ a b \mu_{ab} μ ab (96) . Then the inverse variance becomes
( σ h ^ 2 2 ) − 1 = 1 4 h − 4 ∑ a , b , c , d μ a b ( μ a c + μ b d + + μ a d + μ b c + ) μ c d = 1 4 h − 4 ( ∑ b , c π b c π b c + ∑ b , d π b d π b d ) = 1 2 h − 4 ∑ b π b b = 1 2 h − 4 N μ + . (110) \begin{aligned}
\left(\sigma_{\hat{h}^{2}}^{2}\right)^{-1}
&= \frac{1}{4}\mathcal{h}^{-4}\sum_{a,b,c,d}\mu_{ab}\left(\mu_{ac}^{+}\mu_{bd}^{+}+\mu_{ad}^{+}\mu_{bc}^{+}\right)\mu_{cd}\\
&= \frac{1}{4}\mathcal{h}^{-4}\left(\sum_{b,c}\pi_{bc}\pi_{bc}+\sum_{b,d}\pi_{bd}\pi_{bd}\right)\\
&= \frac{1}{2}\mathcal{h}^{-4}\sum_b\pi_{bb}\\
&= \frac{1}{2}\mathcal{h}^{-4}N_{\mu}^{+}. \tag*{(110)}
\end{aligned} ( σ h ^ 2 2 ) − 1 = 4 1 h − 4 a , b , c , d ∑ μ ab ( μ a c + μ b d + + μ a d + μ b c + ) μ c d = 4 1 h − 4 b , c ∑ π b c π b c + b , d ∑ π b d π b d = 2 1 h − 4 b ∑ π bb = 2 1 h − 4 N μ + . (110) Here, the projection operator π a b \pi_{ab} π ab (97) , from which it follows that its trace is N μ + N_{\mu}^{+} N μ + μ a b \mu_{ab} μ ab
There are two possibilities for the variance. Looking at the rhs of (110) :
If the pulsar term is present, meaning that η > 0 \eta>0 η > 0 (87) , then μ a b \mu_{ab} μ ab N μ + = N p u l N_{\mu}^{+}=N_{\mathrm{pul}} N μ + = N pul (110) agrees with that of (108) .
If the pulsar term is absent, corresponding to η = 0 \eta=0 η = 0 (87) , then μ a b = U a b \mu_{ab}=U_{ab} μ ab = U ab (93) shows that N μ + = N U + = min ( N p u l , N D O F ) N_{\mu}^{+}=N_{U}^{+}=\min(N_{\mathrm{pul}},N_{\mathrm{DOF}}) N μ + = N U + = min ( N pul , N DOF )
( σ h ^ 2 2 ) − 1 = 1 2 h − 4 min ( N p u l , N D O F ) . (111) \left(\sigma_{\hat{h}^{2}}^{2}\right)^{-1}=\frac{1}{2}\mathcal{h}^{-4}\min(N_{\mathrm{pul}},N_{\mathrm{DOF}}). \tag*{(111)} ( σ h ^ 2 2 ) − 1 = 2 1 h − 4 min ( N pul , N DOF ) . (111) Here, as the number of pulsars grows, the behavior depends upon the number of DOF to which the detector responds. If N D O F N_{\mathrm{DOF}} N DOF Figure 11 . It demonstrates that for the squared GW strain, “cosmic variance” is a property of the detector rather than of the Universe.
This provides a clear answer to the second question. Consider an array of one-arm LIGO-like (L = 2 L=2 L = 2 N D O F = 2 L + 1 = 5 N_{\mathrm{DOF}}=2L+1=5 N DOF = 2 L + 1 = 5 N p u l ≥ 5 N_{\mathrm{pul}}\geq5 N pul ≥ 5 (111) the variance is
σ h ^ 2 2 = 2 5 h 4 . (112) \sigma_{\hat{h}^{2}}^{2}=\frac{2}{5}h^{4}. \tag*{(112)} σ h ^ 2 2 = 5 2 h 4 . (112) But five one-arm LIGO-like detectors are enough to measure all five independent components of the GW strain h μ ν h_{\mu\nu} h μν h μ ν h_{\mu\nu} h μν 3 × 3 3 \times3 3 × 3 σ h ^ 2 / ⟨ h ^ 2 ⟩ \sigma_{\hat{h}^{2}}/\langle\hat{h}^{2}\rangle σ h ^ 2 / ⟨ h ^ 2 ⟩ σ s / ⟨ s ⟩ \sigma_s/\langle s\rangle σ s / ⟨ s ⟩ [11 ] where the cosmic variance of s ≡ s \equiv s ≡ ( h μ ν h μ ν ) (h_{\mu\nu}h^{\mu\nu}) ( h μν h μν ) s s s
We can also clearly see why a PTA detector provides vanishing variance as N p u l → ∞ N_{\mathrm{pul}} \to\infty N pul → ∞ l = 2 l=2 l = 2 1 + Ω ^ ⋅ p ^ 1+\hat{\Omega}\cdot\hat{p} 1 + Ω ^ ⋅ p ^ (80) . This factor of ( 1 + x ) − 1 = 1 − x + x 2 − x 3 + ⋯ (1+x)^{-1}=1-x+x^{2}-x^{3}+\cdots ( 1 + x ) − 1 = 1 − x + x 2 − x 3 + ⋯ x = Ω ^ ⋅ p ^ x=\hat{\Omega}\cdot\hat{p} x = Ω ^ ⋅ p ^ p ^ \hat{p} p ^ l > 2 l>2 l > 2 Y l m ( p ^ ) Y_{lm}(\hat{p}) Y l m ( p ^ ) (82) . Each independent mode provides additional information, and with enough pulsars, all of those modes can be accessed. Hence, by employing a large enough number of pulsars, (109) shows that the uncertainty in the squared GW strain can be reduced as much as desired.
While the calculations of this section and the previous section require careful attention to detail, the results themselves have an appealingly simple and intuitive physical interpretation. The optimal combination of auto- and cross-correlation terms extracts as much information as is possible from the GWs. Each additional pulsar added to the array provides another such degree of freedom, reducing the variance in proportion. In the absence of the pulsar term, this continues until the number of pulsars reaches the number of degrees of freedom probed by the detector. Then the variance saturates, because the detector cannot extract further information from the local measurements. On the other hand, if the pulsar term is present, then each additional pulsar brings access to a new set of degrees of freedom: the GW fluctuations in the space-time region surrounding that new pulsar. In this case, the variance continues to decrease without bound as new pulsars are added to the PTA.
Case (ii): Using only auto-correlations
In this section, we construct an optimal estimator h ^ 2 = ∑ a w a a ρ a a \hat{h}^{2}=\sum_a w_{aa}\rho_{aa} h ^ 2 = ∑ a w aa ρ aa h 2 h^{2} h 2 ρ a a \rho_{aa} ρ aa a = 1 , 2 , … , N p u l a=1,2,\ldots,N_{\mathrm{pul}} a = 1 , 2 , … , N pul h 2 h^{2} h 2 h 2 h^{2} h 2
We evaluate the inverse variance of the estimator by starting with the general expression (100) and including only the auto-correlation terms in the sums. Thus,
( σ h ^ 2 2 ) − 1 = ∑ a , b μ a a C a b − 1 μ b b , (113) \left(\sigma_{\hat{h}^{2}}^{2}\right)^{-1}=\sum_{a,b}\mu_{aa}\mathcal{C}^{-1}_{ab}\mu_{bb}, \tag*{(113)} ( σ h ^ 2 2 ) − 1 = a , b ∑ μ aa C ab − 1 μ bb , (113) where ∑ a , b \sum_{a,b} ∑ a , b (101) , and C a b − 1 ≡ C a a , b b − 1 \mathcal{C}^{-1}_{ab}\equiv\mathcal{C}^{-1}_{aa,bb} C ab − 1 ≡ C aa , bb − 1 N p u l × N p u l N_{\mathrm{pul}}\times N_{\mathrm{pul}} N pul × N pul [52 ]
C a b = C a a , b b = 2 h 4 μ a b 2 = 2 h 4 U a b 2 ( 1 + η 2 δ a b ) 2 = 2 h 4 ( U a b 2 + ( 2 η 2 + η 4 ) U 2 ( 0 ) δ a b ) . (114) \begin{aligned}
\mathcal{C}_{ab}&=\mathcal{C}_{aa,bb}\\
&=2\mathcal{h}^{4}\mu_{ab}^{2}\\
&=2\mathcal{h}^{4}U_{ab}^{2}(1+\eta^{2}\delta_{ab})^{2}\\
&=2\mathcal{h}^{4}\left(U_{ab}^{2}+(2\eta^{2}+\eta^{4})U^{2}(0)\delta_{ab}\right). \tag*{(114)}
\end{aligned} C ab = C aa , bb = 2 h 4 μ ab 2 = 2 h 4 U ab 2 ( 1 + η 2 δ ab ) 2 = 2 h 4 ( U ab 2 + ( 2 η 2 + η 4 ) U 2 ( 0 ) δ ab ) . (114) The second line follows from (11) , the third line follows from (87) , and the final line follows from (94) , because all of the diagonal elements of U a b U_{ab} U ab U ( 0 ) U(0) U ( 0 ) U a b 2 U_{ab}^{2} U ab 2 U a b U_{ab} U ab U U U
To evaluate the inverse variance, use (95) to write the diagonal elements of μ a b \mu_{ab} μ ab μ a a = ( 1 + η 2 ) U ( 0 ) \mu_{aa}=(1+\eta^{2})U(0) μ aa = ( 1 + η 2 ) U ( 0 ) (113) is
( σ h ^ 2 2 ) − 1 = ( 1 + η 2 ) 2 U 2 ( 0 ) ∑ a , b C a b − 1 , (115) \left(\sigma_{\hat{h}^{2}}^{2}\right)^{-1}=(1+\eta^{2})^{2}U^{2}(0)\sum_{a,b}\mathcal{C}^{-1}_{ab}, \tag*{(115)} ( σ h ^ 2 2 ) − 1 = ( 1 + η 2 ) 2 U 2 ( 0 ) a , b ∑ C ab − 1 , (115) which is proportional to the grand sum of C a b − 1 \mathcal{C}^{-1}_{ab} C ab − 1 1 \mathbf{1} 1 N p u l N_{\mathrm{pul}} N pul C a b \mathcal{C}_{ab} C ab (115) can be computed from its eigenvalue. We now employ the same argument, showing that if there are many pulsars uniformly distributed on the sky, then 1 \mathbf{1} 1 C a b \mathcal{C}_{ab} C ab C a b − 1 \mathcal{C}^{-1}_{ab} C ab − 1
It is easy to see that 1 \mathbf{1} 1 C a b \mathcal{C}_{ab} C ab U a b 2 U_{ab}^{2} U ab 2 (114) , since 1 \mathbf{1} 1 δ a b \delta_{ab} δ ab (88) shows that U a b 2 U_{ab}^{2} U ab 2 γ a b \gamma_{ab} γ ab p ^ a \hat{p}_{a} p ^ a p ^ b \hat{p}_{b} p ^ b a a a b b b p ^ b \hat{p}_{b} p ^ b p ^ a \hat{p}_{a} p ^ a p ^ a \hat{p}_{a} p ^ a
To compute the action of U a b 2 U_{ab}^{2} U ab 2 δ a b \delta_{ab} δ ab 1 / N p u l 1/N_{\mathrm{pul}} 1/ N pul (88) to write U a b 2 U_{ab}^{2} U ab 2 p ^ b \hat{p}_b p ^ b
1 N p u l ∑ b U a b 2 ≃ 1 4 π ∫ d p ^ b U a b 2 = 1 4 π ∫ d p ^ b ∑ l , l ′ Q l Q l ′ P l ( cos γ a b ) P l ′ ( cos γ a b ) = 1 2 ∫ − 1 1 d x ∑ l , l ′ Q l Q l ′ P l ( x ) P l ′ ( x ) = ∑ l , l ′ Q l Q l ′ δ l l ′ 2 l + 1 = ∑ l Q l 2 2 l + 1 . (116) \frac{1}{N_{\mathrm{pul}}}\sum_b U_{ab}^2 \simeq\frac{1}{4\pi}\int d\hat{p}_b\,U_{ab}^2
= \frac{1}{4\pi}\int d\hat{p}_b \sum_{l,l'} Q_l Q_{l'} P_l(\cos\gamma_{ab})P_{l'}(\cos\gamma_{ab})
= \frac{1}{2}\int_{-1}^{1}dx\sum_{l,l'}Q_lQ_{l'}P_l(x)P_{l'}(x)
= \sum_{l,l'}Q_lQ_{l'}\frac{\delta_{ll'}}{2l+1}
= \sum_l\frac{Q_l^2}{2l+1}. \tag*{(116)} N pul 1 b ∑ U ab 2 ≃ 4 π 1 ∫ d p ^ b U ab 2 = 4 π 1 ∫ d p ^ b l , l ′ ∑ Q l Q l ′ P l ( cos γ ab ) P l ′ ( cos γ ab ) = 2 1 ∫ − 1 1 d x l , l ′ ∑ Q l Q l ′ P l ( x ) P l ′ ( x ) = l , l ′ ∑ Q l Q l ′ 2 l + 1 δ l l ′ = l ∑ 2 l + 1 Q l 2 . (116) 1 N p u l ∑ b δ a b = 1 N p u l . (117) \frac{1}{N_{\mathrm{pul}}}\sum_b\delta_{ab}=\frac{1}{N_{\mathrm{pul}}}. \tag*{(117)} N pul 1 b ∑ δ ab = N pul 1 . (117) 1 N p u l ∑ b C a b ≃ 2 h 4 ( ∑ l Q l 2 2 l + 1 + 1 N p u l ( 2 η 2 + η 4 ) U 2 ( 0 ) ) ≡ λ N p u l . (118) \frac{1}{N_{\mathrm{pul}}}\sum_b C_{ab}\simeq2h^4\left(\sum_l\frac{Q_l^2}{2l+1}+\frac{1}{N_{\mathrm{pul}}}(2\eta^2+\eta^4)U^2(0)\right)\equiv\frac{\lambda}{N_{\mathrm{pul}}}. \tag*{(118)} N pul 1 b ∑ C ab ≃ 2 h 4 ( l ∑ 2 l + 1 Q l 2 + N pul 1 ( 2 η 2 + η 4 ) U 2 ( 0 ) ) ≡ N pul λ . (118) Thus, 1 \mathbf{1} 1 C ≡ C a b \mathcal{C}\equiv C_{ab} C ≡ C ab λ \lambda λ
Using this last result together with the results of Sec. III D, we can now write down an expression for the variance of the auto-correlation-only estimator of h 2 h^2 h 2 (115) , we immediately have
σ h ^ 2 2 = 1 ( 1 + η 2 ) 2 U 2 ( 0 ) 1 ∑ a , b C a b − 1 . (119) \sigma_{\hat{h}^2}^2=\frac{1}{(1+\eta^2)^2U^2(0)}\frac{1}{\sum_{a,b}C_{ab}^{-1}}. \tag*{(119)} σ h ^ 2 2 = ( 1 + η 2 ) 2 U 2 ( 0 ) 1 ∑ a , b C ab − 1 1 . (119) Then, using (38) , we can replace ( ∑ a , b C a b − 1 ) − 1 \left(\sum_{a,b}C_{ab}^{-1}\right)^{-1} ( ∑ a , b C ab − 1 ) − 1 λ / N p u l \lambda/N_{\mathrm{pul}} λ / N pul
σ h ^ 2 2 ≃ 2 h 4 ( 1 + η 2 ) 2 U 2 ( 0 ) ∑ l Q l 2 2 l + 1 + 2 h 4 ( 2 η 2 + η 4 ) N p u l ( 1 + η 2 ) 2 . (120) \sigma_{\hat{h}^2}^2\simeq\frac{2h^4}{(1+\eta^2)^2U^2(0)}\sum_l\frac{Q_l^2}{2l+1}+\frac{2h^4(2\eta^2+\eta^4)}{N_{\mathrm{pul}}(1+\eta^2)^2}. \tag*{(120)} σ h ^ 2 2 ≃ ( 1 + η 2 ) 2 U 2 ( 0 ) 2 h 4 l ∑ 2 l + 1 Q l 2 + N pul ( 1 + η 2 ) 2 2 h 4 ( 2 η 2 + η 4 ) . (120) The variance (120) is nonzero in the limit N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞ h 2 h^2 h 2
We now evaluate the variance (120) of the squared-strain estimator which employs only PTA auto-correlations, in the many-pulsar limit. Set Q l = ( 2 l + 1 ) C l Q_l=(2l+1)C_l Q l = ( 2 l + 1 ) C l C l C_l C l (60) . Then
1 U 2 ( 0 ) ∑ l Q l 2 2 l + 1 = 1 U 2 ( 0 ) ∑ l ( 2 l + 1 ) C l 2 = 1 2 h 4 σ cos 2 ( 0 ) U 2 ( 0 ) = 1 12 . (121) \frac{1}{U^2(0)}\sum_l\frac{Q_l^2}{2l+1}
=\frac{1}{U^2(0)}\sum_l(2l+1)C_l^2
=\frac{1}{2h^4}\frac{\sigma_{\cos}^2(0)}{U^2(0)}
=\frac{1}{12}. \tag*{(121)} U 2 ( 0 ) 1 l ∑ 2 l + 1 Q l 2 = U 2 ( 0 ) 1 l ∑ ( 2 l + 1 ) C l 2 = 2 h 4 1 U 2 ( 0 ) σ c o s 2 ( 0 ) = 12 1 . (121) In the first line, the approximation sign is a reminder that we have averaged over a large number of pulsars uniformly distributed on the sky. The second line follows from the definition of U a b U_{ab} U ab (88) . The third line follows by setting x = cos γ a b x=\cos\gamma_{ab} x = cos γ ab p ^ a \hat{p}_a p ^ a a a a (65) . Now, we return to (114) and consider the term proportional to δ a b \delta_{ab} δ ab 1 / N p u l 1/N_{\mathrm{pul}} 1/ N pul
So, combining (116) and (117) with (114) , it follows that
where we used (61) with γ = γ ′ = 0 \gamma=\gamma'=0 γ = γ ′ = 0 σ cos 2 ( 0 ) / U 2 ( 0 ) = 2 h 4 μ ~ 2 2 ( 0 ) / μ u 2 ( 0 ) = h 4 / 6 \sigma_{\cos}^2(0)/U^2(0)=2h^4\tilde{\mu}_2^2(0)/\mu_u^2(0)=h^4/6 σ c o s 2 ( 0 ) / U 2 ( 0 ) = 2 h 4 μ ~ 2 2 ( 0 ) / μ u 2 ( 0 ) = h 4 /6 (77) and (78) . Then, setting η = 1 \eta=1 η = 1 (120) to properly incorporate the pulsar term, and making use of (121) to evaluate the first term, we obtain
σ h ^ 2 2 ≃ h 4 24 ( 1 + 36 N p u l ) . (122) \sigma_{\hat{h}^2}^2\simeq\frac{h^4}{24}\left(1+\frac{36}{N_{\mathrm{pul}}}\right). \tag*{(122)} σ h ^ 2 2 ≃ 24 h 4 ( 1 + N pul 36 ) . (122) for the variance of the auto-correlation-only PTA estimator of h 2 h^2 h 2 σ h ^ 2 − 2 \sigma_{\hat{h}^2}^{-2} σ h ^ 2 − 2 Figure 12 . As N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞ h 4 / 24 h^4/24 h 4 /24
Figure 12. Inverse variance of strain estimators formed entirely from auto-correlations, as a function of the number of (uniformly distributed) pulsars, for different types of GW detectors, determined numerically. All have nonvanishing cosmic variance as N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞ 2 h 4 / σ h ^ 2 2 2h^4/\sigma_{\hat{h}^2}^2 2 h 4 / σ h ^ 2 2 (122) both asymptote to 48. The next two solid curves show the corresponding quantities for detectors having only a single l l l Q l = δ l L Q_l = \delta_{lL} Q l = δ l L L = 3 L = 3 L = 3 L = 2 L = 2 L = 2 4 × 7 = 28 4 \times7 = 28 4 × 7 = 28 4 × 5 = 20 4 \times5 = 20 4 × 5 = 20 η = 0 \eta= 0 η = 0 L = 2 L = 2 L = 2 L = 3 L = 3 L = 3 N p u l = ( L + 1 ) ( 2 L + 1 ) N_{\rm pul} = (L + 1)(2L + 1) N pul = ( L + 1 ) ( 2 L + 1 )
It is also instructive to evaluate the variance of this h 2 h^2 h 2 N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞ Q l = δ l 2 Q_l=\delta_{l2} Q l = δ l 2 U ( 0 ) = 1 U(0)=1 U ( 0 ) = 1 η = 0 \eta=0 η = 0 (120) is then
σ h ^ 2 2 ≃ 2 5 h 4 . (123) \sigma_{\hat{h}^2}^2\simeq\frac{2}{5}h^4. \tag*{(123)} σ h ^ 2 2 ≃ 5 2 h 4 . (123) which agrees exactly with our expectations from (112) for the estimator of h 2 h^2 h 2 l = 3 l=3 l = 3 Q l = δ l 3 Q_l=\delta_{l3} Q l = δ l 3 Figure 12 .
The reader will note from Figure 12 that if there is no pulsar term present, then the variance of the finite-mode detectors saturates if the number of pulsars is large enough. For the detector of mode L L L Q l = δ l L Q_l = \delta_{lL} Q l = δ l L N p u l = ( L + 1 ) ( 2 L + 1 ) N_{\rm pul} = (L + 1)(2L + 1) N pul = ( L + 1 ) ( 2 L + 1 ) L = 2 L = 2 L = 2 N p u l = 15 N_{\rm pul} = 15 N pul = 15
We can determine the number of detectors needed to saturate the variance if the detectors only respond to a finite number of DOF and the pulsar term is absent. It suffices to decompose the matrix whose entries are the squares of the entries of U a b U_{ab} U ab U a b 2 U_{ab}^2 U ab 2 (88) , with coefficients formed from quadratic combinations of the Q l Q_l Q l [53 , 54 ] . For Q l = δ l L Q_l = \delta_{lL} Q l = δ l L U a b 2 U_{ab}^2 U ab 2 ( L + 1 ) ( 2 L + 1 ) (L + 1)(2L + 1) ( L + 1 ) ( 2 L + 1 ) (91) . If modes l ≤ L max l \le L_{\max} l ≤ L m a x ( 2 L max + 1 ) 2 (2L_{\max} + 1)^2 ( 2 L m a x + 1 ) 2
Case (iii): Using only cross-correlations
This section examines the optimal estimator h ^ 2 \hat{h}^2 h ^ 2 ρ a b \rho_{ab} ρ ab (99) are restricted to distinct pulsars a < b a < b a < b
We computed the variance Eq. (100) of this estimator numerically, for N p u l N_{\rm pul} N pul Figure 13 . They show that σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞
σ h ^ 2 2 ≈ 1.61 N p u l + 75 N p u l h 4 . (124) \sigma_{\hat{h}^2}^2 \approx1.61 \frac{\sqrt{N_{\rm pul}+75}}{N_{\rm pul}} h^4 . \tag*{(124)} σ h ^ 2 2 ≈ 1.61 N pul N pul + 75 h 4 . (124) Thus, as the number of (randomly distributed) pulsars grows, the variance in the estimate of h ^ 2 \hat{h}^2 h ^ 2 ∝ N p u l − 1 / 2 \propto N_{\rm pul}^{-1/2} ∝ N pul − 1/2 N p u l N_{\rm pul} N pul h 2 h^2 h 2 ∝ 1 / N p u l \propto1/N_{\rm pul} ∝ 1/ N pul (109) and Figure 10 .
To confirm the N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞ Figure 13 , we compute the variance σ h ^ 2 2 \sigma_{\hat{h}^2}^2 σ h ^ 2 2 μ a b \mu_{ab} μ ab (100) is effectively averaging over nearby entries in the inverse covariance matrix C − 1 \mathcal{C}^{-1} C − 1
justified [55 ] in replacing C − 1 \mathsf{C}^{-1} C − 1 s j k − 1 s_{jk}^{-1} s j k − 1 (75) . We are also justified in replacing μ a b \mu_{ab} μ ab μ c d \mu_{cd} μ c d (59) . This is because auto-correlation terms such as δ a b μ u ( 0 ) \delta_{ab}\mu_u(0) δ ab μ u ( 0 ) (5) , vanish for distinct pulsar pairs a < b a < b a < b (59) and (75) (with N b i n s N_{\mathrm{bins}} N bins N p a i r s N_{\mathrm{pairs}} N pairs (100) , and converting the Riemann sums into integrals by taking the limit N p a i r s → ∞ N_{\mathrm{pairs}} \to\infty N pairs → ∞
( σ ^ h ^ 2 2 ) − 1 ≃ ∑ a b , c d μ a b C a b , c d − 1 μ c d = ∫ 0 π sin γ d γ ∫ 0 π sin γ ′ d γ ′ μ u ( γ ) s − 1 ( γ , γ ′ ) μ u ( γ ′ ) = ∫ − 1 1 d x ∫ − 1 1 d x ′ ∑ l , l ′ , l ′ ′ ≥ 2 L ( 2 l + 1 ) ( 2 l ′ + 1 ) ( 2 l ′ ′ + 1 ) C l C l ′ ′ 8 h ^ 4 C l ′ 2 P l ( x ) P l ′ ( x ) P l ′ ( x ′ ) P l ′ ′ ( x ′ ) . (125) \begin{aligned}
(\widehat{\sigma}_{\hat{h}^{2}}^{2})^{-1} &\simeq\sum_{ab,cd} \mu_{ab} C^{-1}_{ab,cd} \mu_{cd} \\
&= \int_{0}^{\pi} \sin\gamma\,d\gamma\int_{0}^{\pi} \sin\gamma'\,d\gamma'\,\mu_u(\gamma)s^{-1}(\gamma,\gamma')\mu_u(\gamma') \\
&= \int_{-1}^{1} dx \int_{-1}^{1} dx' \sum_{\substack{l,l',l''\ge2}}^{L} (2l+1)(2l'+1)(2l''+1)\frac{C_l C_{l''}}{8\widehat{h}^{4}C_{l'}^{2}}P_l(x)P_{l'}(x)P_{l'}(x')P_{l''}(x'). \tag*{(125)}
\end{aligned} ( σ h ^ 2 2 ) − 1 ≃ ab , c d ∑ μ ab C ab , c d − 1 μ c d = ∫ 0 π sin γ d γ ∫ 0 π sin γ ′ d γ ′ μ u ( γ ) s − 1 ( γ , γ ′ ) μ u ( γ ′ ) = ∫ − 1 1 d x ∫ − 1 1 d x ′ l , l ′ , l ′′ ≥ 2 ∑ L ( 2 l + 1 ) ( 2 l ′ + 1 ) ( 2 l ′′ + 1 ) 8 h 4 C l ′ 2 C l C l ′′ P l ( x ) P l ′ ( x ) P l ′ ( x ′ ) P l ′′ ( x ′ ) . (125) Here, we have changed variables to x ≡ cos γ x \equiv\cos\gamma x ≡ cos γ x ′ ≡ cos γ ′ x' \equiv\cos\gamma' x ′ ≡ cos γ ′
The sums over l l l l ′ l' l ′ l ′ ′ l'' l ′′ (125) arise from expressions (59) and (71) , which are harmonic sums over all modes l ≥ 2 l \ge2 l ≥ 2 σ h ^ 2 2 \sigma_{\hat{h}^{2}}^{2} σ h ^ 2 2 l , l ′ , l ′ ′ = L l,l',l'' = L l , l ′ , l ′′ = L L ≥ 2 L \ge2 L ≥ 2 L → ∞ L \to\infty L → ∞
The integrals in (125) may be evaluated using the orthogonality relation (65) for Legendre polynomials. This converts the integrals over x x x x ′ x' x ′ 2 δ l l ′ / ( 2 l + 1 ) 2\delta_{ll'}/(2l+1) 2 δ l l ′ / ( 2 l + 1 ) 2 δ l ′ l ′ ′ / ( 2 l ′ + 1 ) 2\delta_{l'l''}/(2l'+1) 2 δ l ′ l ′′ / ( 2 l ′ + 1 ) l ′ l' l ′ l ′ ′ l'' l ′′ (125) becomes
( σ ^ h ^ 2 2 ) − 1 ≃ ∑ l = 2 L 2 l + 1 2 h ^ 4 = 1 2 h ^ 4 ( ( L + 1 ) 2 − 4 ) . (126) (\widehat{\sigma}_{\hat{h}^{2}}^{2})^{-1} \simeq\sum_{l=2}^{L}\frac{2l+1}{2\widehat{h}^{4}}=\frac{1}{2\widehat{h}^{4}}\left((L+1)^2-4\right). \tag*{(126)} ( σ h ^ 2 2 ) − 1 ≃ l = 2 ∑ L 2 h 4 2 l + 1 = 2 h 4 1 ( ( L + 1 ) 2 − 4 ) . (126) As more harmonics are included, L L L (126) grow larger. Thus, for a PTA, σ h ^ 2 2 \sigma_{\hat{h}^{2}}^{2} σ h ^ 2 2 N p u l → ∞ N_{\mathrm{pul}} \to\infty N pul → ∞ Figure 13 suggests.
Equation (126) also applies to non-PTA detectors. To see this, note that C l C_l C l (125) and so does not affect the value of (126) , provided C l ≠ 0 C_l \ne0 C l = 0 C l C_l C l C l C_l C l C l C_l C l l l l (126) , whereas if C l C_l C l (126) holds for the non-PTA detectors of Sec. VI A, which are defined by correlation-matrix expansion coefficients Q l Q_l Q l 2 ≤ l ≤ L 2 \le l \le L 2 ≤ l ≤ L L = 2 L=2 L = 2
σ h ^ 2 2 ≃ 2 5 h ^ 4 . (127) \sigma_{\hat{h}^{2}}^{2} \simeq\frac{2}{5}\widehat{h}^{4}. \tag*{(127)} σ h ^ 2 2 ≃ 5 2 h 4 . (127) This is the same result that we have seen before for both the auto + cross and auto-only correlation estimators, see (112) and (123) . Plots of the inverse variance for the cross-correlation-only squared-strain estimator h ^ 2 \widehat{h}^{2} h 2 Figure 14 for the different detector types discussed above.
Figure 14. Plots of 2 h ^ 4 / σ h ^ 2 2 2\widehat{h}^{4}/\sigma_{\hat{h}^{2}}^{2} 2 h 4 / σ h ^ 2 2 h ^ 2 \widehat{h}^{2} h 2 2 ≤ l ≤ L 2 \le l \le L 2 ≤ l ≤ L L → ∞ L \to\infty L → ∞ 2 / 5 2/5 2/5
Uncertainty of the squared-strain estimators for the four PTAs
We now take the sky locations of the pulsars currently being monitored by the different PTAs, as given in
Appendix H
For these, we can predict the (best-case, noise-free) fractional uncertainties, σ h ^ 2 / ⟨ h ^ 2 ⟩ \sigma_{\hat{h}^2}/\langle\hat{h}^2\rangle σ h ^ 2 / ⟨ h ^ 2 ⟩ Table I . For comparison, the last row of the table gives the fractional uncertainty σ s / ⟨ s ⟩ \sigma_s/\langle s\rangle σ s / ⟨ s ⟩ (112) , where s = time average ( h μ ν h μ ν ) s=\text{time average}(h_{\mu\nu}h^{\mu\nu}) s = time average ( h μν h μν ) (112) , we assume that five or more of the pulsars are replaced by ideal one-arm LIGO-like detectors (Q l = δ l 2 Q_l=\delta_{l2} Q l = δ l 2 s s s
It is interesting to contrast the (inverses of the) fractional uncertainties in h ^ 2 \hat{h}^2 h ^ 2 presence of the signal, divided by the rms value of the statistic in the absence of the signal. It appears to be practically the same thing as the inverse of the fractional uncertainty, ⟨ h ^ 2 ⟩ / σ h ^ 2 \langle\hat{h}^2\rangle/\sigma_{\hat{h}^2} ⟨ h ^ 2 ⟩ / σ h ^ 2 σ h ^ 2 \sigma_{\hat{h}^2} σ h ^ 2 not the rms value of the “detection statistic” h ^ 2 \hat{h}^2 h ^ 2 h ^ 2 \hat{h}^2 h ^ 2 h ^ 2 \hat{h}^2 h ^ 2
Thus, we can employ h ^ 2 \hat{h}^2 h ^ 2 h ^ 2 \hat{h}^2 h ^ 2 assuming the presence of a GW signal . For the first of these applications, the variance of h ^ 2 \hat{h}^2 h ^ 2 h ^ 2 \hat{h}^2 h ^ 2 Table I .
There is a useful graphical way to think about the second application: estimating the squared GW strain from observational data. This parameter estimation problem is illustrated in Figure 15 , which shows the results of a single numerical simulation. The simulation generates one realization of a Gaussian ensemble with h 2 = 1 h^2=1 h 2 = 1 h ^ 2 \hat{h}^2 h ^ 2
Figure 15. A “one-sigma self-consistent” h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 ± σ h ^ 2 ( h 2 ) \pm\sigma_{\hat{h}^2}(h^2) ± σ h ^ 2 ( h 2 ) h 2 h^2 h 2 h 2 ± σ h ^ 2 ( h 2 ) h^2\pm\sigma_{\hat{h}^2}(h^2) h 2 ± σ h ^ 2 ( h 2 ) h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 = 1 h^2=1 h 2 = 1
realization gave h ^ 2 = 1.35 \hat{h}^{2}=1.35 h ^ 2 = 1.35
As discussed earlier, there is a complication: the width of the ± 1 σ \pm1\sigma ± 1 σ true value h 2 h^{2} h 2 h 2 h^{2} h 2 Figure 15 shows our proposed “self-consistent” approach. Since we do not know the actual value of h 2 h^{2} h 2 all possible values of h 2 h^{2} h 2
Examination of Figure 15 shows that only some values of h 2 h^{2} h 2 ± 1 σ \pm1\sigma ± 1 σ h 2 h^{2} h 2 h ^ 2 \hat{h}^{2} h ^ 2 h ^ 2 ∈ [ 1.6 , 4.4 ] \hat{h}^{2} \in[1.6, 4.4] h ^ 2 ∈ [ 1.6 , 4.4 ] ± 1 σ \pm1\sigma ± 1 σ h 2 h^{2} h 2
Testing the Correlation Model Against Data
The most important result of the present paper and of Ref. [11 ] is this: even with data from many noise-free pulsars, and even with signals coming from a perfectly Gaussian GW background, the pulsar-averaged Hellings and Downs correlation in our Universe will not follow the Hellings and Downs curve exactly. The following question then arises: after experiments have binned and (optimally) averaged the correlations, are the deviations away from the Hellings and Downs curve consistent with expectations? A simple way to quantify and answer this question is via χ 2 \chi^{2} χ 2
In the following subsections, we define two types of χ 2 \chi^{2} χ 2 χ 2 \chi^{2} χ 2 § 11 A with the “unprojected” statistic, which assumes that both the overall scale and spectral shape of the GW background are known exactly, a priori . This means that the true values of h 2 h^{2} h 2 h 4 h^{4} h 4 χ 2 \chi^{2} χ 2 χ 2 \chi^{2} χ 2
In § 11 B, we define a “projected” χ 2 \chi^{2} χ 2 h 2 h^{2} h 2 h 2 h^{2} h 2 h 2 h^{2} h 2 χ 2 \chi^{2} χ 2 χ 2 \chi^{2} χ 2
The projected χ 2 \chi^{2} χ 2 χ 2 \chi^{2} χ 2 h 2 h^{2} h 2 Figure 15 . It arises because χ 2 \chi^{2} χ 2 C \mathcal{C} C h ^ 4 \hat{h}^{4} h ^ 4 h 2 h^{2} h 2 χ 2 \chi^{2} χ 2 all possible values of h 2 h^{2} h 2 χ 2 ( h 2 ) \chi^{2}(h^{2}) χ 2 ( h 2 ) ⟨ χ 2 ⟩ ± σ χ 2 ( h 2 ) \langle\chi^{2}\rangle\pm\sigma_{\chi^{2}}(h^{2}) ⟨ χ 2 ⟩ ± σ χ 2 ( h 2 ) h 2 h^{2} h 2 h ^ 2 \hat{h}^{2} h ^ 2 § 11 C 4.
For the remainder of this preamble, we review the notation and some related issues. As described above, our starting point is a set of correlations which have been binned and optimally averaged. The notation reflects this: indices j , k , … j,k,\ldots j , k , … ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j j j j n p a i r s , j n_{\mathrm{pairs},j} n pairs , j γ a b \gamma_{ab} γ ab a b ab ab j j j γ b i n , j \gamma_{\mathrm{bin},j} γ bin , j μ b i n , j = μ u ( γ b i n , j ) \mu_{\mathrm{bin},j}=\mu_{\mathrm{u}}(\gamma_{\mathrm{bin},j}) μ bin , j = μ u ( γ bin , j )
Binning is arbitrary: how many bins should be used, and where should the bin boundaries be placed? Among the possibilities are two important extreme cases: (a) combine all pulsar pairs into a single bin, and (b) put every pulsar pair into its own separate bin. We will see that (b) provides the highest discriminating power and the best statistical test. However, the intermediate cases are also of interest, particularly to convince the scientific community regarding detection claims.
In reading the remainder of this section, it may be helpful to imagine that the choice of bins has been “fixed in advance” by some external agency, say a review board or detection committee. That choice might be one of the extremes (a) or (b) described above, or lie somewhere in between.
To parse the equations, it is also helpful to keep in mind that there are different vector spaces that enter. At the top level, we have a vector space whose dimension N b i n s N_{\mathrm{bins}} N bins j j j k k k l l l m m m n p a i r s , j n_{\mathrm{pairs},j} n pairs , j j j j a b ab ab c d cd c d … \ldots … g h gh g h a < b a < b a < b c < d c < d c < d … \ldots … g < h g < h g < h a = b a=b a = b c = d c=d c = d ρ a a \rho_{aa} ρ aa γ = 0 \gamma=0 γ = 0
In the remainder of this section, we construct χ 2 \chi^2 χ 2 S S S S = { a b ∣ a ≤ b } S=\{ab|a\leq b\} S = { ab ∣ a ≤ b } S = { a b ∣ a = b } S=\{ab|a=b\} S = { ab ∣ a = b } S = { a b ∣ a < b } S=\{ab|a<b\} S = { ab ∣ a < b } χ 2 \chi^2 χ 2
Here, we restrict our attention to noise-free correlation measurements. In Sec. IX, we discuss how the χ 2 \chi^2 χ 2
The unprojected χ 2 \chi^2 χ 2
Suppose that the squared GW strain h 2 h^2 h 2 χ 2 \chi^2 χ 2
χ 2 = ∑ j , k ( ρ o p t , j − ⟨ ρ o p t , j ⟩ ) B j k − 1 ( ρ o p t , k − ⟨ ρ o p t , k ⟩ ) , (128) \chi^2=\sum_{j,k}\left(\rho_{\mathrm{opt},j}-\langle\rho_{\mathrm{opt},j}\rangle\right)B^{-1}_{jk}\left(\rho_{\mathrm{opt},k}-\langle\rho_{\mathrm{opt},k}\rangle\right), \tag*{(128)} χ 2 = j , k ∑ ( ρ opt , j − ⟨ ρ opt , j ⟩ ) B j k − 1 ( ρ opt , k − ⟨ ρ opt , k ⟩ ) , (128) where the expectation value ⟨ ρ o p t , j ⟩ = h 2 μ b i n , j \langle\rho_{\mathrm{opt},j}\rangle=h^2\mu_{\mathrm{bin},j} ⟨ ρ opt , j ⟩ = h 2 μ bin , j B j k − 1 B^{-1}_{jk} B j k − 1
B j k B_{jk} B j k j j j k k k (29) and (30) :
B j k = μ b i n , j μ b i n , k ( μ j T C j j − 1 μ j ) ( μ k T C k k − 1 μ k ) μ j T C j j − 1 C j k C k k − 1 μ k . (129) B_{jk}=\frac{\mu_{\mathrm{bin},j}\mu_{\mathrm{bin},k}}{\left(\mu_j^{\mathsf T}C_{jj}^{-1}\mu_j\right)\left(\mu_k^{\mathsf T}C_{kk}^{-1}\mu_k\right)}\mu_j^{\mathsf T}C_{jj}^{-1}C_{jk}C_{kk}^{-1}\mu_k. \tag*{(129)} B j k = ( μ j T C j j − 1 μ j ) ( μ k T C k k − 1 μ k ) μ bin , j μ bin , k μ j T C j j − 1 C j k C k k − 1 μ k . (129) As we remarked above, this equation contains quantities with different (vector) dimensions. For example, the vector μ b i n , j \mu_{\mathrm{bin},j} μ bin , j j = 1 , … , N b i n s j=1,\ldots,N_{\mathrm{bins}} j = 1 , … , N bins j j j C j j C_{jj} C j j n p a i r s , j n_{\mathrm{pairs},j} n pairs , j a b ab ab j j j μ j \mu_j μ j C j k C_{jk} C j k j j j k k k C C C (15) .
In the remainder of this section, to simplify equations and calculations, and to gain geometric insight, we adopt standard notation for vectors and tensors in metric spaces. We exploit the fact that B j k B_{jk} B j k N b i n s × N b i n s N_{\mathrm{bins}}\times N_{\mathrm{bins}} N bins × N bins N b i n s N_{\mathrm{bins}} N bins
g j k ≡ B j k , (130) g_{jk}\equiv B_{jk}, \tag*{(130)} g j k ≡ B j k , (130) on the vector space indexed by angular separation bins. This notation will be familiar to many readers because it is widely used for general relativity (GR) [26 ] . However, in contrast with GR, where g j k g_{jk} g j k 4 × 4 4\times4 4 × 4 ( − , + , + , + ) (-,+,+,+) ( − , + , + , + ) N b i n s × N b i n s N_{\mathrm{bins}}\times N_{\mathrm{bins}} N bins × N bins ( + , + , … , + ) (+,+,\ldots,+) ( + , + , … , + ) g j k ≡ B j k g_{jk}\equiv B_{jk} g j k ≡ B j k
g j k ≡ B j k − 1 , (131) g^{jk}\equiv B^{-1}_{jk}, \tag*{(131)} g j k ≡ B j k − 1 , (131) are used to lower and raise indices, where B − 1 B^{-1} B − 1 [26 ] for equations. That is, if the same index label appears once in the covariant position and once in the contravariant position, then that index is summed over all angular bins.
To illustrate our conventions, we define the two contravariant vectors
μ b i n j ≡ g j k μ b i n , k = ∑ k B j k − 1 μ b i n , k and ρ o p t j ≡ g j k ρ o p t , k = ∑ k B j k − 1 ρ o p t , k . (132) \mu_{\mathrm{bin}}^j\equiv g^{jk}\mu_{\mathrm{bin},k}=\sum_k B^{-1}_{jk}\mu_{\mathrm{bin},k}
\qquad\text{and}\qquad
\rho_{\mathrm{opt}}^j\equiv g^{jk}\rho_{\mathrm{opt},k}=\sum_k B^{-1}_{jk}\rho_{\mathrm{opt},k}. \tag*{(132)} μ bin j ≡ g j k μ bin , k = k ∑ B j k − 1 μ bin , k and ρ opt j ≡ g j k ρ opt , k = k ∑ B j k − 1 ρ opt , k . (132) The χ 2 \chi^2 χ 2 (128) may then be written as
χ 2 = ( ρ o p t , j − h 2 μ b i n , j ) ( ρ o p t j − h 2 μ b i n j ) . (133) \chi^2=\left(\rho_{\mathrm{opt},j}-h^2\mu_{\mathrm{bin},j}\right)\left(\rho_{\mathrm{opt}}^j-h^2\mu_{\mathrm{bin}}^j\right). \tag*{(133)} χ 2 = ( ρ opt , j − h 2 μ bin , j ) ( ρ opt j − h 2 μ bin j ) . (133) Equivalently, if we define the fluctuations of the observed correlations away from the Hellings and Downs curve as
n j ≡ ρ o p t , j − ⟨ ρ o p t , j ⟩ = ρ o p t , j − h 2 μ b i n , j , (134) n_j\equiv\rho_{\mathrm{opt},j}-\langle\rho_{\mathrm{opt},j}\rangle=\rho_{\mathrm{opt},j}-h^2\mu_{\mathrm{bin},j}, \tag*{(134)} n j ≡ ρ opt , j − ⟨ ρ opt , j ⟩ = ρ opt , j − h 2 μ bin , j , (134) then the χ 2 \chi^2 χ 2 (128) and (133) is simply the squared length
χ 2 = n j n j . (135) \chi^2 = n_j n^j. \tag*{(135)} χ 2 = n j n j . (135) Note that the fluctuations have zero mean ⟨ n j ⟩ = 0 \langle n_j\rangle= 0 ⟨ n j ⟩ = 0
⟨ n j n k ⟩ = B j k = g j k . (136) \langle n_j n_k\rangle= B_{jk} = g_{jk}. \tag*{(136)} ⟨ n j n k ⟩ = B j k = g j k . (136) These follow directly from (134) and (29) .
The expected value of χ 2 \chi^2 χ 2 (135) and using (136) . We obtain
⟨ χ 2 ⟩ = g j k ⟨ n j n k ⟩ = g j k g j k = δ j j = N b i n s , (137) \langle\chi^2\rangle= g^{jk}\langle n_j n_k\rangle= g^{jk}g_{jk} = \delta^j_j = N_{\rm bins}, \tag*{(137)} ⟨ χ 2 ⟩ = g j k ⟨ n j n k ⟩ = g j k g j k = δ j j = N bins , (137) where N b i n s N_{\rm bins} N bins N b i n s N_{\rm bins} N bins
To further characterize χ 2 \chi^2 χ 2 ⟨ χ 2 ⟩ \langle\chi^2\rangle ⟨ χ 2 ⟩ χ 2 \chi^2 χ 2
σ χ 2 2 ≡ ⟨ ( χ 2 ) 2 ⟩ − ⟨ χ 2 ⟩ 2 = g j k g ℓ m ( ⟨ n j n k n ℓ n m ⟩ − ⟨ n j n k ⟩ ⟨ n ℓ n m ⟩ ) , (138) \sigma_{\chi^2}^2 \equiv\langle(\chi^2)^2\rangle-\langle\chi^2\rangle^2 = g^{jk}g^{\ell m}\left(\langle n_j n_k n_\ell n_m\rangle-\langle n_j n_k\rangle\langle n_\ell n_m\rangle\right), \tag*{(138)} σ χ 2 2 ≡ ⟨( χ 2 ) 2 ⟩ − ⟨ χ 2 ⟩ 2 = g j k g ℓ m ( ⟨ n j n k n ℓ n m ⟩ − ⟨ n j n k ⟩ ⟨ n ℓ n m ⟩ ) , (138) where the first equality defines the variance, and the second equality follows by substitution of (135) . The fourth-order expectation value that appears in (138) is deceptive. Since the GW background is described by a Gaussian ensemble, one might suppose that Isserlis’s theorem [19 ] could be applied to ⟨ n j n k n ℓ n m ⟩ \langle n_j n_k n_\ell n_m\rangle ⟨ n j n k n ℓ n m ⟩ n j n_j n j
Evaluating the fourth-order expectation value using the results of Appendix § 12.6 gives
⟨ n j n k n ℓ n m ⟩ = ⟨ n j n k ⟩ ⟨ n ℓ n m ⟩ + ⟨ n j n ℓ ⟩ ⟨ n k n m ⟩ + ⟨ n j n m ⟩ ⟨ n k n ℓ ⟩ + E j k ℓ m . (139) \langle n_j n_k n_\ell n_m\rangle= \langle n_j n_k\rangle\langle n_\ell n_m\rangle+ \langle n_j n_\ell\rangle\langle n_k n_m\rangle+ \langle n_j n_m\rangle\langle n_k n_\ell\rangle+ \mathbb{E}_{j k \ell m}. \tag*{(139)} ⟨ n j n k n ℓ n m ⟩ = ⟨ n j n k ⟩ ⟨ n ℓ n m ⟩ + ⟨ n j n ℓ ⟩ ⟨ n k n m ⟩ + ⟨ n j n m ⟩ ⟨ n k n ℓ ⟩ + E j k ℓ m . (139) The first three terms on the rhs are quadratic expectation values given by (136) . If the n j n_j n j
The final term in (139) arises entirely from the non-Gaussian behavior of n j n_j n j χ 2 \chi^2 χ 2
E j k ℓ m ≡ ∑ a b , c d , e f , g h w j , a b w k , c d w ℓ , e f w m , g h E a b , c d , e f , g h . (140) \mathbb{E}_{j k \ell m} \equiv\sum_{ab,cd,ef,gh} w_{j,ab}w_{k,cd}w_{\ell,ef}w_{m,gh}\mathbb{E}_{ab,cd,ef,gh}. \tag*{(140)} E j k ℓ m ≡ ab , c d , e f , g h ∑ w j , ab w k , c d w ℓ , e f w m , g h E ab , c d , e f , g h . (140) with E a b , c d , e f , g h \mathbb{E}_{ab,cd,ef,gh} E ab , c d , e f , g h w j , a b w_{j,ab} w j , ab (28) .
We now evaluate the variance of χ 2 \chi^2 χ 2 (138) , we use (139) and (136) to obtain
σ χ 2 2 = g j k g ℓ m ( g j ℓ g k m + g j m g k ℓ + E j k ℓ m ) = 2 N b i n s + E , (141) \sigma_{\chi^2}^2 = g^{jk}g^{\ell m}\left(g_{j\ell}g_{km}+g_{jm}g_{k\ell}+\mathbb{E}_{j k \ell m}\right)=2N_{\rm bins}+\mathbb{E}, \tag*{(141)} σ χ 2 2 = g j k g ℓ m ( g j ℓ g k m + g j m g k ℓ + E j k ℓ m ) = 2 N bins + E , (141) where we have defined
E ≡ g j k g ℓ m E j k ℓ m . (142) \mathbb{E}\equiv g^{jk}g^{\ell m}\mathbb{E}_{j k \ell m}. \tag*{(142)} E ≡ g j k g ℓ m E j k ℓ m . (142) If the n j n_j n j E \mathbb{E} E (141) , and χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 N b i n s N_{\rm bins} N bins N b i n s N_{\rm bins} N bins 2 N b i n s 2N_{\rm bins} 2 N bins E \mathbb{E} E χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 N b i n s N_{\rm bins} N bins
Note that all of these expressions simplify in the extreme case (b) described above, where every pulsar pair lies in its own bin. In that case, the weights w j , a b w_{j,ab} w j , ab [56 ] , and B j k ≡ g j k B_{jk}\equiv g_{jk} B j k ≡ g j k C a b , c d C_{ab,cd} C ab , c d
The “projected” χ 2 \chi^2 χ 2
We now define a “projected” χ 2 \chi^2 χ 2 h ^ 2 \hat h^2 h ^ 2 h 2 h^2 h 2 h 2 h^2 h 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 h 2 h^2 h 2 χ 2 \chi^2 χ 2
There is, however, a complication. The projected χ 2 \chi^2 χ 2 h 2 h^2 h 2 B j k − 1 B_{jk}^{-1} B j k − 1 C C C (129) ], which in turn depends on h ^ 4 \hat h^4 h ^ 4 h 2 h^2 h 2 χ 2 \chi^2 χ 2 h 2 h^2 h 2 χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 ) ± σ χ 2 ( h 2 ) \pm\sigma_{\chi^2}(h^2) ± σ χ 2 ( h 2 ) h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 § 11 C 4.
The projected χ 2 \chi^2 χ 2
χ 2 ( h 2 ) ≡ min h ~ 2 ≥ 0 { ∑ j , k ( ρ o p t , j − h ~ 2 μ b i n , j ) B j k − 1 ( h 2 ) ( ρ o p t , k − h ~ 2 μ b i n , k ) } . (143) \chi^2(h^2) \equiv\min_{\tilde{h}^2\geq0}\left\{\sum_{j,k}\left(\rho_{\mathrm{opt},j}-\tilde{h}^2\mu_{\mathrm{bin},j}\right)B^{-1}_{jk}(h^2)\left(\rho_{\mathrm{opt},k}-\tilde{h}^2\mu_{\mathrm{bin},k}\right)\right\}. \tag*{(143)} χ 2 ( h 2 ) ≡ h ~ 2 ≥ 0 min ⎩ ⎨ ⎧ j , k ∑ ( ρ opt , j − h ~ 2 μ bin , j ) B j k − 1 ( h 2 ) ( ρ opt , k − h ~ 2 μ bin , k ) ⎭ ⎬ ⎫ . (143) where we have explicitly indicated the h 2 h^2 h 2 B j k − 1 B^{-1}_{jk} B j k − 1 χ 2 \chi^2 χ 2 h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 h ~ 2 ≥ 0 \tilde{h}^2\geq0 h ~ 2 ≥ 0 (143) is minimized.
We can easily obtain an explicit expression for h ^ 2 \hat{h}^2 h ^ 2 ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j h ~ 2 \tilde{h}^2 h ~ 2 (143) , setting the derivative to zero, and solving the resulting linear equation to obtain h ~ 2 \tilde{h}^2 h ~ 2
h ^ 2 = 1 μ b i n 2 ∑ j , k μ b i n , j B j k − 1 ρ o p t , k , (144) \hat{h}^2=\frac{1}{\mu_{\mathrm{bin}}^2}\sum_{j,k}\mu_{\mathrm{bin},j}B^{-1}_{jk}\rho_{\mathrm{opt},k}, \tag*{(144)} h ^ 2 = μ bin 2 1 j , k ∑ μ bin , j B j k − 1 ρ opt , k , (144) where the positive scalar quantity
μ b i n 2 = ∑ j , k μ b i n , j B j k − 1 μ b i n , k (145) \mu_{\mathrm{bin}}^2=\sum_{j,k}\mu_{\mathrm{bin},j}B^{-1}_{jk}\mu_{\mathrm{bin},k} \tag*{(145)} μ bin 2 = j , k ∑ μ bin , j B j k − 1 μ bin , k (145) is the squared norm of the vector μ b i n , j \mu_{\mathrm{bin},j} μ bin , j (144) for h ^ 2 \hat{h}^2 h ^ 2 (98) which we used to find the best h 2 h^2 h 2 σ h ^ 2 2 = 1 / μ b i n 2 \sigma^2_{\hat{h}^2}=1/\mu_{\mathrm{bin}}^2 σ h ^ 2 2 = 1/ μ bin 2
Equation (144) provides a squared-strain estimate h ^ 2 \hat{h}^2 h ^ 2 (144) reduces to the “best” h 2 h^2 h 2 (99) . It is easy to see that the estimators formed from the binned averages cannot have a smaller variance than the best estimator. This is because the general form of the best estimator allows it to equal the binned estimator, simply by selecting appropriate weights.
An explicit expression for χ 2 \chi^2 χ 2 h ~ 2 \tilde{h}^2 h ~ 2 (143) to the value found in (144) . This minimizes the quantity in curly brackets in (143) , showing that the projected χ 2 \chi^2 χ 2
χ 2 ( h 2 ) = ∑ j , k ( ρ o p t , j − h ^ 2 μ b i n , j ) B j k − 1 ( h 2 ) ( ρ o p t , k − h ^ 2 μ b i n , k ) . (146) \chi^2(h^2)=\sum_{j,k}\left(\rho_{\mathrm{opt},j}-\hat{h}^2\mu_{\mathrm{bin},j}\right)B^{-1}_{jk}(h^2)\left(\rho_{\mathrm{opt},k}-\hat{h}^2\mu_{\mathrm{bin},k}\right). \tag*{(146)} χ 2 ( h 2 ) = j , k ∑ ( ρ opt , j − h ^ 2 μ bin , j ) B j k − 1 ( h 2 ) ( ρ opt , k − h ^ 2 μ bin , k ) . (146) In the absence of noise, ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 κ 4 \kappa^4 κ 4 (27) , (28) , and (144) , (145) ] cancel out. In the presence of instrumental or pulsar noise, there is no such cancellation. In this case, ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 (146) or in the following equations, it is not a problem. Our philosophy/approach for interpreting the χ 2 \chi^2 χ 2 χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 ) h 2 h^2 h 2
As before, χ 2 \chi^2 χ 2
χ 2 ( h 2 ) = ∑ j , k n ^ j B j k − 1 ( h 2 ) n ^ k , (147) \chi^2(h^2)=\sum_{j,k}\hat{n}_jB^{-1}_{jk}(h^2)\hat{n}_k, \tag*{(147)} χ 2 ( h 2 ) = j , k ∑ n ^ j B j k − 1 ( h 2 ) n ^ k , (147) but now this vector is the estimated fluctuations n ^ j \hat{n}_j n ^ j actual fluctuations n j n_j n j
n ^ j ≡ ρ o p t , j − h ^ 2 μ b i n , j = ∑ k P j k ρ o p t , k , (148) \hat{n}_j\equiv\rho_{\mathrm{opt},j}-\hat{h}^2\mu_{\mathrm{bin},j}=\sum_kP_{jk}\rho_{\mathrm{opt},k}, \tag*{(148)} n ^ j ≡ ρ opt , j − h ^ 2 μ bin , j = k ∑ P j k ρ opt , k , (148) where we have defined the projection operator
P j k ≡ δ j k − u j ∑ ℓ B k ℓ − 1 u ℓ for u ℓ ≡ μ b i n , ℓ μ b i n 2 , (149) P_{jk}\equiv\delta_{jk}-u_j\sum_{\ell}B^{-1}_{k\ell}u_\ell\quad\text{for }u_\ell\equiv\frac{\mu_{\mathrm{bin},\ell}}{\sqrt{\mu_{\mathrm{bin}}^2}}, \tag*{(149)} P j k ≡ δ j k − u j ℓ ∑ B k ℓ − 1 u ℓ for u ℓ ≡ μ bin 2 μ bin , ℓ , (149) with μ b i n 2 \mu_{\mathrm{bin}}^2 μ bin 2 (145) . The unit vector u j u_j u j μ b i n , j \mu_{\mathrm{bin},j} μ bin , j
∑ j , k u j B j k − 1 u k = 1. (150) \sum_{j,k}u_jB^{-1}_{jk}u_k=1. \tag*{(150)} j , k ∑ u j B j k − 1 u k = 1. (150) Thus, P j k P_{jk} P j k u k u_k u k n ^ j = ∑ k P j k n k \hat{n}_j=\sum_kP_{jk}n_k n ^ j = ∑ k P j k n k n j n_j n j h 2 μ b i n , j h^2\mu_{\mathrm{bin},j} h 2 μ bin , j χ 2 \chi^2 χ 2 (147) is the squared length (with respect to the metric B j k B_{jk} B j k n j n_j n j P k j P^j_k P k j h 2 h^2 h 2 B k l − 1 B^{-1}_{kl} B k l − 1 κ − 4 \kappa^{-4} κ − 4 u j u_j u j u ℓ u_\ell u ℓ h 2 \mathcal{h}^2 h 2 μ b i n 2 \sqrt{\mu_{\mathrm{bin}}^2} μ bin 2 (149) .
In contrast with the n j n_j n j (134) , the n ^ j \hat{n}_j n ^ j estimators of the fluctuations away from the mean. We will see that they have one fewer degree of freedom than the n j n_j n j n ^ j \hat{n}_j n ^ j n j n_j n j a priori knowledge of the true value of h 2 h^2 h 2 n ^ j \hat{n}_j n ^ j ⟨ n ^ j ⟩ = 0 \langle\hat{n}_j\rangle=0 ⟨ n ^ j ⟩ = 0
⟨ n ^ j n ^ k ⟩ = B j k − u j u k = ∑ ℓ P j ℓ B ℓ k . (151) \langle\hat{n}_j\hat{n}_k\rangle=B_{jk}-u_ju_k=\sum_\ell P_{j\ell}B_{\ell k}. \tag*{(151)} ⟨ n ^ j n ^ k ⟩ = B j k − u j u k = ℓ ∑ P j ℓ B ℓ k . (151) The quantity B j k − u j u k B_{jk}-u_ju_k B j k − u j u k ( N b i n s − 1 ) (N_{\mathrm{bins}}-1) ( N bins − 1 ) u j u_j u j
We can calculate the ensemble mean and variance of χ 2 \chi^2 χ 2 § 11 A. From (147) and (151) , the expected value of χ 2 \chi^2 χ 2
⟨ χ 2 ⟩ = ∑ j , k B j k − 1 ⟨ n ^ j n ^ k ⟩ = ∑ j , k B j k − 1 ( B j k − u j u k ) = N b i n s − 1. (152) \langle\chi^2\rangle=\sum_{j,k}B^{-1}_{jk}\langle\hat{n}_j\hat{n}_k\rangle=\sum_{j,k}B^{-1}_{jk}(B_{jk}-u_ju_k)=N_{\mathrm{bins}}-1. \tag*{(152)} ⟨ χ 2 ⟩ = j , k ∑ B j k − 1 ⟨ n ^ j n ^ k ⟩ = j , k ∑ B j k − 1 ( B j k − u j u k ) = N bins − 1. (152) From (147) , the variance in χ 2 \chi^2 χ 2
σ χ 2 2 = ∑ j , k , ℓ , m B j k − 1 B ℓ m − 1 ( ⟨ n ^ j n ^ k n ^ ℓ n ^ m ⟩ − ⟨ n ^ j n ^ k ⟩ ⟨ n ^ ℓ n ^ m ⟩ ) . (153) \sigma_{\chi^2}^2=\sum_{j,k,\ell,m}B^{-1}_{jk}B^{-1}_{\ell m}\left(\langle\hat{n}_j\hat{n}_k\hat{n}_\ell\hat{n}_m\rangle-\langle\hat{n}_j\hat{n}_k\rangle\langle\hat{n}_\ell\hat{n}_m\rangle\right). \tag*{(153)} σ χ 2 2 = j , k , ℓ , m ∑ B j k − 1 B ℓ m − 1 ( ⟨ n ^ j n ^ k n ^ ℓ n ^ m ⟩ − ⟨ n ^ j n ^ k ⟩ ⟨ n ^ ℓ n ^ m ⟩ ) . (153) To evaluate the fourth-order expectation value of the n ^ j \hat{n}_j n ^ j n ^ j = ∑ k P j k ρ o p t , k \hat{n}_j=\sum_kP_{jk}\rho_{\mathrm{opt},k} n ^ j = ∑ k P j k ρ opt , k (148) and the results of Appendix § 12.6 to evaluate the fourth-order expectation value of the ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j
⟨ n ^ j n ^ k n ^ ℓ n ^ m ⟩ = ⟨ n ^ j n ^ k ⟩ ⟨ n ^ ℓ n ^ m ⟩ + ⟨ n ^ j n ^ ℓ ⟩ ⟨ n ^ k n ^ m ⟩ + ⟨ n ^ j n ^ m ⟩ ⟨ n ^ k n ^ ℓ ⟩ + E ^ j k l m . (154) \langle\hat{n}_j\hat{n}_k\hat{n}_\ell\hat{n}_m\rangle=\langle\hat{n}_j\hat{n}_k\rangle\langle\hat{n}_\ell\hat{n}_m\rangle+\langle\hat{n}_j\hat{n}_\ell\rangle\langle\hat{n}_k\hat{n}_m\rangle+\langle\hat{n}_j\hat{n}_m\rangle\langle\hat{n}_k\hat{n}_\ell\rangle+\hat{E}_{jklm}. \tag*{(154)} ⟨ n ^ j n ^ k n ^ ℓ n ^ m ⟩ = ⟨ n ^ j n ^ k ⟩ ⟨ n ^ ℓ n ^ m ⟩ + ⟨ n ^ j n ^ ℓ ⟩ ⟨ n ^ k n ^ m ⟩ + ⟨ n ^ j n ^ m ⟩ ⟨ n ^ k n ^ ℓ ⟩ + E ^ j k l m . (154) where the quadratic expectation values are given by (151) , and
E ^ j k l m ≡ ∑ j ′ , k ′ , ℓ ′ , m ′ P j j ′ P k k ′ P ℓ ℓ ′ P m m ′ E j ′ k ′ ℓ ′ m ′ . (155) \hat{E}_{jklm}\equiv\sum_{j',k',\ell',m'}P_{jj'}P_{kk'}P_{\ell\ell'}P_{mm'}E_{j'k'\ell'm'}. \tag*{(155)} E ^ j k l m ≡ j ′ , k ′ , ℓ ′ , m ′ ∑ P j j ′ P k k ′ P ℓ ℓ ′ P m m ′ E j ′ k ′ ℓ ′ m ′ . (155) Starting with (153) and making use of (154) and (152) , we obtain
σ χ 2 2 = 2 ( N b i n s − 1 ) + E ^ , (156) \sigma_{\chi^2}^2=2(N_{\mathrm{bins}}-1)+\hat{E}, \tag*{(156)} σ χ 2 2 = 2 ( N bins − 1 ) + E ^ , (156) where
E ^ ≡ ∑ j , k , ℓ , m B j k − 1 B ℓ m − 1 E ^ j k l m (157) \hat{E}\equiv\sum_{j,k,\ell,m}B^{-1}_{jk}B^{-1}_{\ell m}\hat{E}_{jklm} \tag*{(157)} E ^ ≡ j , k , ℓ , m ∑ B j k − 1 B ℓ m − 1 E ^ j k l m (157) is the trace of the projected non-Gaussian term. Note that for the noise-free case, the expected value and variance of the projected χ 2 \chi^2 χ 2 h 2 h^2 h 2 E ^ \hat{E} E ^ (157) depends only on the ratio h 8 / h 8 \mathcal{h}^8/\mathcal{h}^8 h 8 / h 8 h 2 h^2 h 2
The mean and variance of the projected χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 § 11 A. We will see that their values and distributions are also very similar. If we compare (152) and (156) to (137) and (141) , we see that N b i n s N_{\mathrm{bins}} N bins N b i n s − 1 N_{\mathrm{bins}}-1 N bins − 1 E j k l m E_{jklm} E j k l m E ^ j k l m \hat{E}_{jklm} E ^ j k l m (155) . This reduction by one degree of freedom takes place whenever one estimates the variance of a data set using the sample mean (here h 2 h^2 h 2 h 2 h^2 h 2 1 / ( N − 1 ) 1/(N-1) 1/ ( N − 1 ) 1 / N 1/N 1/ N
Variance σ χ 2 2 \sigma_{\chi^2}^2 σ χ 2 2 χ 2 \chi^2 χ 2
Here, we investigate the variance of the unprojected and projected χ 2 \chi^2 χ 2 § 11 A and VIII B. The first has variance given by (141)
σ χ 2 2 = 2 N b i n s + E , (158) \sigma_{\chi^2}^2=2N_{\mathrm{bins}}+E, \tag*{(158)} σ χ 2 2 = 2 N bins + E , (158) where E E E (142) and N b i n s N_{\mathrm{bins}} N bins χ 2 \chi^2 χ 2 (156) . It may be obtained from (158) and (142) by replacing N b i n s N_{\mathrm{bins}} N bins N b i n s − 1 N_{\mathrm{bins}}-1 N bins − 1 E E E E ^ \hat{E} E ^ (157) .
Recall that the E E E § 12.6 ) arises because pulsar-pulsar correlations are non-Gaussian. If the pulsar-pulsar correlations were Gaussian, then E E E (158) by the square of (137) ] the fractional variation would be
E = 0 ⇒ σ χ 2 2 ⟨ χ 2 ⟩ 2 = 2 N b i n s N b i n s 2 = 2 N b i n s . (159) E=0\quad\Rightarrow\quad\frac{\sigma_{\chi^2}^2}{\langle\chi^2\rangle^2}=\frac{2N_{\mathrm{bins}}}{N_{\mathrm{bins}}^2}=\frac{2}{N_{\mathrm{bins}}}. \tag*{(159)} E = 0 ⇒ ⟨ χ 2 ⟩ 2 σ χ 2 2 = N bins 2 2 N bins = N bins 2 . (159) This behavior, if E E E χ 2 \chi^2 χ 2 N b i n s N_{\mathrm{bins}} N bins χ 2 \chi^2 χ 2 N b i n s − 1 / 2 N_{\mathrm{bins}}^{-1/2} N bins − 1/2 E ^ \hat{E} E ^ N b i n s N_{\mathrm{bins}} N bins N b i n s − 1 N_{\mathrm{bins}}-1 N bins − 1
However, we will see that when there are more than a few pulsars, the E \mathbb{E} E E ^ \hat{\mathbb{E}} E ^ dominant contributors to the variance. Hence, the fractional uncertainties of the χ 2 \chi^2 χ 2 ρ a b \rho_{ab} ρ ab different scaling behavior than for the Gaussian case of (159) . The minimum number of pulsars required for E \mathbb{E} E E ^ \hat{\mathbb{E}} E ^ h ^ / h \hat{h}/h h ^ / h (158) , since E \mathbb{E} E E ^ \hat{\mathbb{E}} E ^ h ^ 8 / h 8 \hat{h}^{8}/h^{8} h ^ 8 / h 8 h ^ 8 \hat{h}^{8} h ^ 8 1 / h 8 1/h^{8} 1/ h 8 h ^ / h \hat{h}/h h ^ / h [57 ] . Hence, assuming coefficients of order unity leads to a correct estimate for this minimum number of pulsars. When the number of pulsars exceeds this minimum, then the first term on the rhs of (158) may be neglected.
For the remainder of this subsection, we focus on the most stringent statistical tests: each pulsar pair is placed into its own individual angular bin, so that N b i n s N_{\rm bins} N bins
Case (i): N b i n s = N p u l ( N p u l + 1 ) / 2 N_{\rm bins}=N_{\rm pul}(N_{\rm pul}+1)/2 N bins = N pul ( N pul + 1 ) /2
Case (ii): N b i n s = N p u l N_{\rm bins}=N_{\rm pul} N bins = N pul
Case (iii): N b i n s = N p u l ( N p u l − 1 ) / 2 N_{\rm bins}=N_{\rm pul}(N_{\rm pul}-1)/2 N bins = N pul ( N pul − 1 ) /2
The unprojected (128) and projected χ 2 \chi^2 χ 2 (146) may then be written
χ 2 = ∑ a b ∈ S ∑ c d ∈ S ( ρ a b − h 2 μ a b ) C a b , c d − 1 ( ρ c d − h 2 μ c d ) , (160) \chi^2=\sum_{ab\in S}\sum_{cd\in S}(\rho_{ab}-h^2\mu_{ab})C^{-1}_{ab,cd}(\rho_{cd}-h^2\mu_{cd}), \tag*{(160)} χ 2 = ab ∈ S ∑ c d ∈ S ∑ ( ρ ab − h 2 μ ab ) C ab , c d − 1 ( ρ c d − h 2 μ c d ) , (160) χ 2 ( h 2 ) = ∑ a b ∈ S ∑ c d ∈ S ( ρ a b − h ^ 2 μ a b , c d ( h 2 ) ) C a b , c d − 1 ( h 2 ) ( ρ c d − h ^ 2 μ c d ) (161) \chi^2(h^2)=\sum_{ab\in S}\sum_{cd\in S}\left(\rho_{ab}-\hat{h}^{2}\mu_{ab,cd}(h^2)\right)C^{-1}_{ab,cd}(h^2)\left(\rho_{cd}-\hat{h}^{2}\mu_{cd}\right) \tag*{(161)} χ 2 ( h 2 ) = ab ∈ S ∑ c d ∈ S ∑ ( ρ ab − h ^ 2 μ ab , c d ( h 2 ) ) C ab , c d − 1 ( h 2 ) ( ρ c d − h ^ 2 μ c d ) (161) respectively, where S S S
The second equation explicitly shows the h 2 h^2 h 2 C − 1 C^{-1} C − 1 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 (160) assumes that the true value of h 2 h^2 h 2 a priori , independent of the observed data: χ 2 \chi^2 χ 2 h 2 h^2 h 2 (161) depends upon h 2 h^2 h 2 h 2 h^2 h 2 C − 1 C^{-1} C − 1 χ 2 \chi^2 χ 2 h 2 h^2 h 2 h ^ 2 \hat{h}^{2} h ^ 2 h 2 h^2 h 2 S S S (161) also depends, in general, upon h 2 h^2 h 2 χ 2 \chi^2 χ 2 χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 ) h 2 h^2 h 2
As mentioned earlier, the χ 2 \chi^2 χ 2 σ χ 2 2 \sigma_{\chi^2}^{2} σ χ 2 2
We have defined six different χ 2 \chi^2 χ 2 N p u l N_{\rm pul} N pul σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ Figure 16 and the
Figure 16. Left panel: non-Gaussian contributions to the variance of the χ 2 \chi^2 χ 2 E \mathbb{E} E E − E ^ \mathbb{E}-\hat{\mathbb{E}} E − E ^ h ^ / h = 1 \hat{h}/h=1 h ^ / h = 1 E \mathbb{E} E σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ χ 2 \chi^2 χ 2 [57 ] . Again, the curves for auto + cross and cross-correlations-only are indistinguishable, as are those for the unprojected and projected χ 2 \chi^2 χ 2
rows labeled by (1) and (3) in Table II . The two auto-correlation-only χ 2 \chi^2 χ 2 N p u l N_{\rm pul} N pul
To determine the variance of the χ 2 \chi^2 χ 2 E \mathcal{E} E E ^ \widehat{\mathcal{E}} E (142) and (157) , of the cumulant tensor E a b ; c d ; e f ; g h \mathcal{E}_{ab;cd;ef;gh} E ab ; c d ; e f ; g h (132) , the metric used to compute the trace (142) is the inverse C a b , c d − 1 \mathsf{C}^{-1}_{ab,cd} C ab , c d − 1 C a b , c d \mathsf{C}_{ab,cd} C ab , c d (11) . Thus,
E ≡ ∑ a b ∈ S ∑ c d ∈ S ∑ e f ∈ S ∑ g h ∈ S C a b , c d − 1 C e f , g h − 1 E a b ; c d ; e f ; g h . (162) \mathcal{E} \equiv\sum_{ab\in\mathcal{S}}\sum_{cd\in\mathcal{S}}\sum_{ef\in\mathcal{S}}\sum_{gh\in\mathcal{S}} \mathsf{C}^{-1}_{ab,cd}\mathsf{C}^{-1}_{ef,gh}\mathcal{E}_{ab;cd;ef;gh}. \tag*{(162)} E ≡ ab ∈ S ∑ c d ∈ S ∑ e f ∈ S ∑ g h ∈ S ∑ C ab , c d − 1 C e f , g h − 1 E ab ; c d ; e f ; g h . (162) A similar expression holds for E ^ \widehat{\mathcal{E}} E E a b ; c d ; e f ; g h \mathcal{E}_{ab;cd;ef;gh} E ab ; c d ; e f ; g h
E ^ a b ; c d ; e f ; g h = ∑ o p ∈ S ∑ q r ∈ S ∑ s t ∈ S ∑ u v ∈ S P a b , o p P c d , q r P e f , s t P g h , u v E o p ; q r ; s t ; u v . (163) \widehat{\mathcal{E}}_{ab;cd;ef;gh}=\sum_{op\in\mathcal{S}}\sum_{qr\in\mathcal{S}}\sum_{st\in\mathcal{S}}\sum_{uv\in\mathcal{S}}P_{ab,op}P_{cd,qr}P_{ef,st}P_{gh,uv}\mathcal{E}_{op;qr;st;uv}. \tag*{(163)} E ab ; c d ; e f ; g h = o p ∈ S ∑ q r ∈ S ∑ s t ∈ S ∑ uv ∈ S ∑ P ab , o p P c d , q r P e f , s t P g h , uv E o p ; q r ; s t ; uv . (163) The projection operator (149) is given by
P a b , c d = δ a c δ b d − μ b i n − 2 μ a b ∑ e f ∈ S C c d , e f − 1 μ e f , (164) P_{ab,cd}=\delta_{ac}\delta_{bd}-\mu_{\rm bin}^{-2}\mu_{ab}\sum_{ef\in\mathcal{S}}\mathsf{C}^{-1}_{cd,ef}\mu_{ef}, \tag*{(164)} P ab , c d = δ a c δ b d − μ bin − 2 μ ab e f ∈ S ∑ C c d , e f − 1 μ e f , (164) where
μ b i n 2 ≡ ∑ a b ∈ S ∑ c d ∈ S μ a b C a b , c d − 1 μ c d . (165) \mu_{\rm bin}^{2}\equiv\sum_{ab\in\mathcal{S}}\sum_{cd\in\mathcal{S}}\mu_{ab}\mathsf{C}^{-1}_{ab,cd}\mu_{cd}. \tag*{(165)} μ bin 2 ≡ ab ∈ S ∑ c d ∈ S ∑ μ ab C ab , c d − 1 μ c d . (165) Equation (164) for μ b i n 2 \mu_{\rm bin}^{2} μ bin 2 (145) for the case where each pulsar pair occupies its own bin.
For the calculations that follow, we apply the projection operators in (163) to the inverse matrices C a b , c d − 1 \mathsf{C}^{-1}_{ab,cd} C ab , c d − 1 C e f , g h − 1 \mathsf{C}^{-1}_{ef,gh} C e f , g h − 1 E o p ; q r ; s t ; u v \mathcal{E}_{op;qr;st;uv} E o p ; q r ; s t ; uv
E ^ = ∑ a b ∈ S ∑ c d ∈ S ∑ e f ∈ S ∑ g h ∈ S C ^ a b , c d − 1 C ^ e f , g h − 1 E a b ; c d ; e f ; g h , (166) \widehat{\mathcal{E}}=\sum_{ab\in\mathcal{S}}\sum_{cd\in\mathcal{S}}\sum_{ef\in\mathcal{S}}\sum_{gh\in\mathcal{S}}\widehat{\mathsf{C}}^{-1}_{ab,cd}\widehat{\mathsf{C}}^{-1}_{ef,gh}\mathcal{E}_{ab;cd;ef;gh}, \tag*{(166)} E = ab ∈ S ∑ c d ∈ S ∑ e f ∈ S ∑ g h ∈ S ∑ C ab , c d − 1 C e f , g h − 1 E ab ; c d ; e f ; g h , (166) where
C ^ a b , c d − 1 ≡ ∑ o p ∈ S ∑ q r ∈ S C o p , q r − 1 P o p , a b P q r , c d (167) \widehat{\mathsf{C}}^{-1}_{ab,cd}\equiv\sum_{op\in\mathcal{S}}\sum_{qr\in\mathcal{S}}\mathsf{C}^{-1}_{op,qr}P_{op,ab}P_{qr,cd} \tag*{(167)} C ab , c d − 1 ≡ o p ∈ S ∑ q r ∈ S ∑ C o p , q r − 1 P o p , ab P q r , c d (167) is the projected version of C a b , c d − 1 \mathsf{C}^{-1}_{ab,cd} C ab , c d − 1 (166) to evaluate E ^ \widehat{\mathcal{E}} E
Numerical evaluation of the sums in (162) and (166) is straightforward, and takes a few minutes of CPU time for a few hundred pulsars. Two different sets of indices/subscripts are used. Pulsar indices correspond to the pulsars themselves; pair indices are obtained by constructing an index set corresponding to all pulsar pairs a b ∈ S ab\in\mathcal{S} ab ∈ S μ a b \mu_{ab} μ ab (5) . Next C a b , c d \mathsf{C}_{ab,cd} C ab , c d (11) for all pairs in the index set, and inverted. (It can have small eigenvalues, so some care may be needed.) The corresponding four-pulsar-index tensors are then formed, using the pair index set mapping. For E ^ \widehat{\mathcal{E}} E P a b , c d P_{ab,cd} P ab , c d C a b , c d − 1 \mathsf{C}^{-1}_{ab,cd} C ab , c d − 1 μ a b \mu_{ab} μ ab (164) . This is contracted with C a b , c d − 1 \mathsf{C}^{-1}_{ab,cd} C ab , c d − 1 C ^ a b , c d − 1 \widehat{\mathsf{C}}^{-1}_{ab,cd} C ab , c d − 1 (167) . Next, E a b ; c d ; e f ; g h \mathcal{E}_{ab;cd;ef;gh} E ab ; c d ; e f ; g h μ a b \mu_{ab} μ ab (162) and (166) are carried out using pulsar indices to obtain either E \mathcal{E} E E ^ \widehat{\mathcal{E}} E
Plots of E \mathcal{E} E E − E ^ \mathcal{E}-\widehat{\mathcal{E}} E − E N p u l N_{\rm pul} N pul Figure 16 . For case (i) we used algebraic expressions. For cases (ii) and (iii) we numerically evaluated (162) and (166) as described above, extrapolating the numerical values to large numbers of pulsars using polynomial fits to the numerical values. The following subsections give details of the calculations for these three different cases.
Case (i): Using both auto- and cross-correlations
If the χ 2 \chi^2 χ 2 E \mathcal{E} E E ^ \widehat{\mathcal{E}} E a b ∈ S = { a b ∣ a ≤ b } ab\in\mathcal{S}=\{ab\mid a\leq b\} ab ∈ S = { ab ∣ a ≤ b }
∑ a b ∈ S ⟶ ∑ a = 1 N p u l ∑ b = 1 N p u l ≡ ∑ a , b . (168) \sum_{ab\in\mathcal{S}}\longrightarrow\sum_{a=1}^{N_{\rm pul}}\sum_{b=1}^{N_{\rm pul}}\equiv\sum_{a,b}. \tag*{(168)} ab ∈ S ∑ ⟶ a = 1 ∑ N pul b = 1 ∑ N pul ≡ a , b ∑ . (168) In Sec. VII A, we discuss the consequences of replacing the set S \mathcal{S} S N p u l ( N p u l + 1 ) / 2 N_{\rm pul}(N_{\rm pul}+1)/2 N pul ( N pul + 1 ) /2 N p u l 2 N_{\rm pul}^{2} N pul 2
The advantage of this larger indexing set is that we can explicitly invert the covariance matrix C a b , c d \mathsf{C}_{ab,cd} C ab , c d μ a b \mu_{ab} μ ab N p u l × N p u l N_{\rm pul}\times N_{\rm pul} N pul × N pul (102) that
C a b , c d − 1 = 1 4 K − 4 ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) . (169) \mathsf{C}^{-1}_{ab,cd}=\frac{1}{4}\mathcal{K}^{-4}\left(\mu^{-1}_{ac}\mu^{-1}_{bd}+\mu^{-1}_{ad}\mu^{-1}_{bc}\right). \tag*{(169)} C ab , c d − 1 = 4 1 K − 4 ( μ a c − 1 μ b d − 1 + μ a d − 1 μ b c − 1 ) . (169) noting that this inverse is only defined on the space of symmetric N p u l × N p u l N_{\rm pul} \times N_{\rm pul} N pul × N pul χ 2 \chi^2 χ 2
We first evaluate the projection operator. Substituting (169) into (165) gives the squared norm
μ b i n 2 ≡ ∑ a , b , c , d μ a b C a b , c d − 1 μ c d = 1 2 ℏ − 4 N p u l , (170) \mu_{\rm bin}^2 \equiv\sum_{a,b,c,d} \mu_{ab}\mathcal{C}^{-1}_{ab,cd}\mu_{cd} = \frac{1}{2}\hbar^{-4}N_{\rm pul}, \tag*{(170)} μ bin 2 ≡ a , b , c , d ∑ μ ab C ab , c d − 1 μ c d = 2 1 ℏ − 4 N pul , (170) where the calculation is identical to (108) . From (164) , this implies that the projection operator is
P a b , c d = δ a c δ b d − 1 N p u l μ a b μ c d − 1 . (171) P_{ab,cd} = \delta_{ac}\delta_{bd} - \frac{1}{N_{\rm pul}}\mu_{ab}\mu^{-1}_{cd}. \tag*{(171)} P ab , c d = δ a c δ b d − N pul 1 μ ab μ c d − 1 . (171) Note that the second term is symmetric in a b ab ab c d cd c d P a b , c d P_{ab,cd} P ab , c d (167) to project C a b , c d − 1 \mathcal{C}^{-1}_{ab,cd} C ab , c d − 1 (169) is explicitly symmetric in both index pairs. However, should one wish to extend the action of P a b , c d P_{ab,cd} P ab , c d δ a c δ b d \delta_{ac}\delta_{bd} δ a c δ b d (171) with the symmetric projector ( δ a c δ b d + δ a d δ b c ) / 2 (\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})/2 ( δ a c δ b d + δ a d δ b c ) /2
As a final preparatory step, we use (171) to evaluate the projection operation in (167) . This gives
C ^ a b , c d − 1 = C a b , c d − 1 − 1 2 N p u l ℏ − 4 μ a b − 1 μ c d − 1 (172) \widehat{\mathcal{C}}^{-1}_{ab,cd} = \mathcal{C}^{-1}_{ab,cd} - \frac{1}{2N_{\rm pul}}\hbar^{-4}\mu^{-1}_{ab}\mu^{-1}_{cd} \tag*{(172)} C ab , c d − 1 = C ab , c d − 1 − 2 N pul 1 ℏ − 4 μ ab − 1 μ c d − 1 (172) for the projected inverse of C a b , c d \mathcal{C}_{ab,cd} C ab , c d
We now use these expressions to compute the non-Gaussian contribution to the variance of χ 2 \chi^2 χ 2 E a b , c d , e f , g h \mathbb{E}_{ab,cd,ef,gh} E ab , c d , e f , g h (162) and using (169) , or substituting (F8) into (166) and using (172) we obtain the following exact expressions:
E = ( 2 N p u l 3 + 5 N p u l 2 + 5 N p u l ) ( ℏ h ^ ) 8 , (173) \mathbb{E} = \left(2N_{\rm pul}^{3}+5N_{\rm pul}^{2}+5N_{\rm pul}\right)\left(\frac{\hbar}{\hat h}\right)^{8}, \tag*{(173)} E = ( 2 N pul 3 + 5 N pul 2 + 5 N pul ) ( h ^ ℏ ) 8 , (173) E ^ = ( 2 N p u l 3 + 5 N p u l 2 − 7 N p u l − 12 + 12 N p u l ) ( ℏ h ^ ) 8 . (174) \widehat{\mathbb{E}} = \left(2N_{\rm pul}^{3}+5N_{\rm pul}^{2}-7N_{\rm pul}-12+\frac{12}{N_{\rm pul}}\right)\left(\frac{\hbar}{\hat h}\right)^{8}. \tag*{(174)} E = ( 2 N pul 3 + 5 N pul 2 − 7 N pul − 12 + N pul 12 ) ( h ^ ℏ ) 8 . (174) We have verified that these algebraic expressions agree with numerical results obtained by placing pulsars at random sky points.
These non-Gaussian contributions to the variance of χ 2 \chi^2 χ 2
E − E ^ = 12 ( N p u l + 1 − 1 N p u l ) ( ℏ h ^ ) 8 . (175) \mathbb{E}-\widehat{\mathbb{E}}=12\left(N_{\rm pul}+1-\frac{1}{N_{\rm pul}}\right)\left(\frac{\hbar}{\hat h}\right)^{8}. \tag*{(175)} E − E = 12 ( N pul + 1 − N pul 1 ) ( h ^ ℏ ) 8 . (175) When there are many pulsars, so N p u l ≫ 1 N_{\rm pul}\gg1 N pul ≫ 1 E ≃ 2 N p u l 3 ( ℏ / h ^ ) 8 \mathbb{E}\simeq2N_{\rm pul}^{3}(\hbar/\hat h)^{8} E ≃ 2 N pul 3 ( ℏ/ h ^ ) 8 E − E ^ ≃ 12 N p u l ( ℏ / h ^ ) 8 \mathbb{E}-\widehat{\mathbb{E}}\simeq12N_{\rm pul}(\hbar/\hat h)^{8} E − E ≃ 12 N pul ( ℏ/ h ^ ) 8 E \mathbb{E} E E − E ^ \mathbb{E}-\widehat{\mathbb{E}} E − E ( ℏ / h ^ ) 8 (\hbar/\hat h)^{8} ( ℏ/ h ^ ) 8 N p u l N_{\rm pul} N pul Figure 16 .
The numbers of pulsars used in current PTAs is large enough that only the leading term is needed to approximate the variance (also assuming, as we have done in this section, that the number of angular bins is as large as possible). With these assumptions, the fractional fluctuations in the χ 2 \chi^2 χ 2 E \mathbb{E} E (158) . Using (137) and (8.47), we obtain
σ χ 2 ⟨ χ 2 ⟩ ≃ E 1 / 2 N b i n s ≃ E ^ 1 / 2 N b i n s − 1 ≃ 2 1 / 2 N p u l 3 / 2 N b i n s ( ℏ h ^ ) 4 ≃ 2 5 / 4 N b i n s 1 / 4 ( ℏ h ^ ) 4 , (176) \frac{\sigma_{\chi^2}}{\langle\chi^2\rangle}\simeq\frac{\mathbb{E}^{1/2}}{N_{\rm bins}}\simeq\frac{\widehat{\mathbb{E}}^{1/2}}{N_{\rm bins}-1}\simeq\frac{2^{1/2}N_{\rm pul}^{3/2}}{N_{\rm bins}}\left(\frac{\hbar}{\hat h}\right)^4\simeq\frac{2^{5/4}}{N_{\rm bins}^{1/4}}\left(\frac{\hbar}{\hat h}\right)^4, \tag*{(176)} ⟨ χ 2 ⟩ σ χ 2 ≃ N bins E 1/2 ≃ N bins − 1 E 1/2 ≃ N bins 2 1/2 N pul 3/2 ( h ^ ℏ ) 4 ≃ N bins 1/4 2 5/4 ( h ^ ℏ ) 4 , (176) where the last approximate equality follows from (8.33), which gives N b i n s N_{\rm bins} N bins N p u l N_{\rm pul} N pul E \mathbb{E} E χ 2 \chi^2 χ 2 (159) . See Table II , which compares the fractional uncertainties for the (unprojected) χ 2 \chi^2 χ 2
Case (ii): Using only auto-correlations
If the χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 ρ a a \rho_{aa} ρ aa γ = 0 \gamma=0 γ = 0 h 2 μ a a = 2 h 2 μ u ( 0 ) h^2\mu_{aa}=2h^2\mu_u(0) h 2 μ aa = 2 h 2 μ u ( 0 ) χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 h ^ 2 μ a a = 2 h ^ 2 μ u ( 0 ) \hat h^2\mu_{aa}=2\hat h^2\mu_u(0) h ^ 2 μ aa = 2 h ^ 2 μ u ( 0 ) h ^ 2 \hat h^2 h ^ 2
There are several simplifications that arise when using only auto correlations. First, the covariance matrix simplifies to
C a a , b b = ℏ 4 ( μ a b μ a b + μ a b μ a b ) = 2 ℏ 4 μ a b 2 . (177) \mathcal{C}_{aa,bb}=\hbar^4(\mu_{ab}\mu_{ab}+\mu_{ab}\mu_{ab})=2\hbar^4\mu_{ab}^{2}. \tag*{(177)} C aa , bb = ℏ 4 ( μ ab μ ab + μ ab μ ab ) = 2 ℏ 4 μ ab 2 . (177) This is an N p u l × N p u l N_{\rm pul}\times N_{\rm pul} N pul × N pul μ a b \mu_{ab} μ ab not proportional to the matrix product of μ a b \mu_{ab} μ ab E \mathbb{E} E
E a a , b b , c c , d d = 16 ℏ 8 ( μ a b μ b c μ c d μ d a + μ a c μ c d μ d b μ b a + μ a d μ d b μ b c μ c a ) . (178) \mathbb{E}_{aa,bb,cc,dd}=16\hbar^{8}(\mu_{ab}\mu_{bc}\mu_{cd}\mu_{da}+\mu_{ac}\mu_{cd}\mu_{db}\mu_{ba}+\mu_{ad}\mu_{db}\mu_{bc}\mu_{ca}). \tag*{(178)} E aa , bb , cc , dd = 16 ℏ 8 ( μ ab μ b c μ c d μ d a + μ a c μ c d μ d b μ ba + μ a d μ d b μ b c μ c a ) . (178) To further simplify the notation in what follows, we define
C a b ≡ C a a , b b , P a b ≡ P a a , b b , E a b c d ≡ E a a , b b , c c , d d , (179) \mathcal{C}_{ab}\equiv\mathcal{C}_{aa,bb},\quad P_{ab}\equiv P_{aa,bb},\quad\mathcal{E}_{abcd}\equiv\mathcal{E}_{aa,bb,cc,dd}, \tag*{(179)} C ab ≡ C aa , bb , P ab ≡ P aa , bb , E ab c d ≡ E aa , bb , cc , dd , (179) where the projection operator P a a , b b P_{aa,bb} P aa , bb (164) . Since μ a a = 2 μ u ( 0 ) 1 a \mu_{aa}=2\mu_u(0)\mathbf{1}_a μ aa = 2 μ u ( 0 ) 1 a
P a b = δ a b − 1 a ( C − 1 1 ) b 1 T C − 1 1 (180) P_{ab}=\delta_{ab}-\frac{\mathbf{1}_a(\mathcal{C}^{-1}\mathbf{1})_b}{\mathbf{1}^{\mathsf{T}}\mathcal{C}^{-1}\mathbf{1}} \tag*{(180)} P ab = δ ab − 1 T C − 1 1 1 a ( C − 1 1 ) b (180) where C − 1 = C a b − 1 \mathcal{C}^{-1}=\mathcal{C}^{-1}_{ab} C − 1 = C ab − 1 C = C a b \mathcal{C}=\mathcal{C}_{ab} C = C ab 1 \mathbf{1} 1 (22) . Note that the trace of (180) correctly gives N p u l − 1 N_{\mathrm{pul}}-1 N pul − 1 C − 1 \mathcal{C}^{-1} C − 1 (162) and (166) simplify to
E = 16 ℏ 8 ∑ a , b , c , d C a b − 1 C c d − 1 ( 2 μ a b μ b c μ c d μ d a + μ a d μ d b μ b c μ c a ) , \mathcal{E}=16\hbar^8\sum_{a,b,c,d}\mathcal{C}^{-1}_{ab}\mathcal{C}^{-1}_{cd}\left(2\mu_{ab}\mu_{bc}\mu_{cd}\mu_{da}+\mu_{ad}\mu_{db}\mu_{bc}\mu_{ca}\right), E = 16 ℏ 8 a , b , c , d ∑ C ab − 1 C c d − 1 ( 2 μ ab μ b c μ c d μ d a + μ a d μ d b μ b c μ c a ) , E ^ = 16 ℏ 8 ∑ a , b , c , d C ^ a b − 1 C ^ c d − 1 ( 2 μ a b μ b c μ c d μ d a + μ a d μ d b μ b c μ c a ) , (181) \widehat{\mathcal{E}}=16\hbar^8\sum_{a,b,c,d}\widehat{\mathcal{C}}^{-1}_{ab}\widehat{\mathcal{C}}^{-1}_{cd}\left(2\mu_{ab}\mu_{bc}\mu_{cd}\mu_{da}+\mu_{ad}\mu_{db}\mu_{bc}\mu_{ca}\right), \tag*{(181)} E = 16 ℏ 8 a , b , c , d ∑ C ab − 1 C c d − 1 ( 2 μ ab μ b c μ c d μ d a + μ a d μ d b μ b c μ c a ) , (181) where C ^ a b − 1 \widehat{\mathcal{C}}^{-1}_{ab} C ab − 1 C a b − 1 \mathcal{C}^{-1}_{ab} C ab − 1
C ^ a b − 1 ≡ ∑ c , d C c d − 1 P c a P d b . (182) \widehat{\mathcal{C}}^{-1}_{ab}\equiv\sum_{c,d}\mathcal{C}^{-1}_{cd}P_{ca}P_{db}. \tag*{(182)} C ab − 1 ≡ c , d ∑ C c d − 1 P c a P d b . (182) To obtain the factor of 2 on the rhs’s of (181) , we used the symmetry of C a b − 1 \mathcal{C}^{-1}_{ab} C ab − 1 C ^ a b − 1 \widehat{\mathcal{C}}^{-1}_{ab} C ab − 1 E a b c d \mathcal{E}_{abcd} E ab c d a a a b b b
For this auto-correlation-only case, we do not have a way to simplify the expressions further, so we resort to numerical evaluation, placing simulated pulsars at random points on the sky. In this way, we evaluated E \mathcal{E} E E − E ^ \mathcal{E}-\widehat{\mathcal{E}} E − E χ 2 \chi^2 χ 2 N p u l N_{\mathrm{pul}} N pul Figure 17 . The black solid and dashed curves show E \mathcal{E} E E − E ^ \mathcal{E}-\widehat{\mathcal{E}} E − E N p u l = 1000 N_{\mathrm{pul}}=1000 N pul = 1000 ℏ / ℏ = 1 \hbar/\hbar=1 ℏ/ℏ = 1 N p u l 1 / 2 N_{\mathrm{pul}}^{1/2} N pul 1/2 E \mathcal{E} E N p u l N_{\mathrm{pul}} N pul E − E ^ \mathcal{E}-\widehat{\mathcal{E}} E − E
Figure 17. Auto-correlations only. Left panel: non-Gaussian contributions to the variance of the unprojected and projected χ 2 \chi^2 χ 2 E \mathcal{E} E E − E ^ \mathcal{E}-\widehat{\mathcal{E}} E − E N p u l = 1000 N_{\mathrm{pul}}=1000 N pul = 1000 ℏ / ℏ = 1 \hbar/\hbar=1 ℏ/ℏ = 1 σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ [57 ] . (On this scale, the unprojected and projected χ 2 \chi^2 χ 2 N p u l N_{\mathrm{pul}} N pul χ 2 \chi^2 χ 2 N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞
If this polynomial behavior extends to arbitrarily large values of N p u l N_{\mathrm{pul}} N pul χ 2 \chi^2 χ 2 E ∼ N p u l 2 ( ℏ / ℏ ) 8 ∼ N b i n s 2 ( ℏ / ℏ ) 8 \mathcal{E}\sim N_{\mathrm{pul}}^2(\hbar/\hbar)^8\sim N_{\mathrm{bins}}^2(\hbar/\hbar)^8 E ∼ N pul 2 ( ℏ/ℏ ) 8 ∼ N bins 2 ( ℏ/ℏ ) 8 σ χ 2 / ⟨ χ 2 ⟩ ≈ E 1 / 2 / N b i n s ∼ const \sigma_{\chi^2}/\langle\chi^2\rangle\approx\mathcal{E}^{1/2}/N_{\mathrm{bins}}\sim\operatorname{const} σ χ 2 / ⟨ χ 2 ⟩ ≈ E 1/2 / N bins ∼ const (176) . Thus, there would be a (nonzero) cosmic variance for χ 2 \chi^2 χ 2 1 % 1\% 1% N = 2000 N=2000 N = 2000
For χ 2 \chi^2 χ 2 E \mathcal{E} E
and give details in Appendix § 23 . We hope that this may help others to derive a simple closed form for the PTA detector case.
Consider (non-PTA) GW detectors whose correlation function (88) is defined by Q l = δ l L Q_l=\delta_{lL} Q l = δ l L L L L 2 L + 1 2L+1 2 L + 1 l = L l=L l = L m = − L , … , L m=-L,\ldots,L m = − L , … , L E \mathbb{E} E N p u l N_{\rm pul} N pul N p u l N_{\rm pul} N pul
E = 2 ( 2 L + 1 ) ( 4 L 2 + 9 L + 6 ) ( h ℏ ) 8 when N p u l ≥ ( 2 L + 1 ) ( L + 1 ) . (183) \mathbb{E}=2(2L+1)(4L^2+9L+6)\left(\frac{\mathfrak{h}}{\hbar}\right)^8 \quad\text{when } N_{\rm pul}\ge(2L+1)(L+1). \tag*{(183)} E = 2 ( 2 L + 1 ) ( 4 L 2 + 9 L + 6 ) ( ℏ h ) 8 when N pul ≥ ( 2 L + 1 ) ( L + 1 ) . (183) Here, ( 2 L + 1 ) ( L + 1 ) ≡ N D O F (2L+1)(L+1)\equiv N_{\rm DOF} ( 2 L + 1 ) ( L + 1 ) ≡ N DOF C a b C_{ab} C ab
This result can be obtained in two ways. (1) Fitting a polynomial to numerically determined values of E \mathbb{E} E E \mathbb{E} E § 23 in the limit N p u l → ∞ N_{\rm pul}\to\infty N pul → ∞
Case (iii): Using only cross-correlations
We now consider χ 2 \chi^2 χ 2 E \mathbb{E} E E ^ \widehat{\mathbb{E}} E (162) and (166) .
Plots of E \mathbb{E} E E − E ^ \mathbb{E}-\widehat{\mathbb{E}} E − E χ 2 \chi^2 χ 2 N p u l N_{\rm pul} N pul Figure 18 , together with polynomial fits. The curves are again normalized so that h / ℏ = 1 \mathfrak{h}/\hbar=1 h /ℏ = 1 E \mathbb{E} E N p u l N_{\rm pul} N pul E ≃ 2 N p u l 3 ( h / ℏ ) 8 \mathbb{E}\simeq2N_{\rm pul}^3(\mathfrak{h}/\hbar)^8 E ≃ 2 N pul 3 ( h /ℏ ) 8 N p u l N_{\rm pul} N pul Figure 16 , this is the same fit obtained when both auto- and cross-correlations are used, see also (8.47). For large N p u l N_{\rm pul} N pul E − E ^ \mathbb{E}-\widehat{\mathbb{E}} E − E N p u l N_{\rm pul} N pul ≈ 8.44 ( h / ℏ ) 8 \approx8.44(\mathfrak{h}/\hbar)^8 ≈ 8.44 ( h /ℏ ) 8 ≈ 1.42 \approx1.42 ≈ 1.42 χ 2 \chi^2 χ 2 2 \sqrt{2} 2 6 2 6\sqrt{2} 6 2
Figure 18. Similar to Fig. 17 but for cross-correlations only. The polynomial fits to the underlying numerical results are shown by the gray dotted curves. As before, on the scale of the plot in the right panel, there is no discernible difference between the fractional uncertainties for the unprojected and projected χ 2 \chi^2 χ 2 N p u l − 1 / 2 N_{\rm pul}^{-1/2} N pul − 1/2 N b i n s − 1 / 4 N_{\rm bins}^{-1/4} N bins − 1/4 [57 ] .
Since the number of bins is N b i n s = N p u l ( N p u l − 1 ) / 2 N_{\rm bins}=N_{\rm pul}(N_{\rm pul}-1)/2 N bins = N pul ( N pul − 1 ) /2 χ 2 \chi^2 χ 2 N b i n s − 1 / 4 N_{\rm bins}^{-1/4} N bins − 1/4 N b i n s N_{\rm bins} N bins N p u l N_{\rm pul} N pul Figure 18 , and is again similar to the auto + cross correlations case. In fact, as N p u l → ∞ N_{\rm pul}\to\infty N pul → ∞ E \mathbb{E} E σ χ 2 / ⟨ χ 2 ⟩ \sigma_{\chi^2}/\langle\chi^2\rangle σ χ 2 / ⟨ χ 2 ⟩ χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 Figure 16 .
Fractional uncertainties for the χ 2 \chi^2 χ 2
The previous sections give numerical results for pulsars located at random sky positions. Here, we consider the specific sets of pulsars currently employed by the different PTAs (see Appendix § 24 and Table IV ).
Table II gives the fractional uncertainties for the unprojected χ 2 \chi^2 χ 2
these PTAs. The fractional uncertainties for the projected χ 2 \chi^2 χ 2 N b i n s N_{\mathrm{bins}} N bins N p u l N_{\mathrm{pul}} N pul
One row and one column of the table are for illustration and pedagogical purposes. Row (4) corresponds to the (incorrect) assumption of Gaussian correlations, and is obtained by taking the square root of (159) . The contrast with the other rows shows that the non-Gaussian contributions dominate the Gaussian ones. In a similar vein, the final column shows the fractional uncertainty for a fictional PTA containing an infinite number of pulsars, uniformly distributed on the sky. This demonstrates the ultimate potential performance of a PTA, in the distant future.
The table contains a question mark in the final column of row (2). As discussed in § 11 C 2, this is because we have not been able to determine if χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 N p u l → ∞ N_{\mathrm{pul}} \to\infty N pul → ∞
Appendix § 23 attempts to resolve this question mark, by computing the dominant (non-Gaussian) contribution to χ 2 \chi^2 χ 2 N p u l → ∞ N_{\mathrm{pul}} \to\infty N pul → ∞
The χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 ± σ χ 2 \pm\sigma_{\chi^2} ± σ χ 2 ± 3 σ χ 2 \pm3\sigma_{\chi^2} ± 3 σ χ 2 χ 2 \chi^2 χ 2
In Figure 19 we illustrate how this would work in practice, based on numerical simulations containing 40 pulsars (at fixed random sky locations). We constructed hundreds of representative universes drawn from a Gaussian ensemble with squared strain h 2 = 1 h^2 = 1 h 2 = 1 χ 2 \chi^2 χ 2 h 2 h^2 h 2 χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 ) h 2 h^2 h 2 χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 )
Figure 19. χ 2 \chi^2 χ 2 h 2 = 1 h^2=1 h 2 = 1 h 2 h^2 h 2 h ^ 2 \hat{h}^2 h ^ 2 ± 1 σ \pm1\sigma ± 1 σ Figure 15 . On the vertical axis, the solid curve shows the values of the projected χ 2 ( h 2 ) \chi^2(h^2) χ 2 ( h 2 ) ± 1 σ \pm1\sigma ± 1 σ χ 2 \chi^2 χ 2 ⟨ χ 2 ⟩ ± σ χ 2 \langle\chi^2\rangle\pm\sigma_{\chi^2} ⟨ χ 2 ⟩ ± σ χ 2 σ χ 2 \sigma_{\chi^2} σ χ 2 h 2 h^2 h 2
Including Pulsar and Measurement Noise
Up to this point, our calculations have assumed noise-free pulsar redshift (or timing-residual) data. Thus, they provide a baseline or “best-case scenario” for analyzing the data, e.g., the best-case precision with which we can recover the expected Hellings and Downs correlation or the smallest fractional uncertainty in estimating the squared GW strain h 2 h^2 h 2
In this section, we describe how the inclusion of pulsar and measurement noise modifies previous (noise-free) expressions for the optimal Hellings and Downs correlation and its variance (Secs. III, IV, § 6 ), for the optimal squared-strain estimator and its variance (Sec. VII), and for the χ 2 \chi^2 χ 2 § 11 ). We assume that the noise in each pulsar is time-stationary and that this noise is uncorrelated between different pulsars. Our extension is not general enough to include noise such as random observatory clock fluctuations, which affect different pulsars in a correlated way.
Since the noise enters our calculations via the first four moments of the pulsar-pair correlations ρ a b \rho_{ab} ρ ab ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ C a b , c d = ⟨ ρ a b ρ c d ⟩ − ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ C_{ab,cd}=\langle\rho_{ab}\rho_{cd}\rangle-\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle C ab , c d = ⟨ ρ ab ρ c d ⟩ − ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ D a b , c d , e f D_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h E_{ab,cd,ef,gh} E ab , c d , e f , g h
If noise is present, and the optimal estimators (for the Hellings and Downs correlation or for the squared GW strain) include pulsar auto-correlations, then the noise-free expressions (16) and (98) are modified. To obtain an unbiased estimator, a nonzero constant term must be added to the existing (proportional to ρ a b \rho_{ab} ρ ab ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ (189) . The optimal estimators then include constant “bias subtraction” terms, which remove the effects of autocorrelated noise. An example of such a modified squared-strain estimator h ^ 2 \hat{h}^2 h ^ 2 (199) .
The presence of noise does not modify the definitions of the χ 2 \chi^2 χ 2 (128) , (143) , and (146) , even if those sums include auto-correlations. Note, however, that χ 2 \chi^2 χ 2 ρ o p t ; j \rho_{\mathrm{opt};j} ρ opt ; j h ^ 2 \hat{h}^2 h ^ 2
Another consequence of including the effects of noise is that the optimal estimator h ^ 2 \hat{h}^2 h ^ 2 ρ o p t \rho_{\mathrm{opt}} ρ opt h 2 h^2 h 2 h 2 h^2 h 2 self-consistent values of the squared strain. We do this by evaluating the estimator h ^ 2 \hat{h}^2 h ^ 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 h^2 h 2 σ h ^ 2 \sigma_{\hat{h}^2} σ h ^ 2
Pulsar redshifts in the presence of noise
In the presence of noise, the measured redshift Z a Z_a Z a a a a
Z ~ a ( f ) = h ~ a ( f ) + n ~ a ( f ) , (184) \widetilde{Z}_a(f)=\widetilde{h}_a(f)+\widetilde{n}_a(f), \tag*{(184)} Z a ( f ) = h a ( f ) + n a ( f ) , (184) where Z ~ a ( f ) \widetilde{Z}_a(f) Z a ( f ) h ~ a ( f ) \widetilde{h}_a(f) h a ( f ) n ~ a ( f ) \widetilde{n}_a(f) n a ( f ) Z a ( t ) Z_a(t) Z a ( t ) h a ( t ) h_a(t) h a ( t ) n a ( t ) n_a(t) n a ( t ) a a a b b b
⟨ n ~ a ∗ ( f ) n ~ b ( f ′ ) ⟩ = 4 π N a ( f ) δ a b δ ( f − f ′ ) , ⟨ h ~ a ∗ ( f ) h ~ b ( f ′ ) ⟩ = 4 π H ( f ) μ a b δ ( f − f ′ ) , and ⟨ n ~ a ∗ ( f ) h ~ b ( f ′ ) ⟩ = 0. (185) \begin{aligned}
\langle\widetilde{n}_a^*(f)\widetilde{n}_b(f')\rangle&= 4\pi N_a(f)\delta_{ab}\delta(f-f'),\\
\langle\widetilde{h}_a^*(f)\widetilde{h}_b(f')\rangle&= 4\pi H(f)\mu_{ab}\delta(f-f'), \qquad\text{and}\\
\langle\widetilde{n}_a^*(f)\widetilde{h}_b(f')\rangle&= 0. \tag*{(185)}
\end{aligned} ⟨ n a ∗ ( f ) n b ( f ′ )⟩ ⟨ h a ∗ ( f ) h b ( f ′ )⟩ ⟨ n a ∗ ( f ) h b ( f ′ )⟩ = 4 π N a ( f ) δ ab δ ( f − f ′ ) , = 4 π H ( f ) μ ab δ ( f − f ′ ) , and = 0. (185) Here N a ( f ) N_a(f) N a ( f ) a a a H ( f ) H(f) H ( f ) H ( f ) H(f) H ( f ) N a ( f ) N_a(f) N a ( f ) ( 2 π f ) − 2 α (2\pi f)^{-2\alpha} ( 2 π f ) − 2 α α = 0 \alpha=0 α = 0 α = 1 \alpha=1 α = 1
The correlation between redshifts in pulsars a a a b b b (184) and (185) , giving
⟨ Z ~ a ∗ ( f ) Z ~ b ( f ′ ) ⟩ = 4 π Γ a b ( f ) δ ( f − f ′ ) , (186) \langle\widetilde{Z}_a^*(f)\widetilde{Z}_b(f')\rangle= 4\pi\Gamma_{ab}(f)\delta(f-f'), \tag*{(186)} ⟨ Z a ∗ ( f ) Z b ( f ′ )⟩ = 4 π Γ ab ( f ) δ ( f − f ′ ) , (186) where
Γ a b ( f ) ≡ H ( f ) μ a b + N a ( f ) δ a b . (187) \Gamma_{ab}(f) \equiv H(f)\mu_{ab}+N_a(f)\delta_{ab}. \tag*{(187)} Γ ab ( f ) ≡ H ( f ) μ ab + N a ( f ) δ ab . (187) As we will see below, to incorporate noise into our previous analyses, H ( f ) μ a b H(f)\mu_{ab} H ( f ) μ ab Γ a b ( f ) \Gamma_{ab}(f) Γ ab ( f )
The transition from noise-free expressions to “noise-full” expressions is nontrivial, because Γ a b ( f ) \Gamma_{ab}(f) Γ ab ( f ) μ a b \mu_{ab} μ ab δ a b \delta_{ab} δ ab ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ C a b , c d C_{ab,cd} C ab , c d D a b , c d , e f D_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h E_{ab,cd,ef,gh} E ab , c d , e f , g h h 2 h^2 h 2 h 4 h^4 h 4 h 6 h^6 h 6 h 8 h^8 h 8 H ( f ) H(f) H ( f ) (7) , (10) , (F9), and (F10).
In the presence of noise, additional terms appear, arising from integrals of various products of H ( f ) H(f) H ( f ) N a ( f ) N_a(f) N a ( f ) N b ( f ) N_b(f) N b ( f ) ρ a b \rho_{ab} ρ ab H ( f ) H ( f ′ ) H(f)H(f') H ( f ) H ( f ′ ) H ( f ) N a ( f ′ ) H(f)N_a(f') H ( f ) N a ( f ′ ) N a ( f ) N b ( f ′ ) N_a(f)N_b(f') N a ( f ) N b ( f ′ ) ρ a b \rho_{ab} ρ ab h 2 h^2 h 2 h 4 h^4 h 4 h 6 h^6 h 6 h 8 h^8 h 8
First moment ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩
To calculate the first-order moment ⟨ ρ a b ⟩ \langle\rho_{ab}\rangle ⟨ ρ ab ⟩ ρ a b ≡ Z a Z b ‾ \rho_{ab}\equiv\overline{Z_aZ_b} ρ ab ≡ Z a Z b
ρ a b ≡ Z a Z b ‾ ≡ 1 T ∫ − T / 2 T / 2 d t Z a ( t ) Z b ( t ) = ∫ d f ∫ d f ′ sinc ( π ( f − f ′ ) T ) Z ~ a ∗ ( f ) Z ~ b ( f ′ ) , (188) \rho_{ab}\equiv\overline{Z_aZ_b}\equiv\frac{1}{T}\int_{-T/2}^{T/2}dt\,Z_a(t)Z_b(t)=\int df\int df'\,\operatorname{sinc}\left(\pi(f-f')T\right)\widetilde{Z}_a^*(f)\widetilde{Z}_b(f'), \tag*{(188)} ρ ab ≡ Z a Z b ≡ T 1 ∫ − T /2 T /2 d t Z a ( t ) Z b ( t ) = ∫ df ∫ d f ′ sinc ( π ( f − f ′ ) T ) Z a ∗ ( f ) Z b ( f ′ ) , (188) where T T T ( − ∞ , ∞ ) (-\infty,\infty) ( − ∞ , ∞ ) (186) and (187) , it immediately follows that
⟨ ρ a b ⟩ = 4 π ∫ d f ( 2 π f ) − 2 α Γ a b ( f ) = h 2 μ a b + n a 2 δ a b , (189) \langle\rho_{ab}\rangle=4\pi\int df\,(2\pi f)^{-2\alpha}\Gamma_{ab}(f)=h^2\mu_{ab}+n_a^2\delta_{ab}, \tag*{(189)} ⟨ ρ ab ⟩ = 4 π ∫ df ( 2 π f ) − 2 α Γ ab ( f ) = h 2 μ ab + n a 2 δ ab , (189) where h 2 h^2 h 2 (7) and
n a 2 ≡ 4 π ∫ d f ( 2 π f ) − 2 α N a ( f ) . (190) n_a^2\equiv4\pi\int df\,(2\pi f)^{-2\alpha}N_a(f). \tag*{(190)} n a 2 ≡ 4 π ∫ df ( 2 π f ) − 2 α N a ( f ) . (190) [We have included a factor of ( 2 π f ) − 2 α (2\pi f)^{-2\alpha} ( 2 π f ) − 2 α a ≠ b a\ne b a = b ⟨ ρ a b ⟩ = h 2 μ a b \langle\rho_{ab}\rangle=h^2\mu_{ab} ⟨ ρ ab ⟩ = h 2 μ ab ρ a a \rho_{aa} ρ aa
Second moment (covariance) C a b , c d C_{ab,cd} C ab , c d
For the covariance matrix C a b , c d ≡ ⟨ ρ a b ρ c d ⟩ − ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ C_{ab,cd}\equiv\langle\rho_{ab}\rho_{cd}\rangle-\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle C ab , c d ≡ ⟨ ρ ab ρ c d ⟩ − ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ ρ a b ρ c d \rho_{ab}\rho_{cd} ρ ab ρ c d Z ~ a , … , Z ~ d \widetilde{Z}_a,\ldots,\widetilde{Z}_d Z a , … , Z d
ρ a b ρ c d = ∫ d f ∫ d f ′ ∫ d f ′ ′ ∫ d f ′ ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ ′ − f ′ ′ ′ ) T ) Z ~ a ∗ ( f ) Z ~ b ( f ′ ) Z ~ c ∗ ( f ′ ′ ) Z ~ d ( f ′ ′ ′ ) . (191) \rho_{ab}\rho_{cd}=\int df\int df'\int df''\int df'''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f''-f''')T\right)\widetilde{Z}_a^*(f)\widetilde{Z}_b(f')\widetilde{Z}_c^*(f'')\widetilde{Z}_d(f'''). \tag*{(191)} ρ ab ρ c d = ∫ df ∫ d f ′ ∫ d f ′′ ∫ d f ′′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′′ − f ′′′ ) T ) Z a ∗ ( f ) Z b ( f ′ ) Z c ∗ ( f ′′ ) Z d ( f ′′′ ) . (191) We assume that the pulsar noise and the GW background are described by independent central-limit-theorem Gaussian ensembles. This implies that Z ~ \widetilde{Z} Z
Thus, starting from the ensemble average of (191) , we use Isserlis’s theorem [19 ] to convert the expectation values of products of four redshift measurements in the frequency domain to products of terms containing two redshift measurements. We then use (186) to evaluate these expectation values, eliminating two of the four integrals. Finally, we subtract ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ \langle\rho_{ab}\rangle\langle\rho_{cd}\rangle ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ (191) . We obtain
C a b , c d = ( 4 π ) 2 ∫ d f ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α ( Γ a c ( f ) Γ b d ( f ′ ) + Γ a d ( f ) Γ b c ( f ′ ) ) = h 4 ( μ a c μ b d + μ a d μ b c ) + ( δ a c δ b d + δ a d δ b c ) N a b 2 + ( δ a c M a μ b d + δ b d M b μ a c + δ a d M a μ b c + δ b c M b μ a d ) , (192) \begin{aligned}
C_{ab,cd}=(4\pi)^2\int df\int df'\,\operatorname{sinc}^2\left(\pi(f-f')T\right)(4\pi^2ff')^{-2\alpha}\left(\Gamma_{ac}(f)\Gamma_{bd}(f')+\Gamma_{ad}(f)\Gamma_{bc}(f')\right) \\
&=\mathcal{h}^4\left(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}\right)+\left(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right)N_{ab}^2+\left(\delta_{ac}M_a\mu_{bd}+\delta_{bd}M_b\mu_{ac}+\delta_{ad}M_a\mu_{bc}+\delta_{bc}M_b\mu_{ad}\right), \tag*{(192)}
\end{aligned} C ab , c d = ( 4 π ) 2 ∫ df ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α ( Γ a c ( f ) Γ b d ( f ′ ) + Γ a d ( f ) Γ b c ( f ′ ) ) = h 4 ( μ a c μ b d + μ a d μ b c ) + ( δ a c δ b d + δ a d δ b c ) N ab 2 + ( δ a c M a μ b d + δ b d M b μ a c + δ a d M a μ b c + δ b c M b μ a d ) , (192) where the second line follows from (187) by expanding the factors of Γ a c ( f ) \Gamma_{ac}(f) Γ a c ( f ) Γ b d ( f ′ ) \Gamma_{bd}(f') Γ b d ( f ′ )
N a b 2 ≡ ( 4 π ) 2 ∫ d f ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α N a ( f ) N b ( f ′ ) , (193) N_{ab}^2\equiv(4\pi)^2\int df\int df'\,\operatorname{sinc}^2\left(\pi(f-f')T\right)(4\pi^2ff')^{-2\alpha}N_a(f)N_b(f'), \tag*{(193)} N ab 2 ≡ ( 4 π ) 2 ∫ df ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α N a ( f ) N b ( f ′ ) , (193) and
M a ≡ ( 4 π ) 2 ∫ d f ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α N a ( f ) H ( f ′ ) . (194) M_a\equiv(4\pi)^2\int df\int df'\,\operatorname{sinc}^2\left(\pi(f-f')T\right)(4\pi^2ff')^{-2\alpha}N_a(f)H(f'). \tag*{(194)} M a ≡ ( 4 π ) 2 ∫ df ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α N a ( f ) H ( f ′ ) . (194) Under our assumptions, this is the form of the covariance matrix, which includes both the GW and pulsar-noise contributions. Note that (192) is valid for all possible choices of the pulsars a a a b b b c c c d d d a = c a=c a = c b = d b=d b = d
Current PTA searches for GW backgrounds assume noise sources similar to those described above. The resulting noise terms appear in the likelihood functions used in Bayesian analyses. They are also used to weight pulsar-pair correlation measurements to form the standard cross-correlation detection statistic [42 , 50 , 51 ] .
Pulsar timing data have been noise-dominated (until possibly recently [6 , 7 , 8 , 9 ] ). (Indeed, after this paper was completed, some PTAs reported evidence of GW-induced timing correlations consistent with the Hellings and Downs prediction [58 , 59 , 60 , 61 ] .) So, current methods to estimate the Hellings and Downs correlation ρ ( γ ) \rho(\gamma) ρ ( γ ) (192) proportional to N a b 2 N_{ab}^2 N ab 2 [58 ] and EPTA [59 ] adopted our definition of the Hellings and Downs correlation and the corresponding estimation method. This shifts the means, and increases the variance of the estimates by about a factor of two.) This is a good approximation if the GW signal power is small relative to the noise. But if the GW power is comparable to (or larger than) that of the noise, then it is a mistake to drop the off-diagonal terms. Leaving out these terms, which are proportional to M a M_a M a h 4 \mathcal{h}^4 h 4 [62 ] to include the full form of the covariance matrix, cf. (192) , in the optimal pulsar-pair estimators described above.
Estimating the squared GW strain in the presence of noise
In Sec. VII we showed how to estimate the squared GW strain h 2 h^2 h 2 N ( f ) = N a ( f ) N(f)=N_a(f) N ( f ) = N a ( f ) spectral shape of the GW spectrum H ( f ) H(f) H ( f ) not its overall normalization (amplitude). This simple model is enough to address the most important questions.
From assumption (i), it follows that the covariance matrix (192) simplifies to
C a b , c d = h 4 ( μ a c μ b d + μ a d μ b c ) + n 4 ( δ a c δ b d + δ a d δ b c ) + m 4 ( δ a c μ b d + δ b d μ a c + δ a d μ b c + δ b c μ a d ) , (195) C_{ab,cd}=\mathcal{h}^4\left(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}\right)+n^4\left(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}\right)+m^4\left(\delta_{ac}\mu_{bd}+\delta_{bd}\mu_{ac}+\delta_{ad}\mu_{bc}+\delta_{bc}\mu_{ad}\right), \tag*{(195)} C ab , c d = h 4 ( μ a c μ b d + μ a d μ b c ) + n 4 ( δ a c δ b d + δ a d δ b c ) + m 4 ( δ a c μ b d + δ b d μ a c + δ a d μ b c + δ b c μ a d ) , (195) where
n 2 ≡ 4 π ∫ d f N ( f ) , (196) n^2\equiv4\pi\int df\,N(f), \tag*{(196)} n 2 ≡ 4 π ∫ df N ( f ) , (196) and
n 4 ≡ ( 4 π ) 2 ∫ d f ∫ d f ′ sinc ( π ( f − f ′ ) ) N ( f ) N ( f ′ ) , (197) n^4\equiv(4\pi)^2\int df\int df'\,\operatorname{sinc}\left(\pi(f-f')\right)N(f)N(f'), \tag*{(197)} n 4 ≡ ( 4 π ) 2 ∫ df ∫ d f ′ sinc ( π ( f − f ′ ) ) N ( f ) N ( f ′ ) , (197) and
m 4 ≡ ( 4 π ) 2 ∫ d f ∫ d f ′ sinc ( π ( f − f ′ ) ) N ( f ) H ( f ′ ) . (198) m^4\equiv(4\pi)^2\int df\int df'\,\operatorname{sinc}\left(\pi(f-f')\right)N(f)H(f'). \tag*{(198)} m 4 ≡ ( 4 π ) 2 ∫ df ∫ d f ′ sinc ( π ( f − f ′ ) ) N ( f ) H ( f ′ ) . (198) These last three equations are (190) , (193) , and (194) , with α = 0 \alpha=0 α = 0 n 2 n^{2} n 2 n 4 n^{4} n 4 m 4 m^{4} m 4 n a 2 n^{2}_{a} n a 2 N a b 2 N^{2}_{ab} N ab 2 M a M_{a} M a ℓ 2 / h 2 \ell^{2}/h^{2} ℓ 2 / h 2 n 4 n^{4} n 4 n 4 n^{4} n 4 m 4 m^{4} m 4 n 2 h 2 n^{2}h^{2} n 2 h 2 C a b , c d C_{ab,cd} C ab , c d h 2 h^{2} h 2
The optimal estimator of h 2 h^{2} h 2 (99) minus a constant term (proportional to n 2 n^{2} n 2
h ^ 2 = ( ∑ a b , c d ∈ S μ a b C a b , c d − 1 ( h 2 ) μ c d ) − 1 ∑ e f , g h ∈ S μ e f C e f , g h − 1 ( h 2 ) ( ρ g h − n 2 δ g h ) . (199) \widehat{h}^{2}=
\left(\sum_{ab,cd\in S}\mu_{ab}C^{-1}_{ab,cd}(h^{2})\mu_{cd}\right)^{-1}
\sum_{ef,gh\in S}\mu_{ef}C^{-1}_{ef,gh}(h^{2})(\rho_{gh}-n^{2}\delta_{gh}). \tag*{(199)} h 2 = ab , c d ∈ S ∑ μ ab C ab , c d − 1 ( h 2 ) μ c d − 1 e f , g h ∈ S ∑ μ e f C e f , g h − 1 ( h 2 ) ( ρ g h − n 2 δ g h ) . (199) The constant term is chosen so that h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2 ⟨ h ^ 2 ⟩ = h 2 \langle\widehat{h}^{2}\rangle=h^{2} ⟨ h 2 ⟩ = h 2 h ^ 2 \widehat{h}^{2} h 2 (100) ,
σ h ^ 2 2 = ( ∑ a b , c d ∈ S μ a b C a b , c d − 1 ( h 2 ) μ c d ) − 1 , (200) \sigma^{2}_{\widehat{h}^{2}}=
\left(\sum_{ab,cd\in S}\mu_{ab}C^{-1}_{ab,cd}(h^{2})\mu_{cd}\right)^{-1}, \tag*{(200)} σ h 2 2 = ab , c d ∈ S ∑ μ ab C ab , c d − 1 ( h 2 ) μ c d − 1 , (200) but contains the inverse of the “noise-full” covariance matrix (195) .
These equations lead to an apparently impossible circle of dependence. Both the estimator and its variance are functions of the (unknown) quantity h 2 h^{2} h 2 C a b , c d C_{ab,cd} C ab , c d C a b , c d − 1 C^{-1}_{ab,cd} C ab , c d − 1 h 2 h^{2} h 2 h 2 h^{2} h 2
To explore this complication, we first write (199) and (200) in more compact matrix/vector notation
h ^ 2 = μ T C − 1 ( h 2 ) ( ρ − n 2 δ ) μ T C − 1 ( h 2 ) μ ≡ f ( h 2 ) , (201) \widehat{h}^{2}=\frac{\mu^{\mathsf T}C^{-1}(h^{2})(\rho-n^{2}\delta)}{\mu^{\mathsf T}C^{-1}(h^{2})\mu}\equiv f(h^{2}), \tag*{(201)} h 2 = μ T C − 1 ( h 2 ) μ μ T C − 1 ( h 2 ) ( ρ − n 2 δ ) ≡ f ( h 2 ) , (201) and
σ h ^ 2 2 ( h 2 ) = 1 μ T C − 1 ( h 2 ) μ , (202) \sigma^{2}_{\widehat{h}^{2}}(h^{2})=\frac{1}{\mu^{\mathsf T}C^{-1}(h^{2})\mu}, \tag*{(202)} σ h 2 2 ( h 2 ) = μ T C − 1 ( h 2 ) μ 1 , (202) where δ \delta δ (201) denotes the identity matrix in “pair-indexed vector” form. In the above equations, we explicitly indicate that both the optimal estimator h ^ 2 \widehat{h}^{2} h 2 σ h ^ 2 2 \sigma^{2}_{\widehat{h}^{2}} σ h 2 2 h 2 h^{2} h 2 f f f
The formulas for h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2 (201) and (202) ? The approach we propose is to make the estimate of h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2 self-consistent . By this, we mean that the inferred estimate h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2 (202) .
The circular aspect is confusing. To obtain an (optimal) estimate h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2
know h 2 h^{2} h 2 H ( f ) H(f) H ( f ) H ( f ) H(f) H ( f ) h 2 h^{2} h 2 particular definite value of h 2 h^{2} h 2 all possible values of h 2 h^{2} h 2 h 2 h^{2} h 2 h 2 h^{2} h 2
This approach is illustrated graphically in Figure 20 . To make these plots, we placed 88 pulsars at random sky locations, simulated Gaussian ensembles with three different GW (squared) amplitudes, and then randomly selected a single realization from each ensemble. The horizontal axis shows the possible values of h 2 h^{2} h 2 h ^ 2 \widehat{h}^{2} h 2 h 2 h^{2} h 2 ⟨ h ^ 2 ⟩ = h 2 \langle\widehat{h}^{2}\rangle=h^{2} ⟨ h 2 ⟩ = h 2 ρ \rho ρ h 2 h^{2} h 2 h 2 + σ h ^ 2 ( h 2 ) h^{2}+\sigma_{\widehat{h}^{2}}(h^{2}) h 2 + σ h 2 ( h 2 ) h 2 − σ h ^ 2 ( h 2 ) h^{2}-\sigma_{\widehat{h}^{2}}(h^{2}) h 2 − σ h 2 ( h 2 ) h 2 h^{2} h 2 self-consistent lie in the interval where the solid black curve h ^ 2 = f ( h 2 ) \widehat{h}^{2}=f(h^{2}) h 2 = f ( h 2 ) h 2 ± σ h ^ 2 ( h 2 ) h^{2}\pm\sigma_{\widehat{h}^{2}}(h^{2}) h 2 ± σ h 2 ( h 2 ) h 2 h^{2} h 2 h 2 h^{2} h 2 h 2 h^{2} h 2 h 2 = 0 h^{2}=0 h 2 = 0 [63 ] .
Figure 20. Self-consistent squared-strain (h 2 h^2 h 2 h 2 h^2 h 2 h 2 / n 2 = 10 h^2/n^2=10 h 2 / n 2 = 10 0.33 0.33 0.33 0.01 0.01 0.01 h ^ 2 = f ( h 2 ) \hat{h}^2=f(h^2) h ^ 2 = f ( h 2 ) h 2 ± σ h ^ 2 ( h 2 ) h^2\pm\sigma_{\hat{h}^2}(h^2) h 2 ± σ h ^ 2 ( h 2 ) ⟨ h ^ 2 ⟩ = h 2 \langle\hat{h}^2\rangle=h^2 ⟨ h ^ 2 ⟩ = h 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 = 0 h^2=0 h 2 = 0
Given a set of pulsar sky locations and correlation observations, other approaches to obtain bounds on h 2 h^{2} h 2 h 2 h^2 h 2 ρ \rho ρ h 2 h^2 h 2
Third and fourth moments (cumulants) D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h
Calculations similar to those of Sec. IX B (first moment) and Sec. IX C (second moment) show how noise modifies D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h ρ a b \rho_{ab} ρ ab § 12.6 .
The noise-free expressions are (F7) for D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h
D a b , c d , e f = 4 ( 4 π ) 3 ∫ d f ∫ d f ′ ∫ d f ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) sinc ( π ( f ′ ′ − f ) T ) ( 8 π 3 f f ′ f ′ ′ ) − 2 α ( Γ b ( c ( f ) Γ d ( e ( f ′ ) Γ f ) a ( f ′ ′ ) + Γ a ( c ( f ) Γ d ( e ( f ′ ) Γ f ) b ( f ′ ′ ) ) . (203) \mathcal{D}_{ab,cd,ef}=4(4\pi)^3\int df\int df'\int df''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right)\operatorname{sinc}\left(\pi(f''-f)T\right)(8\pi^3ff'f'')^{-2\alpha}
\left(\Gamma_{b(c}(f)\Gamma_{d(e}(f')\Gamma_{f)a}(f'')+\Gamma_{a(c}(f)\Gamma_{d(e}(f')\Gamma_{f)b}(f'')\right). \tag*{(203)} D ab , c d , e f = 4 ( 4 π ) 3 ∫ df ∫ d f ′ ∫ d f ′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) sinc ( π ( f ′′ − f ) T ) ( 8 π 3 f f ′ f ′′ ) − 2 α ( Γ b ( c ( f ) Γ d ( e ( f ′ ) Γ f ) a ( f ′′ ) + Γ a ( c ( f ) Γ d ( e ( f ′ ) Γ f ) b ( f ′′ ) ) . (203) and
E a b , c d , e f , g h = 8 ( 4 π ) 4 ∫ d f ∫ d f ′ ∫ d f ′ ′ ∫ d f ′ ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) × sinc ( π ( f ′ ′ − f ′ ′ ′ ) T ) sinc ( π ( f ′ ′ ′ − f ) T ) ( 16 π 4 f f ′ f ′ ′ f ′ ′ ′ ) − 2 α × ( Γ b ( c ( f ) Γ d ( e ( f ′ ) Γ f ) ( g ( f ′ ′ ) Γ h ) a ( f ′ ′ ′ ) + Γ b ( e ( f ) Γ f ) ( g ( f ′ ) Γ h ) ( c ( f ′ ′ ) Γ d ) a ( f ′ ′ ′ ) + Γ b ( g ( f ) Γ h ) ( c ( f ′ ) Γ d ) ( e ( f ′ ′ ) Γ f ) a ( f ′ ′ ′ ) + a ↔ b ) . (204) \begin{aligned}
\mathcal{E}_{ab,cd,ef,gh}={}&8(4\pi)^4\int df\int df'\int df''\int df'''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right)\\
&\mathrel{}\times\operatorname{sinc}\left(\pi(f''-f''')T\right)\operatorname{sinc}\left(\pi(f'''-f)T\right)(16\pi^4ff'f''f''')^{-2\alpha}\\
&\mathrel{}\times\left(\Gamma_{b(c}(f)\Gamma_{d(e}(f')\Gamma_{f)(g}(f'')\Gamma_{h)a}(f''')+\Gamma_{b(e}(f)\Gamma_{f)(g}(f')\Gamma_{h)(c}(f'')\Gamma_{d)a}(f''')\right.\\
&\mathrel{}\left.+\Gamma_{b(g}(f)\Gamma_{h)(c}(f')\Gamma_{d)(e}(f'')\Gamma_{f)a}(f''')+a\leftrightarrow b\right). \tag*{(204)}
\end{aligned} E ab , c d , e f , g h = 8 ( 4 π ) 4 ∫ df ∫ d f ′ ∫ d f ′′ ∫ d f ′′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) × sinc ( π ( f ′′ − f ′′′ ) T ) sinc ( π ( f ′′′ − f ) T ) ( 16 π 4 f f ′ f ′′ f ′′′ ) − 2 α × ( Γ b ( c ( f ) Γ d ( e ( f ′ ) Γ f ) ( g ( f ′′ ) Γ h ) a ( f ′′′ ) + Γ b ( e ( f ) Γ f ) ( g ( f ′ ) Γ h ) ( c ( f ′′ ) Γ d ) a ( f ′′′ ) + Γ b ( g ( f ) Γ h ) ( c ( f ′ ) Γ d ) ( e ( f ′′ ) Γ f ) a ( f ′′′ ) + a ↔ b ) . (204) These expressions can be further simplified by using (187) to expand the factors of Γ b c ( f ) \Gamma_{bc}(f) Γ b c ( f ) (203) and (204) . One then obtains integrals of various products of H ( f ) H(f) H ( f ) N a ( f ) N_a(f) N a ( f ) N b ( f ) N_b(f) N b ( f ) h 2 h^2 h 2 h 4 h^4 h 4 ℏ 6 \hbar^6 ℏ 6 h 8 h^8 h 8 n a 2 n_a^2 n a 2 N a b 2 N_{ab}^2 N ab 2 M a M_a M a
Q a ≡ ( 4 π ) 3 ∫ d f ∫ d f ′ ∫ d f ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) sinc ( π ( f ′ ′ − f ) T ) × ( 8 π 3 f f ′ f ′ ′ ) − 2 α N a ( f ) H ( f ′ ) H ( f ′ ′ ) (205) Q_a \equiv(4\pi)^3 \int df \int df' \int df''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right)\operatorname{sinc}\left(\pi(f''-f)T\right)
\\
\times(8\pi^3ff'f'')^{-2\alpha}N_a(f)H(f')H(f'') \tag*{(205)} Q a ≡ ( 4 π ) 3 ∫ df ∫ d f ′ ∫ d f ′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) sinc ( π ( f ′′ − f ) T ) × ( 8 π 3 f f ′ f ′′ ) − 2 α N a ( f ) H ( f ′ ) H ( f ′′ ) (205) and
R a b c ≡ ( 4 π ) 4 ∫ d f ∫ d f ′ ∫ d f ′ ′ ∫ d f ′ ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) × sinc ( π ( f ′ ′ − f ′ ′ ′ ) T ) sinc ( π ( f ′ ′ ′ − f ) T ) ( 16 π 4 f f ′ f ′ ′ f ′ ′ ′ ) − 2 α N a ( f ) N b ( f ′ ) N c ( f ′ ′ ) H ( f ′ ′ ′ ) (206) R_{abc} \equiv(4\pi)^4 \int df \int df' \int df'' \int df'''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right)
\\
\times\operatorname{sinc}\left(\pi(f''-f''')T\right)\operatorname{sinc}\left(\pi(f'''-f)T\right)(16\pi^4ff'f''f''')^{-2\alpha}N_a(f)N_b(f')N_c(f'')H(f''') \tag*{(206)} R ab c ≡ ( 4 π ) 4 ∫ df ∫ d f ′ ∫ d f ′′ ∫ d f ′′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) × sinc ( π ( f ′′ − f ′′′ ) T ) sinc ( π ( f ′′′ − f ) T ) ( 16 π 4 f f ′ f ′′ f ′′′ ) − 2 α N a ( f ) N b ( f ′ ) N c ( f ′′ ) H ( f ′′′ ) (206) There are a total of eight such new terms.
Properties of the noise contribution to the covariance matrix
Does the inclusion of pulsar noise in the covariance matrix affect the analyses described in the previous sections? For a finite number of pulsars and sufficiently loud noise, the answer is yes. But in the limit of a sufficiently large number of pulsar pairs that faithfully cover the sky, the noise contributions (for independent pulsar noise) average to zero, leaving only the GW contribution.
To see this, we can repeat the calculation of Sec. IV including the additional noise terms. As before, the relevant entries of the covariance matrix are those where the rows and columns correspond to pulsar pairs a b ab ab c d cd c d γ \gamma γ 1 1 1 C j j C_{jj} C j j γ j ≡ γ \gamma_j \equiv\gamma γ j ≡ γ C \mathcal{C} C (39) , the variance of the optimal estimator in this narrow bin is the average value of the entries of this block.
So, is this also the case when we include the noise terms in (192) ? Examining those terms by first symmetrizing over the indices c c c d d d μ a c \mu_{ac} μ a c μ u ( γ a c ) \mu_u(\gamma_{ac}) μ u ( γ a c ) δ a c \delta_{ac} δ a c (5) , we obtain
N a b , c d ≡ ( δ a c δ b d + δ a d δ b c ) N a b 2 + ( δ a c M a μ b d + δ b d M b μ a c + δ a d M a μ b c + δ b c M b μ a d ) = 2 [ N a b 2 + ( M a + M b ) μ u ( 0 ) ] δ a ( c δ d ) b + 2 M a δ a ( c μ u ( γ d ) b ) + 2 M b δ b ( c μ u ( γ d ) a ) . (207) N_{ab,cd} \equiv(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})N_{ab}^{2} + (\delta_{ac}M_a\mu_{bd}+\delta_{bd}M_b\mu_{ac}+\delta_{ad}M_a\mu_{bc}+\delta_{bc}M_b\mu_{ad})
\\
= 2\left[N_{ab}^{2}+(M_a+M_b)\mu_u(0)\right]\delta_{a(c}\delta_{d)b}+2M_a\delta_{a(c}\mu_u(\gamma_{d)b})+2M_b\delta_{b(c}\mu_u(\gamma_{d)a}). \tag*{(207)} N ab , c d ≡ ( δ a c δ b d + δ a d δ b c ) N ab 2 + ( δ a c M a μ b d + δ b d M b μ a c + δ a d M a μ b c + δ b c M b μ a d ) = 2 [ N ab 2 + ( M a + M b ) μ u ( 0 ) ] δ a ( c δ d ) b + 2 M a δ a ( c μ u ( γ d ) b ) + 2 M b δ b ( c μ u ( γ d ) a ) . (207) But since all of the above terms involve either a single Kronecker delta or a product of two Kronecker deltas, they give zero in the limit of an infinite number of pulsar pairs distributed uniformly on the sky, as shown explicitly in Appendix C.
Thus, the noise terms do not contribute to the limiting behavior of the full covariance matrix, and
σ o p t 2 ( γ ) = σ o p t , s i g n a l 2 ( γ ) + σ o p t , n o i s e 2 ( γ ) = σ cos 2 ( γ ) , (208) \sigma_{\mathrm{opt}}^{2}(\gamma)=\sigma_{\mathrm{opt,signal}}^{2}(\gamma)+\sigma_{\mathrm{opt,noise}}^{2}(\gamma)=\sigma_{\cos}^{2}(\gamma), \tag*{(208)} σ opt 2 ( γ ) = σ opt , signal 2 ( γ ) + σ opt , noise 2 ( γ ) = σ c o s 2 ( γ ) , (208) just as we found for the noise-free case.
How to apply the optimal estimator
It is helpful to think about how the methods described in this paper would be used in practice. Imagine that we are handed a set of pulsar-pair correlation measurements ρ a b \rho_{ab} ρ ab γ \gamma γ γ \gamma γ w a b w_{ab} w ab (24) or (33) . For this, we need the (inverse of the) matrix block C j j C_{jj} C j j C \mathcal{C} C γ j ≃ γ \gamma_j \simeq\gamma γ j ≃ γ But this covariance matrix cannot be obtained or estimated from the data that we have—a theoretical model is needed.
For the Gaussian ensemble, in the limit of noise-free observations, the theoretical model of the covariance matrix is given by (11) . The values of μ a c \mu_{ac} μ a c μ b d \mu_{bd} μ b d (11) are fully determined by the sky locations of the pulsars via Eqs. (5) and (1) . The value of H 4 \mathcal{H}^{4} H 4 (11) , is not needed: it cancels out of the numerator and denominator of (24) when evaluating the weights, thus allowing us to determine ρ o p t , j \rho_{\mathrm{opt},j} ρ opt , j ρ a b \rho_{ab} ρ ab [64 ] .
The important point to note is that in following this procedure we have assumed that the Gaussian ensemble model is correct. Logically, there are only two possibilities: (a) the data/Universe is well described by the Gaussian ensemble, or (b) the data/Universe is not well described by the Gaussian ensemble. In case (a), the optimal estimator that we form, and the variance that we attribute to it, will both be meaningful and consistent. However, in case (b), neither the weights that we have derived nor the variance of the optimal estimator that we compute, have any justification.
Consider a specific example: a (non-Gaussian!) ensemble of universes containing GW sources which radiate at distinct nonoverlapping frequencies. Such an ensemble is constructed in Sec. III B of [11 ] , where it is demonstrated that the cosmic variance is zero, so the pulsar-averaged correlations always follow the μ u ( γ ) \mu_u(\gamma) μ u ( γ ) μ u ( γ ) \mu_u(\gamma) μ u ( γ )
The conclusion is that the construction provided here is mainly useful as a consistency test, given that we have a finite set of observations that are not uniform on the sky. It can be used to determine the number of pulsar pairs to place in each angular bin, and also used to weight the correlations of those pairs. Finally, it can be used to estimate (for the Gaussian ensemble) the variance that would be expected for that set of pulsar pairs. If the correlation curve that is obtained has fluctuations of about ± 1 σ o p t \pm1\sigma_{\rm opt} ± 1 σ opt h 2 μ u ( γ ) h^2\mu_u(\gamma) h 2 μ u ( γ ) χ 2 \chi^2 χ 2 § 11 A.
In this sense, the results of this paper are useful because they predict the total Gaussian ensemble variance (pulsar plus cosmic) for a finite number of pulsars at specific sky locations, whereas the variance σ c o s 2 \sigma_{\rm cos}^2 σ cos 2 [11 ] is for the limit of an infinite number of uniformly distributed pulsars.
Conclusion
The standard definition of the Hellings and Downs correlation is for a single pair of pulsars. We have extended and generalized that, defining the Hellings and Downs correlation for a set of pulsar pairs whose angular separations γ \gamma γ
Assuming that the GW background is described by a Gaussian ensemble, we have found the optimal way to weight the correlations of the individual pulsar pairs, so that the Hellings and Downs correlation has the smallest possible variations from the mean. These weights are normalized to produce an average that agrees with the single-pair correlation, enabling direct comparison. Moreover, in the absence of pulsar and instrumental noise, it is straightforward to compute the expected variance of these weighted sums.
As pulsar pairs are added at some angle, the variance of the correlation decreases. This variance starts at the total variance σ t o t 2 \sigma_{\rm tot}^2 σ tot 2 [11 ] for a single pulsar pair. It decreases smoothly as pulsar pairs are added, eventually approaching the cosmic variance σ c o s 2 \sigma_{\rm cos}^2 σ cos 2 Figure 6 illustrate this transition. Physically, there is a simple way to understand this. Once there are enough pulsar pairs at a given angular separation, adding additional pulsar pairs does not provide any additional information about the GW-induced correlations. This is because those pairs lie close on the sky to other pairs: their correlations provide redundant information.
We have also calculated the covariance of the optimal estimators for different angular separation bins. In the limit of an infinite number of pulsar pairs lying in narrow bins j j j k k k (57) for the cosmic covariance matrix. This shows that the likely fluctuations away from the Hellings and Downs curve are strongly correlated (or anticorrelated) across three broad angular separation regions. We also computed the cosmic covariance matrix in harmonic space (61) , where it takes diagonal form: a product of Legendre polynomials evaluated at cos γ j \cos\gamma_j cos γ j cos γ k \cos\gamma_k cos γ k
To test if the observed deviations away from the expected Hellings and Downs curve are consistent with expectations, we proposed two sets of χ 2 \chi^2 χ 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 h^2 h 2 h 2 h^2 h 2
We believe that the PTA GW sources in our Universe are well described by a Gaussian ensemble, and we have a good idea of their spectrum. This means that, for a specific set of pulsars at specific sky locations, we can predict the best-case variance in the Hellings and Downs correlation. This best case assumes that the intrinsic pulsar and experimental noise are small, and so provides a lower bound on the uncertainty of observational results. A simple recipe for making this prediction, given the pulsar sky locations, is given at the end of § 6 B. Modifications needed to extend our formalism to include pulsar and measurement noise were described in Sec. IX.
In future PTA searches, using improved telescopes and larger pulsar populations, we expect that the variance of the Hellings and Downs correlation will reveal exactly the cosmic variance. In practice, this means that the optimal estimator of the Hellings and Downs correlation will not agree exactly in shape with the classic Hellings and Downs curve μ u ( γ ) \mu_u(\gamma) μ u ( γ ) χ 2 \chi^2 χ 2
Acknowledgments
The authors thank Curt Cutler for helpful discussions about the variance in the estimator of the squared strain, and EPTA, NANOGrav, and PPTA for providing lists of their pulsar names and sky positions. B. A. is grateful to the members of the IPTA Detection Committee for their help in understanding many aspects of PTA data analysis, and J. D. R. acknowledges support from NSF Physics Frontiers Center Award PFC-2020265 and start-up funds from Texas Tech University.
Outline of Appendices
The appendices contain calculations and arguments which are useful for the main body of the paper. Here, we outline their contents.
Appendix A shows how our results for the Hellings and Downs correlation of pulsar redshifts can be easily modified to treat pulsar timing residuals.
The relationship between the mean and the variance for the Gaussian ensemble depends upon certain integrals of the power spectrum. We discuss this dependence in Appendix B and calculate the relationship for a simple binary-inspiral model of the GW background.
In Appendix C, we examine the Kronecker delta terms which appear in the covariance matrix. We show that if there are many pulsar pairs distributed uniformly on the sky, then these terms constitute a set of measure zero and do not affect the variance.
In Appendix D, we consider the case where there are large numbers of pulsar pairs at a small (integer) number κ \kappa κ κ ( κ − 1 ) / 2 \kappa(\kappa-1)/2 κ ( κ − 1 ) /2 § 23 in [11 ] via (54) .
We exploit this result in Appendix E to show how (with a binned analysis employing a large number of pulsar-pair correlations at two different angles) it is possible to “beat” the cosmic variance. This does not come for free: angular resolution is lost.
In Appendix § 12.6 , we calculate the third- and fourth-order moments of the pulsar-pair correlations ρ a b \rho_{ab} ρ ab χ 2 \chi^2 χ 2 § 11 A. Since ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab [19 ] , obtaining triple and quadruple integrals of the GW power spectrum. We compute these integrals for both redshift and timing-residual measurements, assuming the same simple binary-inspiral background used in Appendix B. Results are given in Figure 23 and Table III.
Figure 23. The coefficients ℏ 2 \hbar^2 ℏ 2 h ~ 2 \tilde{h}^2 h ~ 2 h ˉ 2 \bar{h}^2 h ˉ 2 h 2 h^2 h 2 α = 1 \alpha=1 α = 1 α = 0 \alpha=0 α = 0
In Appendix § 23 , we calculate the non-Gaussian contributions to the variance of the χ 2 \chi^2 χ 2 χ 2 \chi^2 χ 2 N p u l → ∞ N_{\rm pul} \to\infty N pul → ∞
Finally, in Appendix § 24 , we list the sky positions of the 88 pulsars currently monitored by three active PTA collaborations [37 ] . These are used for the example plots of § 6 B, shown in Figure 3 and Figure 9 , and for constructing Table I and Table II .
Mean and Variance of Timing-Residual Correlations (Versus Redshift Correlations)
Much of the literature and experimental/observational work for PTAs is based on pulsar timing residuals rather than redshifts. Pulsar timing residuals may be obtained by integrating the pulse redshifts with respect to time, which inserts a factor of 1 / 2 π f 1/2\pi f 1/2 π f
Following this procedure, formulas identical to (4) and (11) are obtained for the first moment and covariance matrix of timing-residual correlations, by replacing H ( f ) H(f) H ( f ) H ( f ) / 4 π 2 f 2 H(f)/4\pi^2f^2 H ( f ) /4 π 2 f 2 h 2 h^2 h 2 K 4 \mathcal{K}^4 K 4 (7) and (10) . To ensure that our formulas can cover both possibilities, we introduce a constant α \alpha α
Set α = 0 \alpha= 0 α = 0 α = 1 \alpha= 1 α = 1
Relative Scaling Between the Correlation Mean and Variance for Noise-Free Data
The GW Gaussian ensemble is completely specified by the real spectral function H ( f ) ≥ 0 H(f) \ge0 H ( f ) ≥ 0 (7) and (8) . For example, H ( f ) H(f) H ( f ) h 2 h^2 h 2 (4) . The spectral function H ( f ) H(f) H ( f ) h 4 \mathcal{h}^4 h 4 C a b , c d C_{ab,cd} C ab , c d (10) and (11) .
This implies that h 4 \mathcal{h}^4 h 4 C a b , c d C_{ab,cd} C ab , c d h 4 \mathcal{h}^4 h 4 universal : they apply to any Gaussian ensemble.
In contrast, the ratio of the variance to the (squared) mean is model dependent: it depends upon the ratio h 4 / h 4 ≤ 1 \mathcal{h}^4/h^4 \le1 h 4 / h 4 ≤ 1 H ( f ) H(f) H ( f ) H ( f ) H(f) H ( f ) Figure 9 ) are model dependent . To make them, we must assume some form of the spectrum H ( f ) H(f) H ( f )
One simple source model is constructed in Sec. III A in [11 ] . This “confusion-noise model” consists of a large set of GW sources, uniformly distributed in space, all radiating GWs at exactly the same frequency, but with different random phases . The corresponding spectral function H ( f ) H(f) H ( f ) [11 ] it is shown that for this model h 4 / h 4 ∈ [ 1 / 2 , 1 ) \mathcal{h}^4/h^4 \in[1/2,1) h 4 / h 4 ∈ [ 1/2 , 1 ) h 4 / h 4 = 1 / 2 \mathcal{h}^4/h^4 = 1/2 h 4 / h 4 = 1/2
Note that the same values for this ratio are obtained for pulsar redshift correlations (α = 0 \alpha= 0 α = 0 α = 1 \alpha= 1 α = 1 h 4 / h 4 = 1 / 2 \mathcal{h}^4/h^4 = 1/2 h 4 / h 4 = 1/2 Figure 1 . However, the model is unrealistic because in our Universe the sources are distributed over frequency.
Here, we construct a more realistic spectral model, which we then use for Figure 9 and elsewhere. This more realistic model assumes that the GW background is produced by large numbers of compact binary systems, whose energy loss is dominated by GW emission in the Newtonian limit (orbital velocities ≪ \ll ≪ H ( f ) ∝ ∣ f ∣ − 7 / 3 H(f) \propto|f|^{-7/3} H ( f ) ∝ ∣ f ∣ − 7/3
Although it continues to much lower frequencies, for the purpose of computing PTA correlations, the spectrum is effectively “cut off” below a characteristic frequency f 0 > 0 f_0 > 0 f 0 > 0
H ( f ) = { q ∣ f ∣ − 7 / 3 for ∣ f ∣ > f 0 , and 0 for ∣ f ∣ ≤ f 0 , (209) H(f) =
\begin{cases}
q|f|^{-7/3} & \text{for } |f| > f_0, \text{ and}\\
0 & \text{for } |f| \le f_0,
\end{cases} \tag*{(209)} H ( f ) = { q ∣ f ∣ − 7/3 0 for ∣ f ∣ > f 0 , and for ∣ f ∣ ≤ f 0 , (209) where q q q f 0 f_0 f 0 1 / T 1/T 1/ T T T T [65 , 66 , 67 ] . For further details, see the “transmission functions” illustrated in Figure 2 of Ref. [67 ] .
The ratio of the variance to the squared mean is proportional to h 4 / h 4 \mathcal{h}^4/h^4 h 4 / h 4 [68 ] upon the low-frequency cutoff f 0 f_0 f 0 (7) and (10) to obtain
h 4 h 4 = ∫ π f 0 T ∞ d x ∫ π f 0 T ∞ d y [ 1 2 sinc 2 ( x − y ) + 1 2 sinc 2 ( x + y ) ] ( x y ) − 7 / 3 − 2 α ( ∫ π f 0 T ∞ d x x − 7 / 3 − 2 α ) 2 . (210) \frac{\mathcal{h}^4}{h^4} =
\frac{ \int_{\pi f_0T}^{\infty} \mathrm{d}x \int_{\pi f_0T}^{\infty} \mathrm{d}y \left[\frac{1}{2}\operatorname{sinc}^{2}(x-y)+\frac{1}{2}\operatorname{sinc}^{2}(x+y)\right](xy)^{-7/3-2\alpha}}
{ \left(\int_{\pi f_0T}^{\infty} \mathrm{d}x\,x^{-7/3-2\alpha}\right)^2}. \tag*{(210)} h 4 h 4 = ( ∫ π f 0 T ∞ d x x − 7/3 − 2 α ) 2 ∫ π f 0 T ∞ d x ∫ π f 0 T ∞ d y [ 2 1 sinc 2 ( x − y ) + 2 1 sinc 2 ( x + y ) ] ( x y ) − 7/3 − 2 α . (210) Here, we have changed to dimensionless variables x = π T f x = \pi T f x = π T f y = π T f ′ y = \pi T f' y = π T f ′ H ( f ) = H ( − f ) H(f) = H(-f) H ( f ) = H ( − f ) α \alpha α α = 0 \alpha= 0 α = 0 α = 1 \alpha= 1 α = 1
The behavior of h 4 / h 4 < 1 \mathcal{h}^4/h^4 < 1 h 4 / h 4 < 1 f 0 T → 0 f_0T \to0 f 0 T → 0 x x x y y y h 4 / h 4 → 1 \mathcal{h}^4/h^4 \to1 h 4 / h 4 → 1 f 0 T f_0T f 0 T R 4 / h 4 → 1 / 2 \mathcal{R}^4/h^4 \to1/2 R 4 / h 4 → 1/2 f 0 T f_0T f 0 T
The ratio R 2 / h 2 \mathcal{R}^2/h^2 R 2 / h 2 Figure 21 as a function of the effective lower cutoff frequency f 0 T f_0T f 0 T f 0 = 1 / T f_0=1/T f 0 = 1/ T Figure 2 of [67 ] . Taking f 0 T = 1 f_0T=1 f 0 T = 1 Figure 21 implies R 2 / h 2 ≈ 0.5622 \mathcal{R}^2/h^2\approx0.5622 R 2 / h 2 ≈ 0.5622 R 2 / h 2 ≈ 0.3905 \mathcal{R}^2/h^2\approx0.3905 R 2 / h 2 ≈ 0.3905 Figure 9 . For the other plot in the paper (Figure 1 ) that shows a mean value, we use R 4 / h 4 = 1 / 2 \mathcal{R}^4/h^4=1/2 R 4 / h 4 = 1/2 [11 ] .
Proof that terms involving Kronecker deltas do not contribute to the cosmic variance
Here, we show that terms in the covariance matrix C a b , c d \mathcal{C}_{ab,cd} C ab , c d
2 R 4 [ μ u 2 ( 0 ) δ a ( c δ d ) b + μ u ( 0 ) ( δ a ( c μ u ( γ d ) b ) + δ b ( c μ u ( γ d ) a ) ) ] 2\mathcal{R}^4\left[\mu_u^2(0)\delta_{a(c}\delta_{d)b}+\mu_u(0)\left(\delta_{a(c}\mu_u(\gamma_{d)b})+\delta_{b(c}\mu_u(\gamma_{d)a})\right)\right] 2 R 4 [ μ u 2 ( 0 ) δ a ( c δ d ) b + μ u ( 0 ) ( δ a ( c μ u ( γ d ) b ) + δ b ( c μ u ( γ d ) a ) ) ] or
2 [ N a b 2 + ( M a + M b ) μ u ( 0 ) ] δ a ( c δ d ) b + 2 M a δ a ( c μ u ( γ d ) b ) + 2 M b δ b ( c μ u ( γ d ) a ) (211) 2\left[N_{ab}^2+(M_a+M_b)\mu_u(0)\right]\delta_{a(c}\delta_{d)b}+2M_a\delta_{a(c}\mu_u(\gamma_{d)b})+2M_b\delta_{b(c}\mu_u(\gamma_{d)a}) \tag*{(211)} 2 [ N ab 2 + ( M a + M b ) μ u ( 0 ) ] δ a ( c δ d ) b + 2 M a δ a ( c μ u ( γ d ) b ) + 2 M b δ b ( c μ u ( γ d ) a ) (211) from (49) or (207) , do not contribute to the average value of the covariance matrix components over a row, ⟨ C a b , c d ⟩ c d ∈ γ \langle\mathcal{C}_{ab,cd}\rangle_{cd\in\gamma} ⟨ C ab , c d ⟩ c d ∈ γ (50) , where γ \gamma γ a b ab ab c d cd c d
We begin by considering the terms proportional to δ a ( c δ d ) b \delta_{a(c}\delta_{d)b} δ a ( c δ d ) b μ u 2 ( 0 ) \mu_u^2(0) μ u 2 ( 0 ) N a b 2 N_{ab}^2 N ab 2 ( M a + M b ) (M_a+M_b) ( M a + M b ) c c c d d d a a a b b b
2 ⟨ δ a ( c δ d ) b ⟩ c d ∈ γ = 1 / n p a i r s . (212) 2\langle\delta_{a(c}\delta_{d)b}\rangle_{cd\in\gamma}=1/n_{\mathrm{pairs}}. \tag*{(212)} 2 ⟨ δ a ( c δ d ) b ⟩ c d ∈ γ = 1/ n pairs . (212) This is manifestly independent of the fixed pulsar pair a b ab ab
The terms proportional to a single Kronecker delta are also sets of measure zero. This may be understood by looking at just one of four subterms
s a b ≡ ⟨ δ a c μ u ( γ d b ) ⟩ c d ∈ γ . (213) s_{ab}\equiv\langle\delta_{ac}\mu_u(\gamma_{db})\rangle_{cd\in\gamma}. \tag*{(213)} s ab ≡ ⟨ δ a c μ u ( γ d b ) ⟩ c d ∈ γ . (213) We can again ignore the factors μ u ( 0 ) \mu_u(0) μ u ( 0 ) M a M_a M a M b M_b M b μ u ( γ d b ) \mu_u(\gamma_{db}) μ u ( γ d b ) μ u ( 0 ) \mu_u(0) μ u ( 0 ) μ u ( γ d b ) \mu_u(\gamma_{db}) μ u ( γ d b )
s a b ≤ μ u ( 0 ) ⟨ δ a c ⟩ c d ∈ γ . (214) s_{ab}\leq\mu_u(0)\langle\delta_{ac}\rangle_{cd\in\gamma}. \tag*{(214)} s ab ≤ μ u ( 0 ) ⟨ δ a c ⟩ c d ∈ γ . (214) Since δ a c \delta_{ac} δ a c d d d d d d γ \gamma γ c c c c d cd c d c c c N N N N N N N N N c c c c c c c c c a a a
s a b ≤ μ u ( 0 ) / N , (215) s_{ab}\leq\mu_u(0)/N, \tag*{(215)} s ab ≤ μ u ( 0 ) / N , (215) so s a b s_{ab} s ab μ u ( 0 ) / N \mu_u(0)/N μ u ( 0 ) / N s a b s_{ab} s ab / N /N / N N N N s a b s_{ab} s ab N → ∞ N \to\infty N → ∞
We consider real, symmetric, positive-definite matrices built up from smaller matrices, of the form
C = ( C 11 C 12 ⋯ C 1 κ C 21 C 22 ⋯ C 2 κ ⋮ ⋮ ⋱ ⋮ C κ 1 C κ 2 ⋯ C κ κ ) (216) \mathcal{C} =
\begin{pmatrix}
C_{11} & C_{12} & \cdots& C_{1\kappa} \\
C_{21} & C_{22} & \cdots& C_{2\kappa} \\
\vdots& \vdots& \ddots& \vdots\\
C_{\kappa1} & C_{\kappa2} & \cdots& C_{\kappa\kappa}
\end{pmatrix} \tag*{(216)} C = C 11 C 21 ⋮ C κ 1 C 12 C 22 ⋮ C κ 2 ⋯ ⋯ ⋱ ⋯ C 1 κ C 2 κ ⋮ C κκ (216) Here, each of the C j k C_{jk} C j k κ \kappa κ κ 2 \kappa^2 κ 2 the sum of any row of C j k C_{jk} C j k C j k C_{jk} C j k (Since C \mathcal{C} C C j k C_{jk} C j k C j k C_{jk} C j k
Note that the symbol C \mathcal{C} C (3) . However, in many cases of interest, (D1) is a submatrix of the full covariance matrix so that κ \kappa κ 0 ∘ 0^\circ 0 ∘ 180 ∘ 180^\circ 18 0 ∘ C \mathcal{C} C (3) or to some submatrix of it along the diagonal. This also holds for Appendix E.
We introduce the symbol d k d_k d k k = 1 , … , κ k=1,\ldots,\kappa k = 1 , … , κ C j k C_{jk} C j k j j j C \mathcal{C} C C j k C_{jk} C j k d j d_j d j C j k C_{jk} C j k ( d j rows ) × ( d k columns ) (d_j\ \text{rows}) \times(d_k\ \text{columns}) ( d j rows ) × ( d k columns )
N = dim ( C ) = ∑ j = 1 κ d j (217) N = \dim(\mathcal{C}) = \sum_{j=1}^{\kappa} d_j \tag*{(217)} N = dim ( C ) = j = 1 ∑ κ d j (217) is the number of rows or columns of the matrix C \mathcal{C} C
Finally, we define the quantity s j k s_{jk} s j k average value of the entries of the block C j k C_{jk} C j k Note that since the sum of any row of C j k C_{jk} C j k d k s j k d_k s_{jk} d k s j k
Equation (D1) is the form taken by the covariance matrix C \mathcal{C} C κ \kappa κ γ j \gamma_j γ j j = 1 , … , κ j=1,\ldots,\kappa j = 1 , … , κ d j d_j d j C j k C_{jk} C j k
In this appendix, we obtain a simple formula for the “sum of interest,” which is the quantity μ T C − 1 μ \mu^{\mathsf T}\mathcal{C}^{-1}\mu μ T C − 1 μ μ \mu μ N N N j = 1 , … , κ j=1,\ldots,\kappa j = 1 , … , κ μ \mu μ d j d_j d j μ u ( γ j ) \mu_u(\gamma_j) μ u ( γ j )
We begin by constructing a set of κ \kappa κ e 1 , e 2 , … , e κ e_1,e_2,\ldots,e_\kappa e 1 , e 2 , … , e κ N N N C \mathcal{C} C N N N e j e_j e j
The first of these vectors has d 1 d_1 d 1 N − d 1 N-d_1 N − d 1 d 1 d_1 d 1 d 2 d_2 d 2 N − d 1 − d 2 N-d_1-d_2 N − d 1 − d 2
e 1 = d 1 − 1 / 2 ( 1 , … , 1 , 0 , … , 0 , … , 0 ) T , e_1=d_1^{-1/2}(1,\ldots,1,0,\ldots,0,\ldots,0)^{\mathsf T}, e 1 = d 1 − 1/2 ( 1 , … , 1 , 0 , … , 0 , … , 0 ) T , e 2 = d 2 − 1 / 2 ( 0 , … , 0 , 1 , … , 1 , … , 0 , … , 0 ) T , e_2=d_2^{-1/2}(0,\ldots,0,1,\ldots,1,\ldots,0,\ldots,0)^{\mathsf T}, e 2 = d 2 − 1/2 ( 0 , … , 0 , 1 , … , 1 , … , 0 , … , 0 ) T , e κ = d κ − 1 / 2 ( 0 , … , 0 , 0 , … , 0 , … , 1 , … , 1 ) T . (218) e_\kappa=d_\kappa^{-1/2}(0,\ldots,0,0,\ldots,0,\ldots,1,\ldots,1)^{\mathsf T}. \tag*{(218)} e κ = d κ − 1/2 ( 0 , … , 0 , 0 , … , 0 , … , 1 , … , 1 ) T . (218) The ( N (N ( N
e j T e k = δ j k , (219) e_j^{\mathsf T}e_k=\delta_{jk}, \tag*{(219)} e j T e k = δ j k , (219) where δ j k \delta_{jk} δ j k
Because the sum of any row of C j k C_{jk} C j k C \mathcal{C} C e k e_k e k e 1 , e 2 , … , e κ e_1,e_2,\ldots,e_\kappa e 1 , e 2 , … , e κ C \mathcal{C} C e k e_k e k C \mathcal{C} C
C e k = ∑ j = 1 κ S j k e j , (220) \mathcal{C}e_k=\sum_{j=1}^{\kappa}S_{jk}e_j, \tag*{(220)} C e k = j = 1 ∑ κ S j k e j , (220) where
S j k = ( d j d k ) 1 / 2 s j k for j , k = 1 , … , κ . (221) S_{jk}=(d_jd_k)^{1/2}s_{jk}\quad\text{for }j,k=1,\ldots,\kappa. \tag*{(221)} S j k = ( d j d k ) 1/2 s j k for j , k = 1 , … , κ . (221) The square root factors in (D6) arise from counting the contributions arising from each entry of the column vector C e k \mathcal{C}e_k C e k
To evaluate the sum of interest, we exploit the fact that s j k s_{jk} s j k S j k S_{jk} S j k κ × κ \kappa\times\kappa κ × κ N × N N \times N N × N κ × κ \kappa\times\kappa κ × κ S − 1 S^{-1} S − 1 S S S
∑ ℓ = 1 κ S j ℓ S ℓ k − 1 = δ j k . (222) \sum_{\ell=1}^{\kappa}S_{j\ell}S^{-1}_{\ell k}=\delta_{jk}. \tag*{(222)} ℓ = 1 ∑ κ S j ℓ S ℓ k − 1 = δ j k . (222) From (D6) and (D7), it immediately follows that
S j k − 1 = ( d j d k ) − 1 / 2 s j k − 1 for j , k = 1 , … , κ , (223) S^{-1}_{jk}=(d_jd_k)^{-1/2}s^{-1}_{jk}\quad\text{for }j,k=1,\ldots,\kappa, \tag*{(223)} S j k − 1 = ( d j d k ) − 1/2 s j k − 1 for j , k = 1 , … , κ , (223) where s − 1 s^{-1} s − 1 s s s
With the matrix inverse S − 1 S^{-1} S − 1 N N N
v j = ∑ k = 1 κ S k j − 1 e k for j = 1 , … , κ . (224) v_j=\sum_{k=1}^{\kappa}S^{-1}_{kj}e_k \quad\text{for }j=1,\ldots,\kappa. \tag*{(224)} v j = k = 1 ∑ κ S k j − 1 e k for j = 1 , … , κ . (224) We now show that C − 1 e j = v j C^{-1}e_j=v_j C − 1 e j = v j C C C v j v_j v j C C C
C v j = ∑ k = 1 κ S k j − 1 ∑ ℓ = 1 κ S ℓ k e ℓ = ∑ ℓ = 1 κ ( ∑ k = 1 κ S ℓ k S k j − 1 ) e ℓ = e j , (225) \begin{aligned}
Cv_j&=\sum_{k=1}^{\kappa}S^{-1}_{kj}\sum_{\ell=1}^{\kappa}S_{\ell k}e_\ell\\
&=\sum_{\ell=1}^{\kappa}\left(\sum_{k=1}^{\kappa}S_{\ell k}S^{-1}_{kj}\right)e_\ell\\
&=e_j, \tag*{(225)}
\end{aligned} C v j = k = 1 ∑ κ S k j − 1 ℓ = 1 ∑ κ S ℓ k e ℓ = ℓ = 1 ∑ κ ( k = 1 ∑ κ S ℓ k S k j − 1 ) e ℓ = e j , (225) where on the second line we use the definition of the matrix inverse (D7) to replace the quantity in curved brackets with the Kronecker delta. Multiplying (D10) on the left by C − 1 C^{-1} C − 1
C − 1 e j = v j . (226) C^{-1}e_j=v_j. \tag*{(226)} C − 1 e j = v j . (226) Taking the inner product of (D11) from the left with e k T e_k^{\mathsf{T}} e k T
e k T C − 1 e j = S k j − 1 , (227) e_k^{\mathsf{T}}C^{-1}e_j=S^{-1}_{kj}, \tag*{(227)} e k T C − 1 e j = S k j − 1 , (227) from which we can now easily evaluate the quantity of interest.
We write the N N N μ \mu μ e j e_j e j
μ = ∑ j = 1 κ μ j d j e j , (228) \mu=\sum_{j=1}^{\kappa}\mu_j\sqrt{d_j}e_j, \tag*{(228)} μ = j = 1 ∑ κ μ j d j e j , (228) where μ j ≡ μ u ( γ j ) \mu_j\equiv\mu_u(\gamma_j) μ j ≡ μ u ( γ j ) γ j \gamma_j γ j
μ T C − 1 μ = ∑ j = 1 κ ∑ k = 1 κ μ j μ k d j d k e j T C − 1 e k = ∑ j = 1 κ ∑ k = 1 κ μ j μ k ( d j d k ) 1 / 2 S j k − 1 = ∑ j = 1 κ ∑ k = 1 κ μ j μ k s j k − 1 = μ ^ T s − 1 μ ^ , (229) \begin{aligned}
\mu^{\mathsf{T}}C^{-1}\mu
&=\sum_{j=1}^{\kappa}\sum_{k=1}^{\kappa}\mu_j\mu_k\sqrt{d_j}\sqrt{d_k}e_j^{\mathsf{T}}C^{-1}e_k\\
&=\sum_{j=1}^{\kappa}\sum_{k=1}^{\kappa}\mu_j\mu_k(d_jd_k)^{1/2}S^{-1}_{jk}\\
&=\sum_{j=1}^{\kappa}\sum_{k=1}^{\kappa}\mu_j\mu_ks^{-1}_{jk}\\
&=\hat{\mu}^{\mathsf{T}}s^{-1}\hat{\mu}, \tag*{(229)}
\end{aligned} μ T C − 1 μ = j = 1 ∑ κ k = 1 ∑ κ μ j μ k d j d k e j T C − 1 e k = j = 1 ∑ κ k = 1 ∑ κ μ j μ k ( d j d k ) 1/2 S j k − 1 = j = 1 ∑ κ k = 1 ∑ κ μ j μ k s j k − 1 = μ ^ T s − 1 μ ^ , (229) where on the second line we use (D12) to evaluate the matrix elements of C − 1 C^{-1} C − 1 S − 1 S^{-1} S − 1 s − 1 s^{-1} s − 1
The κ \kappa κ μ ^ \hat{\mu} μ ^
μ ^ = ( μ 1 , μ 2 , … , μ κ ) T . (230) \hat{\mu}=(\mu_1,\mu_2,\ldots,\mu_\kappa)^{\mathsf{T}}. \tag*{(230)} μ ^ = ( μ 1 , μ 2 , … , μ κ ) T . (230) Thus, N × N N\times N N × N μ T C − 1 μ \mu^{\mathsf{T}}C^{-1}\mu μ T C − 1 μ κ × κ \kappa\times\kappa κ × κ κ \kappa κ
An important feature of this result is that when there are enough pulsar pairs in a particular angular separation bin, then the variance is not significantly reduced by adding more pulsar pairs to that angular separation bin. Here, “enough pairs” means that the average value of the covariance matrix rows at that angle are approximately constant. This is because (D14) depends only upon the matrix s j k s_{jk} s j k
“Beating” the Cosmic Variance (A Simple Example)
Here, with a simple example [69 ] , we show that the variance of the binned optimal estimator with two distinct angles can be smaller than the cosmic variance for a point estimator at a single angle.
We take an angular-separation bin containing a large number of pulsar pairs with separation angles γ 1 \gamma_1 γ 1 γ 2 \gamma_2 γ 2 γ 1 ≈ 54 ∘ \gamma_1\approx54^\circ γ 1 ≈ 5 4 ∘ σ cos 2 \sigma_{\cos}^2 σ c o s 2 Figure 6 , and γ 2 ≈ 49 ∘ \gamma_2\approx49^\circ γ 2 ≈ 4 9 ∘ μ 2 ≡ μ u ( γ 2 ) = 0 \mu_2\equiv\mu_u(\gamma_2)=0 μ 2 ≡ μ u ( γ 2 ) = 0 smaller variance than the minimum value of the cosmic variance σ cos 2 ( γ 1 ) \sigma_{\cos}^2(\gamma_1) σ c o s 2 ( γ 1 ) both angular separations γ 1 \gamma_1 γ 1 γ 2 \gamma_2 γ 2 Figure 7 .
We assume that there are n p a i r s n_{\mathrm{pairs}} n pairs C C C 2 n p a i r s × 2 n p a i r s 2n_{\mathrm{pairs}}\times2n_{\mathrm{pairs}} 2 n pairs × 2 n pairs n p a i r s → ∞ n_{\mathrm{pairs}}\to\infty n pairs → ∞ 2 × 2 2\times2 2 × 2
s = s j k = ( s 11 s 12 s 12 s 22 ) . (231) s=s_{jk}=\begin{pmatrix}
s_{11} & s_{12}\\
s_{12} & s_{22}
\end{pmatrix}. \tag*{(231)} s = s j k = ( s 11 s 12 s 12 s 22 ) . (231) The four values that appear in s s s n p a i r s × n p a i r s n_{\mathrm{pairs}}\times n_{\mathrm{pairs}} n pairs × n pairs C j k C_{jk} C j k j , k = 1 , 2 j,k=1,2 j , k = 1 , 2
An explicit expression for s j k s_{jk} s j k (54)
s j k = ℏ 4 ∫ 0 π d β sin β ( μ + + ( γ j , β ) μ + + ( γ k , β ) + μ × × ( γ j , β ) μ × × ( γ k , β ) ) . (232) s_{jk}=\hbar^4\int_0^\pi\mathrm{d}\beta\sin\beta\left(\mu_{++}(\gamma_j,\beta)\mu_{++}(\gamma_k,\beta)+\mu_{\times\times}(\gamma_j,\beta)\mu_{\times\times}(\gamma_k,\beta)\right). \tag*{(232)} s j k = ℏ 4 ∫ 0 π d β sin β ( μ ++ ( γ j , β ) μ ++ ( γ k , β ) + μ ×× ( γ j , β ) μ ×× ( γ k , β ) ) . (232) As shown in (78) , for large n p a i r s n_{\mathrm{pairs}} n pairs s 11 ≡ σ 1 2 = σ c o s 2 ( γ 1 ) s_{11}\equiv\sigma_1^2=\sigma_{\mathrm{cos}}^2(\gamma_1) s 11 ≡ σ 1 2 = σ cos 2 ( γ 1 ) s 22 ≡ σ 2 2 = σ c o s 2 ( γ 2 ) s_{22}\equiv\sigma_2^2=\sigma_{\mathrm{cos}}^2(\gamma_2) s 22 ≡ σ 2 2 = σ cos 2 ( γ 2 ) s 12 2 ≤ s 11 s 22 s_{12}^2\le s_{11}s_{22} s 12 2 ≤ s 11 s 22 s s s
s 12 = r σ 1 σ 2 , (233) s_{12}=r\sigma_1\sigma_2, \tag*{(233)} s 12 = r σ 1 σ 2 , (233) where the Schwarz inequality ensures that the real number r ∈ [ − 1 , 1 ] r\in[-1,1] r ∈ [ − 1 , 1 ]
The variance σ o p t 2 \sigma_{\mathrm{opt}}^2 σ opt 2 μ T C − 1 μ \mu^{\mathsf T}C^{-1}\mu μ T C − 1 μ (26) . This is evaluated in (D14), in terms of the matrix s s s
μ b i n 2 σ o p t 2 = μ T C − 1 μ = μ ^ T s − 1 μ ^ = 1 s 11 s 22 − s 12 2 ( μ 1 2 s 22 + μ 2 2 s 11 − 2 μ 1 μ 2 s 12 ) = 1 1 − r 2 ( μ 1 2 σ 1 2 + μ 2 2 σ 2 2 − 2 r μ 1 μ 2 σ 1 σ 2 ) . (234) \begin{aligned}
\frac{\mu_{\mathrm{bin}}^2}{\sigma_{\mathrm{opt}}^2}=\mu^{\mathsf T}C^{-1}\mu=\hat{\mu}^{\mathsf T}s^{-1}\hat{\mu} \\
&=\frac{1}{s_{11}s_{22}-s_{12}^2}\left(\mu_1^2s_{22}+\mu_2^2s_{11}-2\mu_1\mu_2s_{12}\right) \\
&=\frac{1}{1-r^2}\left(\frac{\mu_1^2}{\sigma_1^2}+\frac{\mu_2^2}{\sigma_2^2}-2r\frac{\mu_1\mu_2}{\sigma_1\sigma_2}\right). \tag*{(234)}
\end{aligned} σ opt 2 μ bin 2 = μ T C − 1 μ = μ ^ T s − 1 μ ^ = s 11 s 22 − s 12 2 1 ( μ 1 2 s 22 + μ 2 2 s 11 − 2 μ 1 μ 2 s 12 ) = 1 − r 2 1 ( σ 1 2 μ 1 2 + σ 2 2 μ 2 2 − 2 r σ 1 σ 2 μ 1 μ 2 ) . (234) To obtain the second line, we explicitly inverted the 2 × 2 2\times2 2 × 2 μ ^ = ( μ 1 , μ 2 ) T \hat{\mu}=(\mu_1,\mu_2)^{\mathsf T} μ ^ = ( μ 1 , μ 2 ) T μ 1 ≡ μ u ( γ 1 ) \mu_1\equiv\mu_u(\gamma_1) μ 1 ≡ μ u ( γ 1 ) μ 2 ≡ μ u ( γ 2 ) \mu_2\equiv\mu_u(\gamma_2) μ 2 ≡ μ u ( γ 2 ) σ 1 2 \sigma_1^2 σ 1 2 σ 2 2 \sigma_2^2 σ 2 2 r r r s 11 s_{11} s 11 s 22 s_{22} s 22 s 12 s_{12} s 12
In signal processing and data analysis, the quantities
S o p t ≡ ℏ 2 μ b i n / σ o p t , S 1 ≡ ℏ 2 μ 1 / σ 1 , and S 2 ≡ ℏ 2 μ 2 / σ 2 , (235) S_{\mathrm{opt}}\equiv\hbar^2\mu_{\mathrm{bin}}/\sigma_{\mathrm{opt}},\quad S_1\equiv\hbar^2\mu_1/\sigma_1,\quad\text{and}\quad S_2\equiv\hbar^2\mu_2/\sigma_2, \tag*{(235)} S opt ≡ ℏ 2 μ bin / σ opt , S 1 ≡ ℏ 2 μ 1 / σ 1 , and S 2 ≡ ℏ 2 μ 2 / σ 2 , (235) which appear in (E4) are called expected SNRs. We include the factors of ℏ 2 \hbar^2 ℏ 2 ρ \rho ρ γ \gamma γ [70 ] ).
A simple geometric argument shows that combining data from the two angular-separation bins always
increases the SNR. First, multiply (E4) by ℏ 4 \hbar^4 ℏ 4
S o p t 2 = 1 1 − r 2 ( S 1 2 + S 2 2 − 2 r S 1 S 2 ) . (236) S_{\mathrm{opt}}^2=\frac{1}{1-r^2}\left(S_1^2+S_2^2-2rS_1S_2\right). \tag*{(236)} S opt 2 = 1 − r 2 1 ( S 1 2 + S 2 2 − 2 r S 1 S 2 ) . (236) Then introduce new variables x x x y y y
S 1 S o p t = x − y 2 , S 2 S o p t = x + y 2 . (237) \frac{S_1}{S_{\mathrm{opt}}}=\frac{x-y}{\sqrt{2}},\qquad\frac{S_2}{S_{\mathrm{opt}}}=\frac{x+y}{\sqrt{2}}. \tag*{(237)} S opt S 1 = 2 x − y , S opt S 2 = 2 x + y . (237) Equation (E6) satisfied by the three SNRs may now be written
x 2 1 + r + y 2 1 − r = 1 , (238) \frac{x^2}{1+r}+\frac{y^2}{1-r}=1, \tag*{(238)} 1 + r x 2 + 1 − r y 2 = 1 , (238) which is the equation of an ellipse with major radius 1 + ∣ r ∣ \sqrt{1+\lvert r\rvert} 1 + ∣ r ∣ 1 − ∣ r ∣ \sqrt{1-\lvert r\rvert} 1 − ∣ r ∣ 45 ∘ 45^\circ 4 5 ∘ S 1 S_1 S 1 S 2 S_2 S 2 Figure 22 . Note that 0 ≤ minor radius ≤ 1 ≤ major radius ≤ 2 0\le\text{minor radius}\le1\le\text{major radius}\le\sqrt{2} 0 ≤ minor radius ≤ 1 ≤ major radius ≤ 2 S o p t 2 ≥ S 1 2 S_{\mathrm{opt}}^2\ge S_1^2 S opt 2 ≥ S 1 2 S o p t 2 ≥ S 2 2 S_{\mathrm{opt}}^2\ge S_2^2 S opt 2 ≥ S 2 2 γ 1 \gamma_1 γ 1 γ 2 \gamma_2 γ 2
Figure 22. Equation (E6) constrains the ratios of SNRs, S 1 / S o p t S_1/S_{\mathrm{opt}} S 1 / S opt S 2 / S o p t S_2/S_{\mathrm{opt}} S 2 / S opt r ∈ [ − 1 , 1 ] r\in[-1,1] r ∈ [ − 1 , 1 ] r = 0 r=0 r = 0 r = ± 1 r=\pm1 r = ± 1 ± 45 ∘ \pm45^\circ ± 4 5 ∘ always lies within the square, proving geometrically that S o p t 2 ≥ S 1 2 S_{\mathrm{opt}}^2\ge S_1^2 S opt 2 ≥ S 1 2 S o p t 2 ≥ S 2 2 S_{\mathrm{opt}}^2\ge S_2^2 S opt 2 ≥ S 2 2
Some insight can be gained from two special cases: (i) If the correlation measurements at the two discrete angles are uncorrelated with one another (i.e., r = 0 r=0 r = 0
S o p t 2 = S 1 2 + S 2 2 . (239) S_{\mathrm{opt}}^2=S_1^2+S_2^2. \tag*{(239)} S opt 2 = S 1 2 + S 2 2 . (239) (ii) For perfectly correlated or anticorrelated correlation measurements r = ± 1 r=\pm1 r = ± 1 S 2 = ± S 1 S_2=\pm S_1 S 2 = ± S 1 Figure 22 , and one has
S o p t 2 = S 1 2 = S 2 2 . (240) S_{\mathrm{opt}}^2=S_1^2=S_2^2. \tag*{(240)} S opt 2 = S 1 2 = S 2 2 . (240) In this second case, the SNR of the optimal (binned) estimator has the same value as that for the (narrow-bin) estimator for either angle separately, since no new information is obtained by considering the second angle.
Finally, to show that σ o p t 2 \sigma_{\mathrm{opt}}^2 σ opt 2 γ 1 ≈ 54 ∘ \gamma_1\approx54^\circ γ 1 ≈ 5 4 ∘ γ 2 ≈ 49 ∘ \gamma_2\approx49^\circ γ 2 ≈ 4 9 ∘ μ 2 = 0 \mu_2=0 μ 2 = 0 μ b i n \mu_{\mathrm{bin}} μ bin μ 2 = 0 \mu_2=0 μ 2 = 0 μ 1 \mu_1 μ 1 S o p t 2 ≥ S 1 2 S_{\mathrm{opt}}^2\ge S_1^2 S opt 2 ≥ S 1 2
σ o p t 2 ≤ μ b i n 2 μ 1 2 σ 1 2 < σ 1 2 . (241) \sigma_{\mathrm{opt}}^2\le\frac{\mu_{\mathrm{bin}}^2}{\mu_1^2}\sigma_1^2<\sigma_1^2. \tag*{(241)} σ opt 2 ≤ μ 1 2 μ bin 2 σ 1 2 < σ 1 2 . (241) Since σ 1 2 \sigma_1^2 σ 1 2 σ o p t 2 \sigma_{\mathrm{opt}}^2 σ opt 2 Figure 7 . [For producing the plot in Figure 7 , we choose the normalization condition on the weights to be μ b i n ≡ μ T 1 / n p a i r s = ( μ 1 + μ 2 ) / 2 = μ 1 / 2 \mu_{\mathrm{bin}}\equiv\boldsymbol{\mu}^{\mathsf{T}}\mathbf{1}/n_{\mathrm{pairs}}=(\mu_1+\mu_2)/2=\mu_1/2 μ bin ≡ μ T 1 / n pairs = ( μ 1 + μ 2 ) /2 = μ 1 /2 σ o p t 2 ≤ σ 1 2 / 4 \sigma_{\mathrm{opt}}^2\le\sigma_1^2/4 σ opt 2 ≤ σ 1 2 /4
Third and fourth moments of the correlation
To calculate the variance of the χ 2 \chi^2 χ 2 § 11 A, we need the third and fourth moments of the correlations ρ a b \rho_{ab} ρ ab [11 ] . We used similar methods to compute the first and second moments in § 2 A and II B. For convenience, we repeat those expressions here:
⟨ ρ a b ⟩ = ℏ 2 μ a b , and \langle\rho_{ab}\rangle=\hbar^2\mu_{ab},\qquad\text{and} ⟨ ρ ab ⟩ = ℏ 2 μ ab , and ⟨ ρ a b ρ c d ⟩ = ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ + C a b , c d . (242) \langle\rho_{ab}\rho_{cd}\rangle=\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle+C_{ab,cd}. \tag*{(242)} ⟨ ρ ab ρ c d ⟩ = ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ + C ab , c d . (242) where
C a b , c d = ℏ 4 ( μ a c μ b d + μ a d μ b c ) , (243) C_{ab,cd}=\hbar^4\left(\mu_{ac}\mu_{bd}+\mu_{ad}\mu_{bc}\right), \tag*{(243)} C ab , c d = ℏ 4 ( μ a c μ b d + μ a d μ b c ) , (243) and
ℏ 2 ≡ 4 π ∫ d f ( 2 π f ) − 2 α H ( f ) , and (244) \hbar^2\equiv4\pi\int df\,(2\pi f)^{-2\alpha}H(f),\qquad\text{and} \tag*{(244)} ℏ 2 ≡ 4 π ∫ df ( 2 π f ) − 2 α H ( f ) , and (244) ℏ 4 ≡ ( 4 π ) 2 ∫ d f ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α H ( f ) H ( f ′ ) . (245) \hbar^4\equiv(4\pi)^2\int df\int df'\,\operatorname{sinc}^2\left(\pi(f-f')T\right)(4\pi^2ff')^{-2\alpha}H(f)H(f'). \tag*{(245)} ℏ 4 ≡ ( 4 π ) 2 ∫ df ∫ d f ′ sinc 2 ( π ( f − f ′ ) T ) ( 4 π 2 f f ′ ) − 2 α H ( f ) H ( f ′ ) . (245) The constant α \alpha α α = 0 \alpha=0 α = 0 ℏ \hbar ℏ α = 1 \alpha=1 α = 1 ℏ \hbar ℏ × \times ×
Since ρ a b \rho_{ab} ρ ab Z a Z_a Z a ρ a b \rho_{ab} ρ ab Z Z Z Z Z Z [19 ] to evaluate these expectation values, we obtain
⟨ ρ a b ρ c d ρ e f ⟩ = ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ + ⟨ ρ a b ⟩ C c d , e f + ⟨ ρ c d ⟩ C e f , a b + ⟨ ρ e f ⟩ C a b , c d + D a b , c d , e f , and (246) \langle\rho_{ab}\rho_{cd}\rho_{ef}\rangle=\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle\langle\rho_{ef}\rangle+\langle\rho_{ab}\rangle C_{cd,ef}+\langle\rho_{cd}\rangle C_{ef,ab}+\langle\rho_{ef}\rangle C_{ab,cd}+D_{ab,cd,ef},\qquad\text{and} \tag*{(246)} ⟨ ρ ab ρ c d ρ e f ⟩ = ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ + ⟨ ρ ab ⟩ C c d , e f + ⟨ ρ c d ⟩ C e f , ab + ⟨ ρ e f ⟩ C ab , c d + D ab , c d , e f , and (246) ⟨ ρ a b ρ c d ρ e f ρ g h ⟩ = ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ ⟨ ρ g h ⟩ + ⟨ ρ a b ⟩ ⟨ ρ c d ⟩ C e f , g h + ⟨ ρ a b ⟩ ⟨ ρ e f ⟩ C g h , c d + ⟨ ρ a b ⟩ ⟨ ρ g h ⟩ C c d , e f + ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ C g h , a b + ⟨ ρ c d ⟩ ⟨ ρ g h ⟩ C a b , e f + ⟨ ρ e f ⟩ ⟨ ρ g h ⟩ C a b , c d + ⟨ ρ a b ⟩ D c d , e f , g h + ⟨ ρ c d ⟩ D e f , g h , a b + ⟨ ρ e f ⟩ D g h , a b , c d + ⟨ ρ g h ⟩ D a b , c d , e f + C a b , c d C e f , g h + C a b , e f C c d , g h + C a b , g h C c d , e f + E a b , c d , e f , g h . (247) \begin{aligned}
\langle\rho_{ab}\rho_{cd}\rho_{ef}\rho_{gh}\rangle={}&\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle\langle\rho_{ef}\rangle\langle\rho_{gh}\rangle\\
&+\langle\rho_{ab}\rangle\langle\rho_{cd}\rangle C_{ef,gh}+\langle\rho_{ab}\rangle\langle\rho_{ef}\rangle C_{gh,cd}+\langle\rho_{ab}\rangle\langle\rho_{gh}\rangle C_{cd,ef}\\
&+\langle\rho_{cd}\rangle\langle\rho_{ef}\rangle C_{gh,ab}+\langle\rho_{cd}\rangle\langle\rho_{gh}\rangle C_{ab,ef}+\langle\rho_{ef}\rangle\langle\rho_{gh}\rangle C_{ab,cd}\\
&+\langle\rho_{ab}\rangle D_{cd,ef,gh}+\langle\rho_{cd}\rangle D_{ef,gh,ab}+\langle\rho_{ef}\rangle D_{gh,ab,cd}+\langle\rho_{gh}\rangle D_{ab,cd,ef}\\
&+C_{ab,cd}C_{ef,gh}+C_{ab,ef}C_{cd,gh}+C_{ab,gh}C_{cd,ef}+E_{ab,cd,ef,gh}. \tag*{(247)}
\end{aligned} ⟨ ρ ab ρ c d ρ e f ρ g h ⟩ = ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ ⟨ ρ g h ⟩ + ⟨ ρ ab ⟩ ⟨ ρ c d ⟩ C e f , g h + ⟨ ρ ab ⟩ ⟨ ρ e f ⟩ C g h , c d + ⟨ ρ ab ⟩ ⟨ ρ g h ⟩ C c d , e f + ⟨ ρ c d ⟩ ⟨ ρ e f ⟩ C g h , ab + ⟨ ρ c d ⟩ ⟨ ρ g h ⟩ C ab , e f + ⟨ ρ e f ⟩ ⟨ ρ g h ⟩ C ab , c d + ⟨ ρ ab ⟩ D c d , e f , g h + ⟨ ρ c d ⟩ D e f , g h , ab + ⟨ ρ e f ⟩ D g h , ab , c d + ⟨ ρ g h ⟩ D ab , c d , e f + C ab , c d C e f , g h + C ab , e f C c d , g h + C ab , g h C c d , e f + E ab , c d , e f , g h . (247) where
D a b , c d , e f = 4 ℏ 6 ( μ b c μ d e μ f a + μ a c μ d e μ f b ) , and (248) D_{ab,cd,ef}=4\hbar^6\left(\mu_{bc}\mu_{de}\mu_{fa}+\mu_{ac}\mu_{de}\mu_{fb}\right),\qquad\text{and} \tag*{(248)} D ab , c d , e f = 4 ℏ 6 ( μ b c μ d e μ f a + μ a c μ d e μ f b ) , and (248) E a b , c d , e f , g h = 8 ℏ 8 ( μ b ( c μ d ) ( e μ f ) ( g μ h ) a + μ b ( e μ f ) ( g μ h ) ( c μ d ) a + μ b ( g μ h ) ( c μ d ) ( e μ f ) a + a ↔ b ) , (249) \mathcal{E}_{ab,cd,ef,gh}=8\hbar^{8}\left(\mu_b(c\mu_d)(e\mu_f)(g\mu_h)a+\mu_b(e\mu_f)(g\mu_h)(c\mu_d)a+\mu_b(g\mu_h)(c\mu_d)(e\mu_f)a+a\leftrightarrow b\right), \tag*{(249)} E ab , c d , e f , g h = 8 ℏ 8 ( μ b ( c μ d ) ( e μ f ) ( g μ h ) a + μ b ( e μ f ) ( g μ h ) ( c μ d ) a + μ b ( g μ h ) ( c μ d ) ( e μ f ) a + a ↔ b ) , (249) and
ℏ 6 ≡ ( 4 π ) 3 ∫ d f ∫ d f ′ ∫ d f ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) sinc ( π ( f ′ ′ − f ) T ) \hbar^{6}\equiv(4\pi)^3\int df\int df'\int df''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right)\operatorname{sinc}\left(\pi(f''-f)T\right) ℏ 6 ≡ ( 4 π ) 3 ∫ df ∫ d f ′ ∫ d f ′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) sinc ( π ( f ′′ − f ) T ) × ( 8 π 3 f f ′ f ′ ′ ) − 2 α H ( f ) H ( f ′ ) H ( f ′ ′ ) , and (250) {}\times(8\pi^3ff'f'')^{-2\alpha}H(f)H(f')H(f''),\qquad\text{and} \tag*{(250)} × ( 8 π 3 f f ′ f ′′ ) − 2 α H ( f ) H ( f ′ ) H ( f ′′ ) , and (250) ℏ 8 ≡ ( 4 π ) 4 ∫ d f ∫ d f ′ ∫ d f ′ ′ ∫ d f ′ ′ ′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′ ′ ) T ) \hbar^{8}\equiv(4\pi)^4\int df\int df'\int df''\int df'''\,\operatorname{sinc}\left(\pi(f-f')T\right)\operatorname{sinc}\left(\pi(f'-f'')T\right) ℏ 8 ≡ ( 4 π ) 4 ∫ df ∫ d f ′ ∫ d f ′′ ∫ d f ′′′ sinc ( π ( f − f ′ ) T ) sinc ( π ( f ′ − f ′′ ) T ) × sinc ( π ( f ′ ′ − f ′ ′ ′ ) T ) sinc ( π ( f ′ ′ ′ − f ) T ) ( 16 π 4 f f ′ f ′ ′ f ′ ′ ′ ) − 2 α H ( f ) H ( f ′ ) H ( f ′ ′ ) H ( f ′ ′ ′ ) . (251) {}\times\operatorname{sinc}\left(\pi(f''-f''')T\right)\operatorname{sinc}\left(\pi(f'''-f)T\right)(16\pi^4ff'f''f''')^{-2\alpha}H(f)H(f')H(f'')H(f'''). \tag*{(251)} × sinc ( π ( f ′′ − f ′′′ ) T ) sinc ( π ( f ′′′ − f ) T ) ( 16 π 4 f f ′ f ′′ f ′′′ ) − 2 α H ( f ) H ( f ′ ) H ( f ′′ ) H ( f ′′′ ) . (251) The quantities C a b , c d \mathcal{C}_{ab,cd} C ab , c d D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h μ \mu μ a b ab ab c d cd c d a b ↔ c d ab\leftrightarrow cd ab ↔ c d h 2 h^2 h 2 ℏ 2 \hbar^2 ℏ 2 ℏ 2 \hbar^2 ℏ 2 ℏ 2 \hbar^2 ℏ 2 ℏ 2 ≤ ℏ 2 ≤ ℏ 2 ≤ ℏ 2 \hbar^2\leq\hbar^2\leq\hbar^2\leq\hbar^2 ℏ 2 ≤ ℏ 2 ≤ ℏ 2 ≤ ℏ 2 D a b , c d , e f \mathcal{D}_{ab,cd,ef} D ab , c d , e f E a b , c d , e f , g h \mathcal{E}_{ab,cd,ef,gh} E ab , c d , e f , g h [71 ] of ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab ρ a b \rho_{ab} ρ ab Z a Z_a Z a Z b Z_b Z b
Using the property H ( f ) = H ( − f ) H(f)=H(-f) H ( f ) = H ( − f ) ℏ 6 \hbar^6 ℏ 6 ℏ 8 \hbar^8 ℏ 8 f m f_m f m f n f_n f n
P m n ≡ sinc ( π ( f m + f n ) T ) and M m n ≡ sinc ( π ( f m − f n ) T ) , (252) P_{mn}\equiv\operatorname{sinc}\left(\pi(f_m+f_n)T\right)\qquad\text{and}\qquad M_{mn}\equiv\operatorname{sinc}\left(\pi(f_m-f_n)T\right), \tag*{(252)} P mn ≡ sinc ( π ( f m + f n ) T ) and M mn ≡ sinc ( π ( f m − f n ) T ) , (252) where P P P M M M (7) , (10) , (F9), and (F10) as
h 2 = 8 π ∫ 0 ∞ d f 1 ( 2 π f 1 ) − 2 α H ( f ) , (253) h^2=8\pi\int_0^\infty df_1\,(2\pi f_1)^{-2\alpha}H(f), \tag*{(253)} h 2 = 8 π ∫ 0 ∞ d f 1 ( 2 π f 1 ) − 2 α H ( f ) , (253) ℏ 4 = 1 2 ( 8 π ) 2 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ( 4 π 2 f 1 f 2 ) − 2 α H ( f 1 ) H ( f 2 ) [ M 12 M 21 + P 12 P 21 ] , (254) \hbar^4=\frac{1}{2}(8\pi)^2\int_0^\infty df_1\int_0^\infty df_2\,(4\pi^2f_1f_2)^{-2\alpha}H(f_1)H(f_2)\left[M_{12}M_{21}+P_{12}P_{21}\right], \tag*{(254)} ℏ 4 = 2 1 ( 8 π ) 2 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ( 4 π 2 f 1 f 2 ) − 2 α H ( f 1 ) H ( f 2 ) [ M 12 M 21 + P 12 P 21 ] , (254) ℏ 6 = 1 2 2 ( 8 π ) 3 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ∫ 0 ∞ d f 3 ( 8 π 3 f 1 f 2 f 3 ) − 2 α H ( f 1 ) H ( f 2 ) H ( f 3 ) [ M 12 M 23 M 31 + 3 M 12 P 23 P 31 ] , (255) \hbar^6=\frac{1}{2^2}(8\pi)^3\int_0^\infty df_1\int_0^\infty df_2\int_0^\infty df_3\,(8\pi^3f_1f_2f_3)^{-2\alpha}H(f_1)H(f_2)H(f_3)\left[M_{12}M_{23}M_{31}+3M_{12}P_{23}P_{31}\right], \tag*{(255)} ℏ 6 = 2 2 1 ( 8 π ) 3 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ∫ 0 ∞ d f 3 ( 8 π 3 f 1 f 2 f 3 ) − 2 α H ( f 1 ) H ( f 2 ) H ( f 3 ) [ M 12 M 23 M 31 + 3 M 12 P 23 P 31 ] , (255) ℏ 8 = 1 2 3 ( 8 π ) 4 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ∫ 0 ∞ d f 3 ∫ 0 ∞ d f 4 ( 16 π 4 f 1 f 2 f 3 f 4 ) − 2 α H ( f 1 ) H ( f 2 ) H ( f 3 ) H ( f 4 ) \hbar^8=\frac{1}{2^3}(8\pi)^4\int_0^\infty df_1\int_0^\infty df_2\int_0^\infty df_3\int_0^\infty df_4\,(16\pi^4f_1f_2f_3f_4)^{-2\alpha}H(f_1)H(f_2)H(f_3)H(f_4) ℏ 8 = 2 3 1 ( 8 π ) 4 ∫ 0 ∞ d f 1 ∫ 0 ∞ d f 2 ∫ 0 ∞ d f 3 ∫ 0 ∞ d f 4 ( 16 π 4 f 1 f 2 f 3 f 4 ) − 2 α H ( f 1 ) H ( f 2 ) H ( f 3 ) H ( f 4 ) × [ M 12 M 23 M 34 M 41 + P 12 P 23 P 34 P 41 + 4 M 12 M 23 P 34 P 41 + 2 M 12 P 23 M 34 P 41 ] . (256) {}\times\left[M_{12}M_{23}M_{34}M_{41}+P_{12}P_{23}P_{34}P_{41}+4M_{12}M_{23}P_{34}P_{41}+2M_{12}P_{23}M_{34}P_{41}\right]. \tag*{(256)} × [ M 12 M 23 M 34 M 41 + P 12 P 23 P 34 P 41 + 4 M 12 M 23 P 34 P 41 + 2 M 12 P 23 M 34 P 41 ] . (256) We now evaluate these integrals for two cases of interest.
The first case of interest is the stochastic GW background model of Appendix B, for which H ( f ) H(f) H ( f ) f 0 f_0 f 0 − 7 / 3 -7/3 − 7/3 f 0 f_0 f 0 1 / T 1/T 1/ T T T T
Table 3. The coefficients h 2 h^2 h 2 ℏ 2 \hbar^2 ℏ 2 ℏ 2 \hbar^2 ℏ 2 f 0 = 1 / T f_0=1/T f 0 = 1/ T
variable u u u d u = f − 7 / 3 − 2 α d f \mathrm{d}u=f^{-7/3-2\alpha}\mathrm{d}f d u = f − 7/3 − 2 α d f M M M P P P [72 ] . The results are shown in Figure 23 , and Table III gives the values for the nominal case f 0 T = 1 f_0T=1 f 0 T = 1
The second case of interest assumes that the spectrum H ( f ) H(f) H ( f ) P m n = 0 P_{mn}=0 P mn = 0 M m n = 1 M_{mn}=1 M mn = 1 ℏ 4 / h 4 = 1 / 2 \hbar^4/h^4=1/2 ℏ 4 / h 4 = 1/2 ℏ 6 / h 6 = 1 / 4 \hbar^6/h^6=1/4 ℏ 6 / h 6 = 1/4 ℏ 8 / h 8 = 1 / 8 \hbar^8/h^8=1/8 ℏ 8 / h 8 = 1/8 ℏ 2 / h 2 = 2 − 1 / 2 ≈ 0.7071 \hbar^2/h^2=2^{-1/2}\approx0.7071 ℏ 2 / h 2 = 2 − 1/2 ≈ 0.7071 ℏ 2 / h 2 = 2 − 2 / 3 ≈ 0.6300 \hbar^2/h^2=2^{-2/3}\approx0.6300 ℏ 2 / h 2 = 2 − 2/3 ≈ 0.6300 ℏ 2 / h 2 = 2 − 3 / 4 ≈ 0.5946 \hbar^2/h^2=2^{-3/4}\approx0.5946 ℏ 2 / h 2 = 2 − 3/4 ≈ 0.5946 α = 1 \alpha=1 α = 1 α = 0 \alpha=0 α = 0
Expressions for E \mathbb{E} E E ^ \hat{\mathbb{E}} E ^ N p u l N_{\mathrm{pul}} N pul
In this appendix, we derive analytic expressions for E \mathbb{E} E E ^ \hat{\mathbb{E}} E ^ N p u l N_{\mathrm{pul}} N pul χ 2 \chi^2 χ 2 § 11 C 2.
Our final expressions may be easily evaluated for GW detectors that respond to only a finite number of harmonic modes (l ≤ L max l\le L_{\max} l ≤ L m a x
We do not know how to evaluate these expressions for detectors that are sensitive to all modes, or if pulsar terms are present, both of which are true for PTAs. In these cases, the sums over modes l l l
We begin by finding a large-N p u l N_{\mathrm{pul}} N pul (87) and (88) :
μ a b = ( 1 + η 2 δ a b ) U a b , U a b = ∑ l Q l P l ( cos γ a b ) . (257) \mu_{ab}=(1+\eta^2\delta_{ab})U_{ab},\qquad U_{ab}=\sum_l Q_lP_l(\cos\gamma_{ab}). \tag*{(257)} μ ab = ( 1 + η 2 δ ab ) U ab , U ab = l ∑ Q l P l ( cos γ ab ) . (257) A PTA detector corresponds to Q l = ( 2 l + 1 ) C l Q_l=(2l+1)C_l Q l = ( 2 l + 1 ) C l η = 1 \eta=1 η = 1 C l C_l C l (60) .
To include the pulsar terms in the large-N p u l N_{\mathrm{pul}} N pul
∑ a ≃ N p u l 4 π ∫ d p ^ a . (258) \sum_a \simeq\frac{N_{\mathrm{pul}}}{4\pi}\int\mathrm{d}\hat{p}_a. \tag*{(258)} a ∑ ≃ 4 π N pul ∫ d p ^ a . (258) Here, and throughout this appendix, we use ≃ \simeq ≃ N p u l → ∞ N_{\mathrm{pul}}\to\infty N pul → ∞ 1 1 1 N p u l N_{\mathrm{pul}} N pul δ a b \delta_{ab} δ ab
δ a b ≃ 4 π N p u l δ 2 ( p ^ a , p ^ b ) , (259) \delta_{ab}\simeq\frac{4\pi}{N_{\mathrm{pul}}}\delta^2(\hat{p}_a,\hat{p}_b), \tag*{(259)} δ ab ≃ N pul 4 π δ 2 ( p ^ a , p ^ b ) , (259) where δ 2 ( p ^ a , p ^ b ) \delta^2(\hat{p}_a,\hat{p}_b) δ 2 ( p ^ a , p ^ b ) S 2 S^2 S 2
Lastly, we expand the two-dimensional Dirac delta function as a sum of Legendre polynomials. Because the spherical harmonics are a complete orthonormal set on S 2 S^2 S 2
δ 2 ( p ^ a , p ^ b ) = ∑ l = 0 ∞ ∑ m = − l l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (260) \delta^2(\hat{p}_a,\hat{p}_b)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}Y_{lm}(\hat{p}_a)Y^*_{lm}(\hat{p}_b). \tag*{(260)} δ 2 ( p ^ a , p ^ b ) = l = 0 ∑ ∞ m = − l ∑ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (260) (Note that, in the remainder of this section, double sums of this form are indicated by ∑ l m \sum_{lm} ∑ l m (84) to carry out the sum over m m m
δ a b ≃ 4 π N p u l ∑ l 2 l + 1 4 π P l ( cos γ a b ) . (261) \delta_{ab}\simeq\frac{4\pi}{N_{\mathrm{pul}}}\sum_l\frac{2l+1}{4\pi}P_l(\cos\gamma_{ab}). \tag*{(261)} δ ab ≃ N pul 4 π l ∑ 4 π 2 l + 1 P l ( cos γ ab ) . (261) Thus, defining U ( 0 ) = U a a = ∑ l Q l U(0)=U_{aa}=\sum_lQ_l U ( 0 ) = U aa = ∑ l Q l
μ a b ≃ ∑ l ( Q l + η 2 U ( 0 ) 2 l + 1 N p u l ) P l ( cos γ a b ) = ∑ l m 4 π ( Q l 2 l + 1 + η 2 U ( 0 ) N p u l ) Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) = ∑ l m Λ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (262) \begin{aligned}
\mu_{ab}&\simeq\sum_l\left(Q_l+\eta^2U(0)\frac{2l+1}{N_{\mathrm{pul}}}\right)P_l(\cos\gamma_{ab})\\
&=\sum_{lm}4\pi\left(\frac{Q_l}{2l+1}+\frac{\eta^2U(0)}{N_{\mathrm{pul}}}\right)Y_{lm}(\hat{p}_a)Y^*_{lm}(\hat{p}_b)\\
&=\sum_{lm}\Lambda_lY_{lm}(\hat{p}_a)Y^*_{lm}(\hat{p}_b). \tag*{(262)}
\end{aligned} μ ab ≃ l ∑ ( Q l + η 2 U ( 0 ) N pul 2 l + 1 ) P l ( cos γ ab ) = l m ∑ 4 π ( 2 l + 1 Q l + N pul η 2 U ( 0 ) ) Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) = l m ∑ Λ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (262) where the second line follows from the addition theorem for spherical harmonics, (84) , and the third line defines the expansion coefficients
Λ l ≡ 4 π ( Q l 2 l + 1 + η 2 U ( 0 ) N p u l ) . (263) \Lambda_l \equiv4\pi\left(\frac{Q_l}{2l+1}+\frac{\eta^2 U(0)}{N_{\rm pul}}\right). \tag*{(263)} Λ l ≡ 4 π ( 2 l + 1 Q l + N pul η 2 U ( 0 ) ) . (263) Although we are examining the case N p u l → ∞ N_{\rm pul}\to\infty N pul → ∞ 1 / N p u l 1/N_{\rm pul} 1/ N pul Q l / ( 2 l + 1 ) Q_l/(2l+1) Q l / ( 2 l + 1 ) l l l l l l l l l 1 / N p u l 1/N_{\rm pul} 1/ N pul Λ l \Lambda_l Λ l
Equations (G6) and (G7) are an eigenvector/eigenvalue decomposition of the N p u l × N p u l N_{\rm pul}\times N_{\rm pul} N pul × N pul μ a b \mu_{ab} μ ab N p u l N_{\rm pul} N pul Y l m ( Ω ^ ) Y_{lm}(\hat{\Omega}) Y l m ( Ω ^ ) p ^ a \hat{p}_a p ^ a p ^ b \hat{p}_b p ^ b Λ l \Lambda_l Λ l m m m
We now do an analogous harmonic expansion of C a b C_{ab} C ab (177) shows that its elements are proportional to the squares of the elements of (G1), which is written down explicitly in (114) :
C a b = 2 h 4 ( U a b 2 + ( 2 η 2 + η 4 ) U 2 ( 0 ) δ a b ) . (264) C_{ab}=2\mathcal{h}^4\left(U_{ab}^2+(2\eta^2+\eta^4)U^2(0)\delta_{ab}\right). \tag*{(264)} C ab = 2 h 4 ( U ab 2 + ( 2 η 2 + η 4 ) U 2 ( 0 ) δ ab ) . (264) Each of the terms on the rhs is a function only of the angular separation γ a b \gamma_{ab} γ ab p ^ a \hat{p}_a p ^ a p ^ b \hat{p}_b p ^ b C a b C_{ab} C ab cos γ a b \cos\gamma_{ab} cos γ ab δ a b \delta_{ab} δ ab U a b 2 U_{ab}^2 U ab 2
U a b 2 = ∑ l D l P l ( cos γ a b ) . (265) U_{ab}^2=\sum_l D_lP_l(\cos\gamma_{ab}). \tag*{(265)} U ab 2 = l ∑ D l P l ( cos γ ab ) . (265) where the D l D_l D l (65) . This gives
D l = 2 l + 1 2 ∫ − 1 1 d x U a b 2 ( x ) P l ( x ) = 2 l + 1 2 ∑ l ′ ∑ l ′ ′ Q l ′ Q l ′ ′ ∫ − 1 1 d x P l ′ ( x ) P l ′ ′ ( x ) P l ( x ) = ( 2 l + 1 ) ∑ l ′ ∑ l ′ ′ Q l ′ Q l ′ ′ ( l l ′ l ′ ′ 0 0 0 ) 2 . (266) \begin{aligned}
D_l&=\frac{2l+1}{2}\int_{-1}^{1}dx\,U_{ab}^2(x)P_l(x)\\
&=\frac{2l+1}{2}\sum_{l'}\sum_{l''}Q_{l'}Q_{l''}\int_{-1}^{1}dx\,P_{l'}(x)P_{l''}(x)P_l(x)\\
&=(2l+1)\sum_{l'}\sum_{l''}Q_{l'}Q_{l''}
\begin{pmatrix}
l&l'&l''\\
0&0&0
\end{pmatrix}^{2}. \tag*{(266)}
\end{aligned} D l = 2 2 l + 1 ∫ − 1 1 d x U ab 2 ( x ) P l ( x ) = 2 2 l + 1 l ′ ∑ l ′′ ∑ Q l ′ Q l ′′ ∫ − 1 1 d x P l ′ ( x ) P l ′′ ( x ) P l ( x ) = ( 2 l + 1 ) l ′ ∑ l ′′ ∑ Q l ′ Q l ′′ ( l 0 l ′ 0 l ′′ 0 ) 2 . (266) where the Wigner 3j symbols [53 , 54 ] are defined by an integral over the two-sphere S 2 S^2 S 2 (84) to write the Legendre polynomials in terms of spherical harmonics yields
C a b ≃ ∑ l m λ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (267) C_{ab}\simeq\sum_{lm}\lambda_lY_{lm}(\hat{p}_a)Y_{lm}^{*}(\hat{p}_b). \tag*{(267)} C ab ≃ l m ∑ λ l Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (267) where the expansion coefficients are
λ l ≡ 8 π h 4 ( D l 2 l + 1 + ( 2 η 2 + η 4 ) U 2 ( 0 ) N p u l ) . (268) \lambda_l\equiv8\pi\mathcal{h}^4\left(\frac{D_l}{2l+1}+\frac{(2\eta^2+\eta^4)U^2(0)}{N_{\rm pul}}\right). \tag*{(268)} λ l ≡ 8 π h 4 ( 2 l + 1 D l + N pul ( 2 η 2 + η 4 ) U 2 ( 0 ) ) . (268) When there are many pulsars, (G11) is an eigenvector/eigenvalue decomposition of C a b C_{ab} C ab μ a b \mu_{ab} μ ab
In the many-pulsar limit, we can also expand C a b − 1 C^{-1}_{ab} C ab − 1
C a b − 1 ≃ ( 4 π N p u l ) 2 ∑ { l ∣ λ l ≠ 0 } ∑ m = − l l λ l − 1 Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (269) C^{-1}_{ab}\simeq\left(\frac{4\pi}{N_{\rm pul}}\right)^2\sum_{\{l\mid\lambda_l\ne0\}}\sum_{m=-l}^{l}\lambda_l^{-1}Y_{lm}(\hat{p}_a)Y_{lm}^{*}(\hat{p}_b). \tag*{(269)} C ab − 1 ≃ ( N pul 4 π ) 2 { l ∣ λ l = 0 } ∑ m = − l ∑ l λ l − 1 Y l m ( p ^ a ) Y l m ∗ ( p ^ b ) . (269) The sum is restricted to those values of l l l λ l \lambda_l λ l λ l \lambda_l λ l λ l \lambda_l λ l C a b − 1 C^{-1}_{ab} C ab − 1
It is straightforward to verify that (G13) is the inverse of C a b C_{ab} C ab ∑ b C a b − 1 C b c \sum_b C^{-1}_{ab}C_{bc} ∑ b C ab − 1 C b c b b b δ 2 ( p ^ a , p ^ c ) \delta^2(\hat{p}_a,\hat{p}_c) δ 2 ( p ^ a , p ^ c ) δ a c \delta_{ac} δ a c
We now use these expressions to evaluate E \mathbb{E} E (181) . The two C − 1 C^{-1} C − 1 μ a b \mu_{ab} μ ab a a a b b b c c c d d d
E ≃ 16 h 8 ∑ { l 1 ∣ λ l 1 ≠ 0 } ∑ m 1 ∑ { l 2 ∣ λ l 2 ≠ 0 } ∑ m 2 ∑ l 3 m 3 ∑ l 4 m 4 ∑ l 5 m 5 ∑ l 6 m 6 λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 ( 2 X + Y ) . (270) \mathbb{E}\simeq16\mathcal{h}^8\sum_{\{l_1\mid\lambda_{l_1}\ne0\}}\sum_{m_1}\sum_{\{l_2\mid\lambda_{l_2}\ne0\}}\sum_{m_2}\sum_{l_3m_3}\sum_{l_4m_4}\sum_{l_5m_5}\sum_{l_6m_6}\lambda_{l_1}^{-1}\lambda_{l_2}^{-1}\Lambda_{l_3}\Lambda_{l_4}\Lambda_{l_5}\Lambda_{l_6}(2X+Y). \tag*{(270)} E ≃ 16 h 8 { l 1 ∣ λ l 1 = 0 } ∑ m 1 ∑ { l 2 ∣ λ l 2 = 0 } ∑ m 2 ∑ l 3 m 3 ∑ l 4 m 4 ∑ l 5 m 5 ∑ l 6 m 6 ∑ λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 ( 2 X + Y ) . (270) where
X ≡ ( − 1 ) m 1 + m 2 + m 3 + m 4 + m 5 + m 6 I l 1 m 1 ; l 3 m 3 ; l 6 , − m 6 I l 1 , − m 1 ; l 3 , − m 3 ; l 4 m 4 I l 2 m 2 ; l 4 , − m 4 ; l 5 m 5 I l 2 , − m 2 ; l 3 , − m 3 ; l 6 m 6 , X\equiv(-1)^{m_1+m_2+m_3+m_4+m_5+m_6}I_{l_1m_1;l_3m_3;l_6,-m_6}I_{l_1,-m_1;l_3,-m_3;l_4m_4}I_{l_2m_2;l_4,-m_4;l_5m_5}I_{l_2,-m_2;l_3,-m_3;l_6m_6}, X ≡ ( − 1 ) m 1 + m 2 + m 3 + m 4 + m 5 + m 6 I l 1 m 1 ; l 3 m 3 ; l 6 , − m 6 I l 1 , − m 1 ; l 3 , − m 3 ; l 4 m 4 I l 2 m 2 ; l 4 , − m 4 ; l 5 m 5 I l 2 , − m 2 ; l 3 , − m 3 ; l 6 m 6 , Y ≡ ( − 1 ) m 1 + m 2 + m 3 + m 4 + m 5 + m 6 I l 1 m 1 ; l 3 m 3 ; l 6 , − m 6 I l 1 , − m 1 ; l 4 , − m 4 ; l 5 m 5 I l 2 m 2 ; l 5 , − m 5 ; l 6 m 6 I l 2 , − m 2 ; l 3 , − m 3 ; l 4 m 4 . (271) Y\equiv(-1)^{m_1+m_2+m_3+m_4+m_5+m_6}I_{l_1m_1;l_3m_3;l_6,-m_6}I_{l_1,-m_1;l_4,-m_4;l_5m_5}I_{l_2m_2;l_5,-m_5;l_6m_6}I_{l_2,-m_2;l_3,-m_3;l_4m_4}. \tag*{(271)} Y ≡ ( − 1 ) m 1 + m 2 + m 3 + m 4 + m 5 + m 6 I l 1 m 1 ; l 3 m 3 ; l 6 , − m 6 I l 1 , − m 1 ; l 4 , − m 4 ; l 5 m 5 I l 2 m 2 ; l 5 , − m 5 ; l 6 m 6 I l 2 , − m 2 ; l 3 , − m 3 ; l 4 m 4 . (271) are products of integrals of spherical harmonics, obtained from the integrals that replaced the sums over a a a b b b c c c d d d
The I I I [53 , 54 ] as
I l m ; l ′ m ′ ; l ′ ′ m ′ ′ ≡ ∫ d Ω ^ Y l m ( Ω ^ ) Y l ′ m ′ ( Ω ^ ) Y l ′ ′ m ′ ′ ( Ω ^ ) = ( 2 l + 1 ) ( 2 l ′ + 1 ) ( 2 l ′ ′ + 1 ) 4 π ( l l ′ l ′ ′ 0 0 0 ) ( l l ′ l ′ ′ m m ′ m ′ ′ ) . (272) \begin{aligned}
I_{lm;l'm';l''m''} \equiv\int\mathrm{d}\hat{\Omega}\,Y_{lm}(\hat{\Omega})Y_{l'm'}(\hat{\Omega})Y_{l''m''}(\hat{\Omega}) \\
&= \sqrt{\frac{(2l+1)(2l'+1)(2l''+1)}{4\pi}}
\begin{pmatrix} l & l' & l'' \\ 0 & 0 & 0 \end{pmatrix}
\begin{pmatrix} l & l' & l'' \\ m & m' & m'' \end{pmatrix}. \tag*{(272)}
\end{aligned} I l m ; l ′ m ′ ; l ′′ m ′′ ≡ ∫ d Ω ^ Y l m ( Ω ^ ) Y l ′ m ′ ( Ω ^ ) Y l ′′ m ′′ ( Ω ^ ) = 4 π ( 2 l + 1 ) ( 2 l ′ + 1 ) ( 2 l ′′ + 1 ) ( l 0 l ′ 0 l ′′ 0 ) ( l m l ′ m ′ l ′′ m ′′ ) . (272) Where they are needed to avoid ambiguity, we add commas within the subscripts of the I I I
The integrals I l m ; l ′ m ′ ; l ′ ′ m ′ ′ I_{lm;l'm';l''m''} I l m ; l ′ m ′ ; l ′′ m ′′ l m lm l m l ′ m ′ l'm' l ′ m ′ l ′ ′ m ′ ′ l''m'' l ′′ m ′′ m + m ′ + m ′ ′ = 0 m+m'+m''=0 m + m ′ + m ′′ = 0 Y l m ( θ , ϕ ) ∝ e i m ϕ Y_{lm}(\theta,\phi)\propto e^{im\phi} Y l m ( θ , ϕ ) ∝ e im ϕ Ω ^ → − Ω ^ \hat{\Omega}\to-\hat{\Omega} Ω ^ → − Ω ^ Ω ^ \hat{\Omega} Ω ^ [73 ] . Under this transformation, a point with spherical coordinates ( θ , ϕ ) (\theta,\phi) ( θ , ϕ ) ( π − θ , ϕ + π ) (\pi-\theta,\phi+\pi) ( π − θ , ϕ + π ) d Ω ^ \mathrm{d}\hat{\Omega} d Ω ^ d Ω ^ \mathrm{d}\hat{\Omega} d Ω ^ d ( − Ω ^ ) \mathrm{d}(-\hat{\Omega}) d ( − Ω ^ ) − Ω ^ -\hat{\Omega} − Ω ^ Y l m ∝ e i m ϕ P l m ( cos θ ) Y_{lm}\propto e^{im\phi}P_l^m(\cos\theta) Y l m ∝ e im ϕ P l m ( cos θ ) Y l m ( − Ω ^ ) = Y l m ( π − θ , π + ϕ ) = e i m π ( − 1 ) m + l Y l m ( θ , ϕ ) = ( − 1 ) l Y l m ( θ , ϕ ) = ( − 1 ) l Y l m ( Ω ^ ) Y_{lm}(-\hat{\Omega})=Y_{lm}(\pi-\theta,\pi+\phi)=e^{im\pi}(-1)^{m+l}Y_{lm}(\theta,\phi)=(-1)^lY_{lm}(\theta,\phi)=(-1)^lY_{lm}(\hat{\Omega}) Y l m ( − Ω ^ ) = Y l m ( π − θ , π + ϕ ) = e imπ ( − 1 ) m + l Y l m ( θ , ϕ ) = ( − 1 ) l Y l m ( θ , ϕ ) = ( − 1 ) l Y l m ( Ω ^ )
I l m ; l ′ m ′ ; l ′ ′ m ′ ′ = ( − 1 ) l + l ′ + l ′ ′ I l m ; l ′ m ′ ; l ′ ′ m ′ ′ ⇒ I l m ; l ′ m ′ ; l ′ ′ m ′ ′ = 0 for l + l ′ + l ′ ′ odd . (273) I_{lm;l'm';l''m''}=(-1)^{l+l'+l''}I_{lm;l'm';l''m''}\Rightarrow I_{lm;l'm';l''m''}=0\quad\text{for }l+l'+l''\text{ odd}. \tag*{(273)} I l m ; l ′ m ′ ; l ′′ m ′′ = ( − 1 ) l + l ′ + l ′′ I l m ; l ′ m ′ ; l ′′ m ′′ ⇒ I l m ; l ′ m ′ ; l ′′ m ′′ = 0 for l + l ′ + l ′′ odd . (273) Thus follows the third property: if l + l ′ + l ′ ′ l+l'+l'' l + l ′ + l ′′ I l m ; l ′ m ′ ; l ′ ′ m ′ ′ I_{lm;l'm';l''m''} I l m ; l ′ m ′ ; l ′′ m ′′
We note that the second equality in (G16) does not suffice to define the Wigner 3j symbols for two reasons. First, if l + l ′ + l ′ ′ l+l'+l'' l + l ′ + l ′′ m = m ′ = m ′ ′ = 0 m=m'=m''=0 m = m ′ = m ′′ = 0 l + l ′ + l ′ ′ l+l'+l'' l + l ′ + l ′′ l + l ′ + l ′ ′ l+l'+l'' l + l ′ + l ′′
The powers of ( − 1 ) (-1) ( − 1 ) Y l m ∗ ( Ω ^ ) = ( − 1 ) m Y l , − m ( Ω ^ ) Y^*_{lm}(\hat{\Omega})=(-1)^mY_{l,-m}(\hat{\Omega}) Y l m ∗ ( Ω ^ ) = ( − 1 ) m Y l , − m ( Ω ^ )
∫ d Ω ^ Y l m ∗ ( Ω ^ ) Y l ′ m ′ ( Ω ^ ) Y l ′ ′ m ′ ′ ( Ω ^ ) = ( − 1 ) m I l , − m ; l ′ m ′ ; l ′ ′ m ′ ′ . (274) \int\mathrm{d}\hat{\Omega}\,Y^*_{lm}(\hat{\Omega})Y_{l'm'}(\hat{\Omega})Y_{l''m''}(\hat{\Omega})=(-1)^mI_{l,-m;l'm';l''m''}. \tag*{(274)} ∫ d Ω ^ Y l m ∗ ( Ω ^ ) Y l ′ m ′ ( Ω ^ ) Y l ′′ m ′′ ( Ω ^ ) = ( − 1 ) m I l , − m ; l ′ m ′ ; l ′′ m ′′ . (274) Similar relations exist for integrals containing any product of three complex-conjugated or unconjugated Y l m Y_{lm} Y l m
I l m ; l ′ m ′ ; l ′ ′ m ′ ′ = I l , − m ; l ′ , − m ′ ; l ′ ′ , − m ′ ′ . (275) I_{lm;l'm';l''m''}=I_{l,-m;l',-m';l'',-m''}. \tag*{(275)} I l m ; l ′ m ′ ; l ′′ m ′′ = I l , − m ; l ′ , − m ′ ; l ′′ , − m ′′ . (275) To see this, consider the integral
∫ d Ω ^ Y l m ∗ ( Ω ^ ) Y l ′ m ′ ∗ ( Ω ^ ) Y l ′ ′ m ′ ′ ∗ ( Ω ^ ) = ( − 1 ) m + m ′ + m ′ ′ I l , − m ; l ′ , − m ′ ; l ′ ′ , − m ′ ′ = I l , m ; l ′ , − m ′ ; l ′ ′ , − m ′ ′ . (276) \int\mathrm{d}\hat{\Omega}Y^*_{lm}(\hat{\Omega})Y^*_{l'm'}(\hat{\Omega})Y^*_{l''m''}(\hat{\Omega})=(-1)^{m+m'+m''}I_{l,-m;l',-m';l'',-m''}=I_{l,m;l',-m';l'',-m''}. \tag*{(276)} ∫ d Ω ^ Y l m ∗ ( Ω ^ ) Y l ′ m ′ ∗ ( Ω ^ ) Y l ′′ m ′′ ∗ ( Ω ^ ) = ( − 1 ) m + m ′ + m ′′ I l , − m ; l ′ , − m ′ ; l ′′ , − m ′′ = I l , m ; l ′ , − m ′ ; l ′′ , − m ′′ . (276) where the first equality follows from Y l m ∗ ( Ω ^ ) = ( − 1 ) m Y l , − m ( Ω ^ ) Y^*_{lm}(\hat{\Omega})=(-1)^mY_{l,-m}(\hat{\Omega}) Y l m ∗ ( Ω ^ ) = ( − 1 ) m Y l , − m ( Ω ^ ) I I I m ′ m' m ′
∫ 0 2 π d ϕ e − i ( m + m ′ + m ′ ′ ) ϕ = ∫ 0 − 2 π − d u e i ( m + m ′ + m ′ ′ ) u = ∫ 2 π 0 − d u e i ( m + m ′ + m ′ ′ ) u = ∫ 0 2 π d u e i ( m + m ′ + m ′ ′ ) u . (277) \int_0^{2\pi}\mathrm{d}\phi\,e^{-i(m+m'+m'')\phi}
=\int_0^{-2\pi}-\mathrm{d}u\,e^{i(m+m'+m'')u}
=\int_{2\pi}^{0}-\mathrm{d}u\,e^{i(m+m'+m'')u}
=\int_0^{2\pi}\mathrm{d}u\,e^{i(m+m'+m'')u}. \tag*{(277)} ∫ 0 2 π d ϕ e − i ( m + m ′ + m ′′ ) ϕ = ∫ 0 − 2 π − d u e i ( m + m ′ + m ′′ ) u = ∫ 2 π 0 − d u e i ( m + m ′ + m ′′ ) u = ∫ 0 2 π d u e i ( m + m ′ + m ′′ ) u . (277) The first equality in (G21) follows from a change of variables ϕ = − u \phi=-u ϕ = − u 2 π 2\pi 2 π
The expression (G14) for E may be computed for any detector response that has only a finite number of nonzero Λ l \Lambda_l Λ l λ l \lambda_l λ l L max 12 L_{\max}^{12} L m a x 12 ( l l ′ l ′ ′ m m ′ m ′ ′ ) \begin{pmatrix} l & l' & l'' \\ m & m' & m'' \end{pmatrix} ( l m l ′ m ′ l ′′ m ′′ ) [53 , 54 ] ) are satisfied:
(i) m ∈ { − l , − l + 1 , … , l } , m ′ ∈ { − l ′ , − l ′ + 1 , … , l ′ } , m ′ ′ ∈ { − l ′ ′ , − l ′ ′ + 1 , … , l ′ ′ } , (ii) m + m ′ + m ′ ′ = 0 , (iii) ∣ l − l ′ ∣ ≤ l ′ ′ ≤ l + l ′ . (278) \begin{aligned}
\text{(i)}\quad&m\in\{-l,-l+1,\ldots,l\},\quad m'\in\{-l',-l'+1,\ldots,l'\},\quad m''\in\{-l'',-l''+1,\ldots,l''\},\\
\text{(ii)}\quad&m+m'+m''=0,\\
\text{(iii)}\quad&|l-l'|\le l''\le l+l'. \tag*{(278)}
\end{aligned} (i) (ii) (iii) m ∈ { − l , − l + 1 , … , l } , m ′ ∈ { − l ′ , − l ′ + 1 , … , l ′ } , m ′′ ∈ { − l ′′ , − l ′′ + 1 , … , l ′′ } , m + m ′ + m ′′ = 0 , ∣ l − l ′ ∣ ≤ l ′′ ≤ l + l ′ . (278) Because (G14) arises from a sum over spherical harmonics, condition (i) is always satisfied. Note that the Wigner 3j symbol which appears in (G16) is invariant under any even permutation of the three “columns.” Under an odd permutation, it is invariant if the sum of the top row is even; otherwise it changes sign.
Using the symmetries and selection rules, we can reduce the number of terms in the sum to order L max 9 L_{\max}^{9} L m a x 9 I I I X X X
m 1 + m 3 − m 6 = 0 , − m 1 − m 3 + m 4 = 0 , m 2 − m 4 + m 5 = 0 , − m 2 − m 5 + m 6 = 0. (279) \begin{aligned}
m_1+m_3-m_6&=0,\\
-m_1-m_3+m_4&=0,\\
m_2-m_4+m_5&=0,\\
-m_2-m_5+m_6&=0. \tag*{(279)}
\end{aligned} m 1 + m 3 − m 6 − m 1 − m 3 + m 4 m 2 − m 4 + m 5 − m 2 − m 5 + m 6 = 0 , = 0 , = 0 , = 0. (279) Since the final equation is implied by the previous ones, only three of these equations are independent. For X X X
m 4 = m 6 = m 1 + m 3 , m 5 = m 1 − m 2 + m 3 . (280) m_4=m_6=m_1+m_3,\qquad m_5=m_1-m_2+m_3. \tag*{(280)} m 4 = m 6 = m 1 + m 3 , m 5 = m 1 − m 2 + m 3 . (280) In addition, selection rule (iii) applied allows us to restrict the summations over l 4 l_4 l 4 l 5 l_5 l 5 l 6 l_6 l 6 X X X
∑ l 4 = ∑ l 4 = ∣ l 1 − l 3 ∣ l 1 + l 3 , ∑ l 5 = ∑ l 5 = ∣ l 2 − l 4 ∣ l 2 + l 4 , ∑ l 6 = ∑ l 6 = ∣ l 1 − l 3 ∣ l 1 + l 3 . (281) \sum_{l_4}=\sum_{l_4=|l_1-l_3|}^{l_1+l_3},\qquad
\sum_{l_5}=\sum_{l_5=|l_2-l_4|}^{l_2+l_4},\qquad
\sum_{l_6}=\sum_{l_6=|l_1-l_3|}^{l_1+l_3}. \tag*{(281)} l 4 ∑ = l 4 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 , l 5 ∑ = l 5 = ∣ l 2 − l 4 ∣ ∑ l 2 + l 4 , l 6 ∑ = l 6 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 . (281) A similar set of conditions and restrictions hold for Y Y Y
m 4 = m 2 + m 3 , m 5 = m 1 + m 2 + m 3 , m 6 = m 1 + m 3 . (282) m_4=m_2+m_3,\qquad m_5=m_1+m_2+m_3,\qquad m_6=m_1+m_3. \tag*{(282)} m 4 = m 2 + m 3 , m 5 = m 1 + m 2 + m 3 , m 6 = m 1 + m 3 . (282) and
∑ l 4 = ∑ l 4 = ∣ l 2 − l 3 ∣ l 2 + l 3 , ∑ l 5 = ∑ l 5 = ∣ l 1 − l 4 ∣ l 1 + l 4 , ∑ l 6 = ∑ l 6 = ∣ l 1 − l 3 ∣ l 1 + l 3 . (283) \sum_{l_4}=\sum_{l_4=|l_2-l_3|}^{l_2+l_3},\qquad
\sum_{l_5}=\sum_{l_5=|l_1-l_4|}^{l_1+l_4},\qquad
\sum_{l_6}=\sum_{l_6=|l_1-l_3|}^{l_1+l_3}. \tag*{(283)} l 4 ∑ = l 4 = ∣ l 2 − l 3 ∣ ∑ l 2 + l 3 , l 5 ∑ = l 5 = ∣ l 1 − l 4 ∣ ∑ l 1 + l 4 , l 6 ∑ = l 6 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 . (283) Thus, when evaluating E, the summations over m 4 m_4 m 4 m 5 m_5 m 5 m 6 m_6 m 6 l 4 l_4 l 4 l 5 l_5 l 5 l 6 l_6 l 6
E ≃ 2 A + B , (284) \mathrm{E}\simeq2A+B, \tag*{(284)} E ≃ 2 A + B , (284) where
A ≡ 16 h 8 ∑ { l 1 ∣ λ l 1 ≠ 0 } ∑ { l 2 ∣ λ l 2 ≠ 0 } ∑ l 3 ∑ l 4 = ∣ l 1 − l 3 ∣ l 1 + l 3 ∑ l 5 = ∣ l 2 − l 4 ∣ l 2 + l 4 ∑ l 6 = ∣ l 1 − l 3 ∣ l 1 + l 3 λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 × ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) ( 2 l 4 + 1 ) ( 2 l 5 + 1 ) ( 2 l 6 + 1 ) ( 4 π ) 2 ( l 1 l 3 l 6 0 0 0 ) ( l 1 l 3 l 4 0 0 0 ) ( l 2 l 4 l 5 0 0 0 ) ( l 2 l 5 l 6 0 0 0 ) × ∑ m 1 = − l 1 l 1 ∑ m 2 = − l 2 l 2 ∑ m 3 = − l 3 l 3 ( l 1 l 3 l 6 m 1 m 3 − m 1 − m 3 ) ( l 1 l 3 l 4 − m 1 − m 3 m 1 + m 3 ) × ( l 2 l 4 l 5 m 2 − m 1 − m 3 m 1 − m 2 + m 3 ) ( l 2 l 5 l 6 − m 2 − m 1 + m 2 − m 3 m 1 + m 3 ) . (285) \begin{aligned}
A \equiv{}^{16}\!h^{8} &\sum_{\{l_1\mid\lambda_{l_1}\ne0\}}\sum_{\{l_2\mid\lambda_{l_2}\ne0\}}\sum_{l_3}\sum_{l_4=\lvert l_1-l_3\rvert}^{l_1+l_3}\sum_{l_5=\lvert l_2-l_4\rvert}^{l_2+l_4}\sum_{l_6=\lvert l_1-l_3\rvert}^{l_1+l_3}\lambda_{l_1}^{-1}\lambda_{l_2}^{-1}\Lambda_{l_3}\Lambda_{l_4}\Lambda_{l_5}\Lambda_{l_6}\\
&\times\frac{(2l_1+1)(2l_2+1)(2l_3+1)(2l_4+1)(2l_5+1)(2l_6+1)}{(4\pi)^2}
\begin{pmatrix}l_1&l_3&l_6\\0&0&0\end{pmatrix}
\begin{pmatrix}l_1&l_3&l_4\\0&0&0\end{pmatrix}
\begin{pmatrix}l_2&l_4&l_5\\0&0&0\end{pmatrix}
\begin{pmatrix}l_2&l_5&l_6\\0&0&0\end{pmatrix}\\
&\times\sum_{m_1=-l_1}^{l_1}\sum_{m_2=-l_2}^{l_2}\sum_{m_3=-l_3}^{l_3}
\begin{pmatrix}l_1&l_3&l_6\\m_1&m_3&-m_1-m_3\end{pmatrix}
\begin{pmatrix}l_1&l_3&l_4\\-m_1&-m_3&m_1+m_3\end{pmatrix}\\
&\times\begin{pmatrix}l_2&l_4&l_5\\m_2&-m_1-m_3&m_1-m_2+m_3\end{pmatrix}
\begin{pmatrix}l_2&l_5&l_6\\-m_2&-m_1+m_2-m_3&m_1+m_3\end{pmatrix}. \tag*{(285)}
\end{aligned} A ≡ 16 h 8 { l 1 ∣ λ l 1 = 0 } ∑ { l 2 ∣ λ l 2 = 0 } ∑ l 3 ∑ l 4 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 l 5 = ∣ l 2 − l 4 ∣ ∑ l 2 + l 4 l 6 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 × ( 4 π ) 2 ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) ( 2 l 4 + 1 ) ( 2 l 5 + 1 ) ( 2 l 6 + 1 ) ( l 1 0 l 3 0 l 6 0 ) ( l 1 0 l 3 0 l 4 0 ) ( l 2 0 l 4 0 l 5 0 ) ( l 2 0 l 5 0 l 6 0 ) × m 1 = − l 1 ∑ l 1 m 2 = − l 2 ∑ l 2 m 3 = − l 3 ∑ l 3 ( l 1 m 1 l 3 m 3 l 6 − m 1 − m 3 ) ( l 1 − m 1 l 3 − m 3 l 4 m 1 + m 3 ) × ( l 2 m 2 l 4 − m 1 − m 3 l 5 m 1 − m 2 + m 3 ) ( l 2 − m 2 l 5 − m 1 + m 2 − m 3 l 6 m 1 + m 3 ) . (285) and
B ≡ 16 h 8 ∑ { l 1 ∣ λ l 1 ≠ 0 } ∑ { l 2 ∣ λ l 2 ≠ 0 } ∑ l 3 ∑ l 4 = ∣ l 2 − l 3 ∣ l 2 + l 3 ∑ l 5 = ∣ l 1 − l 4 ∣ l 1 + l 4 ∑ l 6 = ∣ l 1 − l 3 ∣ l 1 + l 3 λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 × ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) ( 2 l 4 + 1 ) ( 2 l 5 + 1 ) ( 2 l 6 + 1 ) ( 4 π ) 2 ( l 1 l 3 l 6 0 0 0 ) ( l 1 l 4 l 5 0 0 0 ) ( l 2 l 5 l 6 0 0 0 ) ( l 2 l 3 l 4 0 0 0 ) × ∑ m 1 = − l 1 l 1 ∑ m 2 = − l 2 l 2 ∑ m 3 = − l 3 l 3 ( − 1 ) m 1 + m 2 ( l 1 l 3 l 6 m 1 m 3 − m 1 − m 3 ) ( l 1 l 4 l 5 − m 1 − m 2 − m 3 m 1 + m 2 + m 3 ) × ( l 2 l 5 l 6 m 2 − m 1 − m 2 − m 3 m 1 + m 3 ) ( l 2 l 3 l 4 − m 2 − m 3 m 2 + m 3 ) . (286) \begin{aligned}
B \equiv{}^{16}\!h^{8} &\sum_{\{l_1\mid\lambda_{l_1}\ne0\}}\sum_{\{l_2\mid\lambda_{l_2}\ne0\}}\sum_{l_3}\sum_{l_4=\lvert l_2-l_3\rvert}^{l_2+l_3}\sum_{l_5=\lvert l_1-l_4\rvert}^{l_1+l_4}\sum_{l_6=\lvert l_1-l_3\rvert}^{l_1+l_3}\lambda_{l_1}^{-1}\lambda_{l_2}^{-1}\Lambda_{l_3}\Lambda_{l_4}\Lambda_{l_5}\Lambda_{l_6}\\
&\times\frac{(2l_1+1)(2l_2+1)(2l_3+1)(2l_4+1)(2l_5+1)(2l_6+1)}{(4\pi)^2}
\begin{pmatrix}l_1&l_3&l_6\\0&0&0\end{pmatrix}
\begin{pmatrix}l_1&l_4&l_5\\0&0&0\end{pmatrix}
\begin{pmatrix}l_2&l_5&l_6\\0&0&0\end{pmatrix}
\begin{pmatrix}l_2&l_3&l_4\\0&0&0\end{pmatrix}\\
&\times\sum_{m_1=-l_1}^{l_1}\sum_{m_2=-l_2}^{l_2}\sum_{m_3=-l_3}^{l_3}(-1)^{m_1+m_2}
\begin{pmatrix}l_1&l_3&l_6\\m_1&m_3&-m_1-m_3\end{pmatrix}
\begin{pmatrix}l_1&l_4&l_5\\-m_1&-m_2-m_3&m_1+m_2+m_3\end{pmatrix}\\
&\times\begin{pmatrix}l_2&l_5&l_6\\m_2&-m_1-m_2-m_3&m_1+m_3\end{pmatrix}
\begin{pmatrix}l_2&l_3&l_4\\-m_2&-m_3&m_2+m_3\end{pmatrix}. \tag*{(286)}
\end{aligned} B ≡ 16 h 8 { l 1 ∣ λ l 1 = 0 } ∑ { l 2 ∣ λ l 2 = 0 } ∑ l 3 ∑ l 4 = ∣ l 2 − l 3 ∣ ∑ l 2 + l 3 l 5 = ∣ l 1 − l 4 ∣ ∑ l 1 + l 4 l 6 = ∣ l 1 − l 3 ∣ ∑ l 1 + l 3 λ l 1 − 1 λ l 2 − 1 Λ l 3 Λ l 4 Λ l 5 Λ l 6 × ( 4 π ) 2 ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) ( 2 l 4 + 1 ) ( 2 l 5 + 1 ) ( 2 l 6 + 1 ) ( l 1 0 l 3 0 l 6 0 ) ( l 1 0 l 4 0 l 5 0 ) ( l 2 0 l 5 0 l 6 0 ) ( l 2 0 l 3 0 l 4 0 ) × m 1 = − l 1 ∑ l 1 m 2 = − l 2 ∑ l 2 m 3 = − l 3 ∑ l 3 ( − 1 ) m 1 + m 2 ( l 1 m 1 l 3 m 3 l 6 − m 1 − m 3 ) ( l 1 − m 1 l 4 − m 2 − m 3 l 5 m 1 + m 2 + m 3 ) × ( l 2 m 2 l 5 − m 1 − m 2 − m 3 l 6 m 1 + m 3 ) ( l 2 − m 2 l 3 − m 3 l 4 m 2 + m 3 ) . (286) To avoid any ambiguity in their second rows, we have added vertical separators to the Wigner 3j symbols. Note that for detectors that respond to an infinite number of l l l l l l
We now prepare to calculate the non-Gaussian contribution E ^ \hat{E} E ^ χ 2 \chi^2 χ 2 (181) . This requires the projection operator P a b P_{ab} P ab C ^ a b − 1 \hat{C}^{-1}_{ab} C ^ ab − 1 (180) and (182) . Using (G13) for C a b − 1 C^{-1}_{ab} C ab − 1 1 \mathbf{1} 1 C − 1 C^{-1} C − 1
∑ b C a b − 1 1 b ≃ 4 π N p u l λ 0 1 a ⇒ 1 T C − 1 1 ≃ 4 π λ 0 , (287) \sum_b C^{-1}_{ab}1_b \simeq\frac{4\pi}{N_{\mathrm{pul}}\lambda_0}1_a \Rightarrow\mathbf{1}^{\mathsf T}C^{-1}\mathbf{1}\simeq\frac{4\pi}{\lambda_0}, \tag*{(287)} b ∑ C ab − 1 1 b ≃ N pul λ 0 4 π 1 a ⇒ 1 T C − 1 1 ≃ λ 0 4 π , (287) where 1 \mathbf{1} 1 (22) . The first relation may be verified by using (G13) to replace C a b − 1 C^{-1}_{ab} C ab − 1 b b b l = m = 0 l=m=0 l = m = 0 (180) leads to
P a b ≃ δ a b − 1 N p u l 1 a 1 b . (288) P_{ab}\simeq\delta_{ab}-\frac{1}{N_{\mathrm{pul}}}1_a1_b. \tag*{(288)} P ab ≃ δ ab − N pul 1 1 a 1 b . (288) (Note that this has the correct trace N p u l − 1 N_{\mathrm{pul}}-1 N pul − 1 (182) and using (G31) again, yields
C ^ a b − 1 ≃ C a b − 1 − 4 π λ 0 1 N p u l 2 1 a 1 b . (289) \hat{C}^{-1}_{ab}\simeq C^{-1}_{ab}-\frac{4\pi}{\lambda_0}\frac{1}{N_{\mathrm{pul}}^2}1_a1_b. \tag*{(289)} C ^ ab − 1 ≃ C ab − 1 − λ 0 4 π N pul 2 1 1 a 1 b . (289) This completes our preparation for the calculation that follows.
To compute the non-Gaussian contribution E ^ \hat{E} E ^ (181) . Since C − 1 C^{-1} C − 1 C ^ − 1 \hat{C}^{-1} C ^ − 1 1 a 1 b 1 c 1 d 1_a1_b1_c1_d 1 a 1 b 1 c 1 d 1 a 1 b C c d − 1 + 1 c 1 d C a b − 1 1_a1_bC^{-1}_{cd}+1_c1_dC^{-1}_{ab} 1 a 1 b C c d − 1 + 1 c 1 d C ab − 1 μ a b \mu_{ab} μ ab C − 1 \mathsf{C}^{-1} C − 1 δ l 1 l 2 δ m 1 m 2 \delta_{l_1l_2}\delta_{m_1m_2} δ l 1 l 2 δ m 1 m 2 m m m
E − E ^ ≃ − 48 ℏ 8 ( 4 π λ 0 ) 2 ∑ l ( 2 l + 1 ) Λ l 4 + 32 ℏ 8 4 π λ 0 ∑ { l 1 ∣ λ l 1 ≠ 0 } ∑ m 1 ∑ l 2 m 2 ∑ l 3 m 3 λ l 1 − 1 Λ l 1 Λ l 2 2 ( 2 Λ l 3 + Λ l 2 ) I l 1 m 1 ; l 2 m 2 ; l 3 m 3 2 . (290) \mathbb{E}-\widehat{\mathbb{E}}\simeq-\frac{48\hbar^{8}}{(4\pi\lambda_{0})^{2}}\sum_{l}(2l+1)\Lambda_{l}^{4}+\frac{32\hbar^{8}}{4\pi\lambda_{0}}\sum_{\{l_{1}\mid\lambda_{l_{1}}\ne0\}}\sum_{m_{1}}\sum_{l_{2}m_{2}}\sum_{l_{3}m_{3}}\lambda_{l_{1}}^{-1}\Lambda_{l_{1}}\Lambda_{l_{2}}^{2}(2\Lambda_{l_{3}}+\Lambda_{l_{2}})I^{2}_{l_{1}m_{1};l_{2}m_{2};l_{3}m_{3}}. \tag*{(290)} E − E ≃ − ( 4 π λ 0 ) 2 48 ℏ 8 l ∑ ( 2 l + 1 ) Λ l 4 + 4 π λ 0 32 ℏ 8 { l 1 ∣ λ l 1 = 0 } ∑ m 1 ∑ l 2 m 2 ∑ l 3 m 3 ∑ λ l 1 − 1 Λ l 1 Λ l 2 2 ( 2 Λ l 3 + Λ l 2 ) I l 1 m 1 ; l 2 m 2 ; l 3 m 3 2 . (290) where the first term on the rhs arises from (1) and the second from (2). As earlier, we can use the Wigner 3j selection rules to eliminate the sum over m 3 m_{3} m 3 m 3 = − m 1 − m 2 m_{3}=-m_{1}-m_{2} m 3 = − m 1 − m 2 l 3 l_{3} l 3 ∣ l 1 − l 2 ∣ \lvert l_{1}-l_{2}\rvert ∣ l 1 − l 2 ∣ l 1 + l 2 l_{1}+l_{2} l 1 + l 2 I 2 I^{2} I 2
E − E ^ ≃ − 48 ℏ 8 ( 4 π λ 0 ) 2 ∑ l ( 2 l + 1 ) Λ l 4 + 32 ℏ 8 4 π λ 0 ∑ { l 1 ∣ λ l 1 ≠ 0 } ∑ l 2 ∑ l 3 = ∣ l 1 − l 2 ∣ l 1 + l 2 λ l 1 − 1 Λ l 1 Λ l 2 Λ l 3 2 ( 2 Λ l 3 + Λ l 2 ) \mathbb{E}-\widehat{\mathbb{E}}\simeq-\frac{48\hbar^{8}}{(4\pi\lambda_{0})^{2}}\sum_{l}(2l+1)\Lambda_{l}^{4}+\frac{32\hbar^{8}}{4\pi\lambda_{0}}\sum_{\{l_{1}\mid\lambda_{l_{1}}\ne0\}}\sum_{l_{2}}\sum_{l_{3}=\lvert l_{1}-l_{2}\rvert}^{l_{1}+l_{2}}\lambda_{l_{1}}^{-1}\Lambda_{l_{1}}\Lambda_{l_{2}}\Lambda_{l_{3}}^{2}(2\Lambda_{l_{3}}+\Lambda_{l_{2}}) E − E ≃ − ( 4 π λ 0 ) 2 48 ℏ 8 l ∑ ( 2 l + 1 ) Λ l 4 + 4 π λ 0 32 ℏ 8 { l 1 ∣ λ l 1 = 0 } ∑ l 2 ∑ l 3 = ∣ l 1 − l 2 ∣ ∑ l 1 + l 2 λ l 1 − 1 Λ l 1 Λ l 2 Λ l 3 2 ( 2 Λ l 3 + Λ l 2 ) × ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ( l 1 l 2 l 3 0 0 0 ) 2 ∑ m 1 = − l 1 l 1 ∑ m 2 = − l 2 l 2 ( l 1 l 2 l 3 m 1 m 2 − m 1 − m 2 ) 2 . (291) {}\times\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}
\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}^{2}
\sum_{m_{1}=-l_{1}}^{l_{1}}\sum_{m_{2}=-l_{2}}^{l_{2}}
\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&-m_{1}-m_{2}\end{pmatrix}^{2}. \tag*{(291)} × 4 π ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) ( l 1 0 l 2 0 l 3 0 ) 2 m 1 = − l 1 ∑ l 1 m 2 = − l 2 ∑ l 2 ( l 1 m 1 l 2 m 2 l 3 − m 1 − m 2 ) 2 . (291) While we have not indicated it, the sum over l 3 l_{3} l 3 l 1 + l 2 + l 3 l_{1}+l_{2}+l_{3} l 1 + l 2 + l 3
To end this section, we consider a simple example: a (non-PTA) detector which is sensitive to only a single harmonic mode (Q l = δ l L Q_l=\delta_{lL} Q l = δ l L η = 0 \eta=0 η = 0 E \mathbb{E} E E ^ \widehat{\mathbb{E}} E
Λ l = 4 π 2 L + 1 δ l L , λ l = 8 π ℏ 4 D l 2 l + 1 = 8 π ℏ 4 ( l L L 0 0 0 ) 2 ⟹ λ 0 = 8 π ℏ 4 2 L + 1 . (292) \Lambda_l=\frac{4\pi}{2L+1}\delta_{lL},\qquad\lambda_l=8\pi\hbar^{4}\frac{D_l}{2l+1}=8\pi\hbar^{4}\begin{pmatrix}l&L&L\\0&0&0\end{pmatrix}^{2}\Longrightarrow\lambda_0=\frac{8\pi\hbar^{4}}{2L+1}. \tag*{(292)} Λ l = 2 L + 1 4 π δ l L , λ l = 8 π ℏ 4 2 l + 1 D l = 8 π ℏ 4 ( l 0 L 0 L 0 ) 2 ⟹ λ 0 = 2 L + 1 8 π ℏ 4 . (292) Symmetry properties of the above Wigner 3j symbol then imply that λ l ≠ 0 \lambda_l\ne0 λ l = 0 l = 0 , 2 , … , 2 L l=0,2,\ldots,2L l = 0 , 2 , … , 2 L C a b \mathsf{C}_{ab} C ab N D O F = ( 2 L + 1 ) ( L + 1 ) N_{\mathrm{DOF}}=(2L+1)(L+1) N DOF = ( 2 L + 1 ) ( L + 1 )
Substituting the expressions in (G36) for Λ l \Lambda_l Λ l λ l \lambda_l λ l
E ≃ 4 ( ℓ ℏ ) 8 ∑ l 1 = 0 L ∑ l 2 = 0 L ( 4 l 1 + 1 ) ( 4 l 2 + 1 ) \mathbb{E}\simeq4\left(\frac{\ell}{\hbar}\right)^{8}\sum_{l_{1}=0}^{L}\sum_{l_{2}=0}^{L}(4l_{1}+1)(4l_{2}+1) E ≃ 4 ( ℏ ℓ ) 8 l 1 = 0 ∑ L l 2 = 0 ∑ L ( 4 l 1 + 1 ) ( 4 l 2 + 1 ) × ∑ m 1 = − 2 l 1 2 l 1 ∑ m 2 = − 2 l 2 2 l 2 ∑ m 3 = − L L [ 2 ( 2 l 1 L L m 1 m 3 − m 1 − m 3 ) 2 ( 2 l 2 L L m 2 − m 1 − m 3 m 1 − m 2 + m 3 ) 2 {}\times\sum_{m_{1}=-2l_{1}}^{2l_{1}}\sum_{m_{2}=-2l_{2}}^{2l_{2}}\sum_{m_{3}=-L}^{L}
\left[
2\begin{pmatrix}2l_{1}&L&L\\m_{1}&m_{3}&-m_{1}-m_{3}\end{pmatrix}^{2}
\begin{pmatrix}2l_{2}&L&L\\m_{2}&-m_{1}-m_{3}&m_{1}-m_{2}+m_{3}\end{pmatrix}^{2}
\right. × m 1 = − 2 l 1 ∑ 2 l 1 m 2 = − 2 l 2 ∑ 2 l 2 m 3 = − L ∑ L [ 2 ( 2 l 1 m 1 L m 3 L − m 1 − m 3 ) 2 ( 2 l 2 m 2 L − m 1 − m 3 L m 1 − m 2 + m 3 ) 2 + ( − 1 ) m 1 + m 2 ( 2 l 1 L L m 1 m 3 − m 1 − m 3 ) ( 2 l 1 L L m 1 m 2 + m 3 − m 1 − m 2 − m 3 ) ( 2 l 2 L L m 2 m 3 − m 2 − m 3 ) ( 2 l 2 L L m 2 m 1 + m 3 − m 1 − m 2 − m 3 ) ] . (293) \left.
{}+(-1)^{m_{1}+m_{2}}
\begin{pmatrix}2l_{1}&L&L\\m_{1}&m_{3}&-m_{1}-m_{3}\end{pmatrix}
\begin{pmatrix}2l_{1}&L&L\\m_{1}&m_{2}+m_{3}&-m_{1}-m_{2}-m_{3}\end{pmatrix}
\begin{pmatrix}2l_{2}&L&L\\m_{2}&m_{3}&-m_{2}-m_{3}\end{pmatrix}
\begin{pmatrix}2l_{2}&L&L\\m_{2}&m_{1}+m_{3}&-m_{1}-m_{2}-m_{3}\end{pmatrix}
\right]. \tag*{(293)} + ( − 1 ) m 1 + m 2 ( 2 l 1 m 1 L m 3 L − m 1 − m 3 ) ( 2 l 1 m 1 L m 2 + m 3 L − m 1 − m 2 − m 3 ) ( 2 l 2 m 2 L m 3 L − m 2 − m 3 ) ( 2 l 2 m 2 L m 1 + m 3 L − m 1 − m 2 − m 3 ) ] . (293) and
E − E ^ ≃ 12 2 L + 1 ( ℏ h ) 8 [ − 1 + 2 ∑ l 1 = 0 L ( 4 l 1 + 1 ) ∑ m 1 = − 2 l 1 2 l 1 ∑ m 2 = − L L ( 2 l 1 L L m 1 m 2 − m 1 − m 2 ) 2 ] (294) \mathbb{E}-\widehat{\mathbb{E}}\simeq\frac{12}{2L+1}\left(\frac{\hbar}{h}\right)^{8}\left[-1+2\sum_{l_{1}=0}^{L}(4l_{1}+1)\sum_{m_{1}=-2l_{1}}^{2l_{1}}\sum_{m_{2}=-L}^{L}\begin{pmatrix}2l_{1} & L & L \\ m_{1} & m_{2} & -m_{1}-m_{2}\end{pmatrix}^{2}\right] \tag*{(294)} E − E ≃ 2 L + 1 12 ( h ℏ ) 8 [ − 1 + 2 l 1 = 0 ∑ L ( 4 l 1 + 1 ) m 1 = − 2 l 1 ∑ 2 l 1 m 2 = − L ∑ L ( 2 l 1 m 1 L m 2 L − m 1 − m 2 ) 2 ] (294) where we “doubled” the indices l 1 l_{1} l 1 l 2 l_{2} l 2
E ≃ 2 ( 2 L + 1 ) ( 4 L 2 + 9 L + 6 ) ( ℏ h ) 8 \mathbb{E}\simeq2(2L+1)(4L^{2}+9L+6)\left(\frac{\hbar}{h}\right)^{8} E ≃ 2 ( 2 L + 1 ) ( 4 L 2 + 9 L + 6 ) ( h ℏ ) 8 and
E − E ^ ≃ 12 ( 4 L 2 + 6 L + 1 2 L + 1 ) ( ℏ h ) 8 . (295) \mathbb{E}-\widehat{\mathbb{E}}\simeq12\left(\frac{4L^{2}+6L+1}{2L+1}\right)\left(\frac{\hbar}{h}\right)^{8}. \tag*{(295)} E − E ≃ 12 ( 2 L + 1 4 L 2 + 6 L + 1 ) ( h ℏ ) 8 . (295) These are amazingly simple formulas, considering the summations of products of the various Wigner 3j symbols that enter (G37) and (G38). We have not found an analytical approach to evaluate the sums leading to (G39).
Sky locations of pulsars currently monitored by PTA collaborations
Table IV gives the sky locations of the 88 pulsars currently monitored by the European Pulsar Timing Array (EPTA) collaboration, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration, and the Parkes Pulsar Timing Array (PPTA) collaboration [37 ] . We used these to create Figure 3 , Figure 8 , and Figure 9 , and for constructing Table I and Table II : see § 6 B, VII C, and VIII C for details. The sky locations are in equatorial coordinates, with right ascension (ra) and declination (dec) given in degrees. They are related to spherical polar coordinates (in radians) via θ = π / 2 − ( π / 180 ∘ ) d e c \theta=\pi/2-(\pi/180^\circ)\mathrm{dec} θ = π /2 − ( π /18 0 ∘ ) dec ϕ = ( π / 180 ∘ ) r a \phi=(\pi/180^\circ)\mathrm{ra} ϕ = ( π /18 0 ∘ ) ra
[16 ] LIGO’s use of a Fabry-Perot cavity means that the effective arm length is hundreds of times larger than the actual arm length. This effective arm length is longer than the shortest GW wavelength which LIGO can detect. Remarkably, a simple approximation that combines the long-wavelength Michelson response with the single-pole approximation to the Fabry-Perot transfer function gives results that agree with the exact response to within a few percent [43 ] .
[17 ] A more general definition of the correlation ρ a b \rho_{ab} ρ ab Z ~ ( f ) \widetilde{Z}(f) Z ( f ) Z ( t ) Z(t) Z ( t )
ρ a b = ∫ d f K ( f ) Z ~ a ( f ) Z ~ b ∗ ( f ) , \rho_{ab}=\int df K(f)\widetilde{Z}_{a}(f)\widetilde{Z}_{b}^{*}(f), ρ ab = ∫ df K ( f ) Z a ( f ) Z b ∗ ( f ) , where K ( f ) = K ∗ ( − f ) K(f)=K^{*}(-f) K ( f ) = K ∗ ( − f ) K K K ρ a b \rho_{ab} ρ ab [13 ] in the context of cross-correlation searches for GW backgrounds using ground-based interferometers.
[18 ] Here, “different distances” means differing by more than a few characteristic GW wavelengths, so tens of years.
[19 ] Isserlis’s theorem is often called “Wick’s theorem” in the physics community, although Wick’s work was three decades later.
[20 ] Note that there are two definitions of the sinc(x x x x = π x=\pi x = π sin ( π x ) / π x \sin(\pi x)/\pi x sin ( π x ) / π x x = 1 x=1 x = 1
[21 ] Note that the angular bin widths d γ d\gamma d γ sin γ \sin\gamma sin γ ∣ d ( cos γ ) ∣ = sin γ d γ |d(\cos\gamma)|=\sin\gamma\,d\gamma ∣ d ( cos γ ) ∣ = sin γ d γ d γ d\gamma d γ γ \gamma γ 0 ∘ 0^\circ 0 ∘ 180 ∘ 180^\circ 18 0 ∘
[22 ] We also allow for singular (i.e., positive-semidefinite) covariance matrices. Although such matrices have zero determinant, one can construct pseudoinverse matrices to invert them; see Sec. VIC for an example.
[25 ] This can be easily proven by requiring that v T C v ≥ 0 v^{T}Cv\geq0 v T C v ≥ 0 v = 0 v=0 v = 0
[27 ] It is also easy to show that the average values of the entries, where the average is done along a column, are column-independent. In fact, with the assumptions we have made, each matrix is characterized by the average value of its entries; the average value along any row or any column is the same.
[36 ] It may be possible to find the exact inverse of s j k s_{jk} s j k N b i n s N_{\mathrm{bins}} N bins [74 ] .
[37 ] Our list of pulsars differs slightly from those that appear in the PTA collaborations’ recent data release [58 , 59 , 60 , 61 ] . For example, compared to our list, [58 ] removed J0614-3329 and added J2234+0944 and J1012+5307. These changes do not significantly modify our figures or results.
[38 ] C. Cutler (private communication). Cutler suggested the idea of considering alternative detector types to understand this, analogous to employing a short dipole antenna to measure the squared amplitude of an electromagnetic field near Earth.
[39 ] An equivalent of the “pulsar term” could be added to our imaginary one-arm LIGO detector. A second detector, formed with two mirrors, would be placed along the line of sight, near the pulsar. Its strain measurement would be relayed back to Earth at the speed of light via a radio or optical link, and then subtracted from the strain measured at the detector near Earth. This would appear as an additional corresponding choice μ b i n = 0 \mu_{\mathrm{bin}} = 0 μ bin = 0 ⟨ ρ o p t ⟩ = 0 \langle\rho_{\mathrm{opt}}\rangle= 0 ⟨ ρ opt ⟩ = 0 σ o p t 2 = 0 \sigma_{\mathrm{opt}}^{2} = 0 σ opt 2 = 0
[70 ] The Santa Claus Principle holds that “there is no Santa Claus,” meaning that one does not get free gifts. Conversely, since here we are giving something up (angular resolution), we must gain something in exchange (higher SNR).
[71 ] The cumulants κ n \kappa_n κ n X X X K ( t ) ≡ ln ⟨ e t X ⟩ = ∑ n = 1 ∞ κ n t n / n ! K(t) \equiv\ln\langle e^{tX}\rangle= \sum_{n=1}^{\infty} \kappa_n t^n/n! K ( t ) ≡ ln ⟨ e tX ⟩ = ∑ n = 1 ∞ κ n t n / n !
[72 ] Efficient evaluation can be done using PYTHON broadcasting and the function NUMPY.EINSUM().
[73 ] These inversion transformations were exploited in [11 ] , for example see the arguments following Eq. (G7) of that reference.
