Multiprobe strategies for Kinematic Lensing with the Roman Space Telescope High Latitude Wide Area Survey

Introduction

Dark energy accounts for roughly 70% of the mass and energy in the universe [1, 2]. Its physical origin, however, remains unexplained. In the absence of a fundamental theory, the standard cosmological model $\Lambda$CDM adopts the simplest phenomenological assumption: a cosmological constant with a fixed equation of state of $w=-1$. This standard model has provided a statistically good fit to observations spanning the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO), and large-scale structure (LSS) individually over decades [1, 3–5].

Recent results from the Dark Energy Spectroscopic Instrument (DESI)¹ challenge the $\Lambda$CDM model. While DESI’s BAO data alone remain consistent with $\Lambda$CDM, the joint analysis of BAO, CMB, and supernova (SN) favors a time-varying equation of state, $w(a)=w_0+w_a(1-a)$ (the Chevallier-Polarski-Linder or CPL parameterization; [6, 7]), deviating from the fixed $w=-1$ assumed in $\Lambda$CDM [8–10]. Since the release of DESI DR1, this result has been the subject of active debate [11–21].

Fully understanding the physics of dark energy and stress-testing the $\Lambda$CDM model requires additional independent measurements. For instance, the Euclid mission² is designed to measure BAO over similar redshift ranges as DESI, with distinctly different observational systematics [22]. Cross-checking DESI’s and Euclid’s BAO measurements will reveal whether unknown pipeline systematics exist. Similarly, the Nancy Grace Roman Space Telescope (Roman)³ and Rubin Observatory’s Legacy Survey of Space and Time (LSST)⁴ are expected to observe thousands of Type Ia SNe in coming years [23, 24], enabling studies to characterize the astrophysical systematics of SN light curves.

Complementary measurements offer another avenue to probe dark energy physics. Cosmic shear, Weak Lensing (WL) by the large-scale structure, probes the dark energy equation of state through lensing geometry and angular projection, providing information complementary to BAO’s standard-ruler measurement. Moreover, it provides additional constraints on structure growth and thus enables a growth-geometry-consistency test of the cosmological model. However, cosmic shear alone is subject to degeneracies between cosmology and nuisance parameters such as intrinsic alignment or photometric redshift uncertainties [25–27]. A typical way to improve cosmic shear constraints is to combine it with galaxy clustering and galaxy-galaxy lensing into a so-called $3\times2\mathrm{pt}$ analysis. This concept has been applied to the Stage-III photometric surveys such as the Dark Energy Survey (DES)⁵, the Kilo-Degree Survey (KiDS⁶), and the Hyper Suprime-Cam Subaru Strategic Program (HSC⁷) [4, 5, 28, 29] to measure dark energy equation of state [31] [30].

Another way to improve cosmic shear constraints is Kinematic Lensing (KL). KL combines spectroscopic and imaging data with the Tully-Fisher relation [32] to extract the lensing signal at a much lower shape-noise level than the traditional WL [33, 34, 35]. Beyond reducing shape noise, KL is also immune to several WL systematics such as photometric redshift uncertainties and intrinsic alignment [36]. The KL concept has been demonstrated in cluster lensing using slit spectroscopy, achieving a shape-noise uncertainty of 0.026 per component [37]. Moreover, forecasts for several upcoming surveys, particularly Roman, suggest that KL has substantial potential to constrain cosmology [34, 38, 39, 40].

One of Roman’s core surveys is the High Latitude Wide Area Survey (HLWAS), which has four tiers: Wide, Medium, Deep, and Ultra-Deep Tier, covering different survey depths and areas. Among the four tiers, the Medium Tier comprises images and slitless grism spectroscopy over 2,300 deg$^2$ [41], providing the essential ingredients for KL measurements. Previously, [34] showed that KL could improve the constraining power on the dark energy equation of state by more than three times compared to WL, using the now-outdated Medium Tier survey configuration [42]. The recent Roman Observations Time Allocation Committee (ROTAC) decided on a shallower but wider HLWAS Medium Tier [41], which we forecast and explore in detail in this paper.

Further, we examine the impact of roll-angle separation between different grism exposures. This is motivated by findings in [43], which revealed a non-trivial relation between the roll-angle separation and the grism spectra success rate. Small roll angle separations increase the difficulty of spectra decontamination, while large roll angles lead to more positions with poor exposure coverage due to more complex tiling, which we should account for when forecasting KL’s performance.

In addition to an in-depth study of KL performance with the new survey design, we explore the multiprobe concepts centered around KL. An intuitive approach is to combine KL with 3D galaxy clustering because the KL sample naturally requires spectroscopic measurements. This cross-correlation also yields the greatest benefit when using overlapping data, such as the Medium Tier of HLWAS [44, 45]. However, developing the full formalism including non-Gaussian covariance matrices for a 3D clustering plus KL analysis is beyond the scope of this paper. As a first step, we explore the information gain of a standard 2D tomographic 3 × 2pt method with KL as the lensing measurement. We denote this multiprobe approach as KL-3 × 2pt.

The paper is structured as follows. In Sect. II, we introduce our model for the simulated analyses. In Sect. III, we describe our galaxy sample selection for both KL and WL. In Sect. IV, we explore the relation between the telescope roll angle and the resulting KL shape noise. In Sect. V we present and compare the updated KL and WL forecasts for HLWAS. In Sect. VI we present the KL-multiprobe forecast. We conclude the paper in Sect. VII.

Theoretical Modeling of observables

We use CosmoLike [46] routines as described in [34] for all lensing-based observables and include the CosmoLike FKEM module [47] to compute non-Limber power spectra for galaxy clustering. We integrate our KL-CosmoLike framework into COBAYA-COSMOLIKE ARCHITECTURE (CoCoA⁸) and make the code available to the public⁹.

In all cases, we compute the posterior probability distribution of cosmological parameters $\boldsymbol{p}_{\mathrm{co}}$ and nuisance parameters $\boldsymbol{p}_{\mathrm{nu}}$ given the data $\mathbf{D}$ via Bayes’ theorem

where $\boldsymbol{p} = (\boldsymbol{p}_{\mathrm{co}},\boldsymbol{p}_{\mathrm{nu}})$ and $\pi(\boldsymbol{p})$ is the prior probability distribution of the parameters. We further assume that the likelihood $L(\mathbf{D}|\boldsymbol{p})$ in eq. (1) is well-approximated by a multivariate Gaussian and define

where $\mathbf{M}$ is the model vector evaluated at $\boldsymbol{p}$ and $\mathbf{Cov}$ the covariance matrix calculated at the fiducial parameter values. We assume the covariance is constant in parameter space; we do not adopt a fully cosmology-dependent covariance approach in this work [48, 49].

Data Vector

We compute all data vectors in Fourier space. For each probe, cosmic shear, galaxy clustering, and galaxy-galaxy lensing, we compute the angular power spectra as

where $\chi$ is the comoving distance, $q_a^i(\chi)$ is the kernel function at the $i$-th tomographic bin for the tracer $a, $P(k,z(\chi_1),z(\chi_2))$ is the 3D matter power spectrum, and $j_\ell^{(a)}(k\chi)$ is the spherical Bessel function with the derivative order associated with the probe. When the radial width of the kernel function is larger than $1/k$ of the relevant mode of the tracer, we apply the Limber approximation [50], where $j_\ell(k\chi) \simeq\delta_{\mathrm{D}}(\ell+ 1/2-k\chi)$ and eq. (3) reduces to

with $\chi_\ell=(\ell+1/2)/k$. The Limber approximation works well when $q_a^i(\chi)$ is smooth over a radial width much larger than the scale of tracer $a$’s fluctuation. This generally holds for the lensing kernel, which is usually wide. However, when the tomographic bin becomes narrower than the scales of the density fluctuation mode, the approximation breaks down.

For the forecasting purpose of this paper, where data vectors are generated with the same framework as model vectors, we adopt the Limber approximation as a sufficient modeling choice. We study the impact of this approximation in the narrow tomographic-bin scenario in Sect. VI B. We note that while Limber is sufficient for forecasting (error bars between Limber and non-Limber are largely the same), non-Limber is required for an actual data analysis to avoid parameter biases.

a. Lensing kernels The lensing kernel for cosmic shear at the $i$-th bin is

where $n_{\mathrm{src}}^i(z)$ represents the redshift distribution of source galaxies in the $i$-th bin, with $\bar n_{\mathrm{src}}^i$ the 2D angular galaxy density within the same redshift bin.

b. Clustering kernels The selection function for clustering is

where $n_{\mathrm{lens}}^i(z)$ is the redshift distribution of lens galaxies in the $i$-th tomographic bin and $\bar n_{\mathrm{lens}}^i$ is the 2D angular galaxy density within the same redshift bin.

Covariance Matrix

The covariance between the two angular power spectra $C_{ab}^{ij}(\ell_1)$ and $C_{cd}^{kl}(\ell_2)$

is the sum of the Gaussian covariance $\operatorname{Cov}^{\mathrm{G}}$ and the non-Gaussian covariance. The non-Gaussian covariance further decomposes into the connected non-Gaussian covariance $\operatorname{Cov}^{\mathrm{cNG}}$ and the super-sample covariance $\operatorname{Cov}^{\mathrm{SSC}}$. The exact formulation for these three covariance terms is detailed in Appendix A of [46]. In this work, we drop the subdominant connected non-Gaussian covariance and keep the super-sample covariance [51, 52].

Systematics Modeling

We model five types of systematic uncertainties: redshift uncertainty, multiplicative shear calibration bias, intrinsic alignment (IA), baryonic feedback, and galaxy bias. The related parameter and prior spaces are summarized in Table II. A detailed description of the systematics modeling is provided in [46, 53].

Cosmology Parameters	Fiducial	Prior
$\Omega_{\mathrm{m}}$	0.3156	$\mathcal{U}(0.095, 0.585)$
$\sigma_8$	0.831	$\mathcal{U}(0.5, 1.1)$
$n_{\mathrm{s}}$	0.9645	$\mathcal{U}(0.84, 1.06)$
$\Omega_{\mathrm{b}}$	0.0491685	$\mathcal{U}(0.005, 0.095)$
$h_0$	0.6727	$\mathcal{U}(0.4, 0.9)$
$w_0$	-1	$\mathcal{U}(-2.1, 0)$
$w_{\mathrm{a}}$	0	$\mathcal{U}(-2.6, 2.6)$
Systematic Parameters	WL		KL
	Fiducial	Prior	Fiducial	Prior
$\Delta_z^i$	0.0	$\mathcal{N}(0, 2\mathrm{e}{-3})$	0.0	$\mathcal{N}(0, 4\mathrm{e}{-4})$
$\sigma_z^i$	0.01	$\mathcal{N}(0.01, 2\mathrm{e}{-3})$	0.002	$\mathcal{N}(0.002, 4\mathrm{e}{-4})$
$m^i$	0.0	$\mathcal{N}(0, 2\mathrm{e}{-3})$	0.0	$\mathcal{N}(0, 4\mathrm{e}{-4})$
$b_1^0$	-	-	0.98	$\mathcal{U}(0.0, 1.7)$
$b_1^1$	-	-	0.34	$\mathcal{U}(0.1, 1.0)$
$A_{\mathrm{IA}}$	5.92	$\mathcal{N}(5.92, 3.0)$	-	-
$\beta_{\mathrm{IA}}$	1.1	$\mathcal{N}(1.1, 1.2)$	-	-
$\eta_{\mathrm{IA}}$	-0.47	$\mathcal{N}(-0.47, 3.8)$	-	-
$\eta_{\mathrm{IA}}^{\mathrm{high\text{-}z}}$	0.0	$\mathcal{N}(0.0, 2.0)$	-	-
$Q_1$	0.0	$\mathcal{N}(0.0, 5.3)$	0.0	$\mathcal{N}(0.0, 16.78)$
$Q_2$	0.0	$\mathcal{N}(0.0, 3.24)$	0.0	$\mathcal{N}(0.0, 5.29)$
$Q_3$	0.0	$\mathcal{N}(0.0, 0.97)$	0.0	$\mathcal{N}(0.0, 1.59)$

We follow [54] to model the redshift uncertainty with a Gaussian model parameterized by the bias $\Delta_z$ and the scatter $\sigma_z$. We assume no bias in the fiducial cosmology, and take $\sigma_z^{\mathrm{WL}}=0.01$ and $\sigma_z^{\mathrm{KL}}=0.002$ [34].

We model the multiplicative bias as a Gaussian centered at 0 with standard deviation $\sigma_{m^i}$ per tomographic bin. We choose $\sigma_{m^i}=0.002$ for WL, supported by recent simulations [55]. For KL, we account for the fact that KL galaxies generally have an image signal-to-noise ratio of 100 (see Figure 2) and are spatially resolved. This implies a smaller multiplicative bias, which motivates our choice of $4\times10^{-4}$. We note that shear calibration bias at either level is subdominant in the overall error budget.

We adopt the non-linear alignment model (NLA; see [56, 57, 58]) only for red elliptical galaxies used in WL, since KL is immune to IA and any IA-like systematics [36]. We model the IA amplitude $A_{\mathrm{IA}}$, the luminosity dependence $\beta_{\mathrm{IA}}$, and the redshift dependence $\eta_{\mathrm{IA}}$ and $\eta_{\mathrm{IA}}^{\mathrm{high-z}}$, where the fiducial values and choice of priors are listed in Table II.

For galaxy-galaxy lensing and galaxy clustering, we refrain from modeling baryonic uncertainties because of conservative scale cuts and conservative marginalization over galaxy bias. We model the impact of baryonic physics on the angular shear-shear power spectrum using a principal component analysis (PCA) framework [59, 60, 61]. We derive the principal components $\mathrm{PC}_i$ with a covariance-driven weighting factor using Horizon-AGN, TNG100, MassiveBlack-II, EAGLE, cosmo-OWLs, and BAHAMAS simulations [62, 63, 64, 65, 66, 67, 68]. The cosmo-OWLs and BAHAMAS suites include three different AGN models that account for different AGN temperatures [68]. The baryon-contaminated data vector is modeled by the linear combination of $n$ principal components

where $\mathbf{Q}=(Q_i)_{i\le n}$ is the amplitude of $\mathbf{PC}_i$ and $\mathbf{M}_{\mathrm{dmo}}$ is the dark-matter-only data vector. We include the first three principal components in our analysis and use Gaussian priors centered at zero for $Q_i$. For the first principal component, we set the prior so that the 68% confidence interval matches the prediction from X-ray measurements [69, 70]. This is motivated by the fact that the first principal component closely traces the gas mass fraction [69]. For $Q_2$ and $Q_3$, we use a broad prior range that covers all hydrodynamic simulations within the 95% confidence interval. The principal components depend on a galaxy sample’s redshift distribution and the specific covariance matrix. We determine priors on the Q’s for KL and WL using the same physical relation, yielding different values.

We parameterize the galaxy bias as a linear function of the effective redshift $b_1(z) = b_1^0 z + b_1^1$. This differs from the more common approach of assuming one linear bias parameter per lens redshift bin. We choose this because this paper studies science cases with more tomographic bins, and we want to reduce the impact of additional systematic parameters that would otherwise affect our results. We adopt the linear relation estimated from fitting the High-z Emission Line Survey (HiZELS) [71, 72] since star-forming galaxies dominate the KL sample.

Galaxy Sample Selection

The Medium Tier of the HLWAS will have $Y J H$ imaging¹⁰ and grism spectroscopy over 2,300 deg$^2$ with imaging $5\sigma$ point-source photometry depth of $Y = 26.5$ and $J = H = 26.4$ (AB) and $1.5 \times10^{-16}$ erg cm$^{-2}$ s$^{-1}$ in grism ($5\sigma$ extended source line threshold, with half-light radius of $0.\!\!^{\prime\prime}3$) [41]. This configuration is different from the one used in [34], which assumes a 2,000 deg$^2$ survey with a $5\sigma$ H-band depth of 26.54 (AB) and a grism sensitivity of $10^{-16}$ erg cm$^{-2}$ s$^{-1}$ at a $7\sigma$ point-source detection limit (see Figure 2-15 in [73]), which corresponds to $(0.86-1.35) \times10^{-16}$ erg cm$^{-2}$ s$^{-1}$ for $5\sigma$ extended source of $r_{1/2} = 0.\!\!^{\prime\prime}3$, depending on wavelength. We derive the KL and the WL samples for the new survey configuration below.

Kinematic Lensing

Following [34], we use the COSMOS Mock Catalog [74] to obtain the galaxy population. To qualify for a KL measurement, a spiral galaxy needs high signal-to-noise emission-line measurements and a sufficiently large apparent size to measure the rotation curve. We therefore impose the following criteria:

1. At least one of the emission lines is resolved in the grism of resolution $R_\lambda= 451 \frac{\lambda}{1\,\mu\mathrm{m}}$ spanning from $1 - 1.93\,\mu\mathrm{m}$.

2. The resolved emission line flux is brighter than the $5\sigma$ extended source depth $1.5 \times10^{-16}$ erg cm$^{-2}$ s$^{-1}$

3. The apparent half-light radius is larger than Roman’s point spread function, which is 0.1 arcsec.

4. $F110W \le26.5$ (AB)

[34] also applies the additional constraint of spatial resolution $R \ge0.4$¹¹ and a $5\sigma$ point source limit on F160W for sample selection. We perform the same procedure and find that these additional criteria do not alter the selection results. Figure 2 characterizes the distribution of the emission-line strength and the image signal-to-noise ratio of the KL sample in the Medium Tier. Additionally, we further impose a 0.5 dropout rate to account for the observing inefficiency and completeness. These, in the end, result in a galaxy number density of 3.9 arcmin$^{-2}$ for our likelihood analysis. For our later $3 \times2$pt analysis, we assume the lens sample matches the source sample [77, 78].

We also estimate the redshift distribution for the Deep Tier following the recommendation in the ROTAC report and the HLWAS documentation¹², which has grism sensitivity of $5.6 \times10^{-17}$ erg cm$^{-2}$ s$^{-1}$ and Y-band imaging depth of 27.7 (AB) in $F110W$ over 19.2 deg$^2$. The resulting sample redshift distribution is shown as the black curve in Figure 1, and the galaxy number density is 9.65 arcmin$^{-2}$.

Weak Lensing

We assume the Medium Tier as the reference WL survey and adopt the WL sample estimated for Data Challenge 1, initialized by the Roman Project Infrastructure Team (private communication). It follows a procedure similar to the previous Roman HLWAS forecasts (e.g. [42]) by computing a realistic survey scenario with the Roman survey exposure time calculator [75] and then applying it to the CANDELS catalog [79] to obtain the sample. We fit the sample with the Smail redshift function [80], fixing $\beta= 2$. This estimation yields a redshift distribution in Figure 1 that peaks at $z \sim0.7$. The total number density is 41.3 arcmin$^{-2}$. We assume the WL shape noise $\sigma_{\epsilon}^{\mathrm{WL}} = 0.37$ per galaxy [81].

Roll-angle Study

The roll angle that optimizes the trade-off between spectra decomposition and survey efficiency is between 8 and 17 degrees [82]. [34] estimates the shape noise of KL using a toy model, assuming a 90-degree roll-angle separation. Intuitively, since the line-of-sight velocity along the major and minor axes relates closely to the inclination and $g_{\times}$, one would expect that an orthogonal spectra configuration would optimize the cosmic shear measurement of KL, and a less-optimal spectra configuration would degrade the KL shape noise. In this section, we examine how the roll-angle separation of grism exposures affects KL shear uncertainty through mock observations. The metric we focus on is the correlation coefficient between the model parameter pairs, $g_{+}$ v.s. inclination $\cos i$, and $g_{\times}$ v.s. intrinsic position angle $\theta_{\mathrm{int}}$, in the Fisher information matrix calculated from a suite of mock image and grism observations. The parameter pairs are chosen to be fully degenerate with each other when only images are considered. We describe the mock observation modeling in Sect. IV A and show Fisher information results in Sect. IV B. We show two particular Fisher information examples in Appendix A to build intuition for how grism data helps break the degeneracy between shear and shape at different roll-angle separations.

Mock KL Measurement

We use KLTOOLS¹³ to build mock Roman image and grism data. We refer to [38] for a detailed description of KLTOOLS for equivalent fiber-spectra-based KL measurement, and summarize the difference between a grism-based measurement below.

We consider three independent cutouts for each mock galaxy: one broadband image from the imaging survey, and two grism images cut around H$\alpha$ emission, observed by the grism survey. The roll angles of the two grism cutouts differ by $\Delta\theta_{\mathrm{RA}}$, and we align the dispersion direction of the first grism image with the $x$-axis of the cutout. We assume independent Gaussian pixel noise for both the broadband image and grism cutouts.

KLTOOLS first generates a dimensionless intensity profile $I(\boldsymbol{x}|\boldsymbol{\theta})$ and a line-of-sight (LoS) rotation velocity field $v_{\mathrm{LoS}}(\boldsymbol{x}|\boldsymbol{\theta})$ (km s$^{-1}$) given the model parameter vector $\boldsymbol{\theta}$, and the cutout coordinate grid $\boldsymbol{x}$. We note that both the intensity profile and the velocity field include shear distortions. We adopt an inclined exponential disk profile for $I(\boldsymbol{x}|\boldsymbol{\theta})$, and the 3D rotation velocity is modeled as [83]

where $V_{\mathrm{circ}}$ is the maximum rotation velocity and $R_{\mathrm{vscale}}$ is the scale radius of the rotation curve.

The observer-frame spectral energy distribution (SED) $F_{\lambda}^{\mathrm{obs}}(\lambda|\boldsymbol{\theta})$ (erg s$^{-1}$ cm$^{-2}$ nm$^{-1}$) is also calculated given $\boldsymbol{\theta}$ and observer-frame wavelength $\lambda$. We model the SED around the H$\alpha$ emission line as a Gaussian line profile with zero continuum, assuming perfect continuum subtraction.

The 3D photon distribution is constructed as

The broadband image $\boldsymbol{X}_{\mathrm{I}}(\boldsymbol{x}|\boldsymbol{\theta})$ can be expressed as

where $\mathcal{P}$ is the point-spread function of Roman, $\mathcal{T}$ is the total system throughput, $A$ is the telescope aperture area, $t$ is the exposure time, $G$ is the detector gain, $*$ is convolution, and $h$ is the Planck constant.

The grism image $\mathbf{X}_{\mathrm{G}}(x|\boldsymbol{\theta})$ can be expressed as

where $d(\lambda)$ is the dispersion law of the Roman grism. $d(\lambda)$ is generally expressed as

where $(x_0, y_0)$ is the position of the galaxy in the direct image, and $dx$ and $dy$ are polynomials calibrated from commissioning or laboratory data. Practically, because the grism cutouts span only a small wavelength range, we keep only the linear term in the polynomial trace around emission lines.

We note that although we only consider $g_+$, $g_\times$, $\cos i$, $\theta_{\rm int}$, and the image and H$\alpha$ half-light radius (hlr) $R_{\mathrm{h,bb}}$, $R_{\mathrm{h,H}\alpha}$ in Sect. IV B, kltools supports a much larger parameter space of the model parameter vector $\boldsymbol{\theta}$, producing realistic model broadband images and 2D grism spectra. We fix other model parameters to their fiducial values, including $V_{\mathrm{circ}} = 251.5\ \mathrm{km\ s}^{-1}$, $R_{\mathrm{vscale}} = 0.''05$, intrinsic H$\alpha$ width $\sigma_{\mathrm{H}\alpha} = 0.175\ \mathrm{nm}$; image and grism spectra fluxes are fixed to an image SNR of 100 and grism spectra SNR of 50 each.

Fisher Information and Shear Uncertainty Dependency

Assuming independent Gaussian likelihoods for each observation, the Fisher information matrix can be written as

where $\mathbf{X}_n$ is the n-th observation in the image and grism cutouts, and $C_n$ is the corresponding pixel noise covariance matrix, which is diagonal assuming uncorrelated pixel noise. The conditional correlation coefficient $r(\theta_i, \theta_j)$ between two model parameters $\theta_i$ and $\theta_j$ is

Note that when only broadband image is available, the $g_+$-$\cos i$ and $g_\times$-$\theta_{\rm int}$ are perfectly degenerate, $r(g_+, \cos i) = -1$, $r(g_\times, \theta_{\rm int}) = 1$. Adding a 2D grism spectrum breaks this degeneracy to $|r| < 1$. Adding a second 2D grism spectrum with a different dispersion direction further reduces the correlation, and we investigate the dependence of $|r|$ on the roll-angle separation between the two grism exposures in Figure 3, leaving more detailed intuition in Appendix A. Galaxies with different intrinsic position angles experience different roll angle dependencies; therefore, we consider the intrinsic position angle-averaged correlation coefficient $\langle|r| \rangle_{\theta_{\rm int}}$ (lines with markers) in Figure 3, with the shaded region showing the spread among different position angles. The face-on scenarios have the lowest correlation coefficient between degenerate parameter pairs, consistent with [35, 38]. The $g_{+}$ and $g_{\times}$ components have different characteristic behavior. At high inclination (low $\cos i$), $g_{+}$ has flat dependency on $\Delta\theta_{\mathrm{RA}}$, and $\Delta\theta_{\mathrm{RA}} = 90$ deg is slightly preferred. However, at low inclination, the $\Delta\theta_{\mathrm{RA}}$ dependence is more obvious and decreases monotonically toward larger $\Delta\theta_{\mathrm{RA}}$. $g_{\times}$, by contrast, shows a consistently flat dependence across inclinations and prefers $\Delta\theta_{\mathrm{RA}} = 90$ deg. When weighted by inverse variance in the real two-point correlation function measurement, the high $\cos i$ scenarios have much higher weight than the edge-on cases. Another factor is the spread of the roll-angle dependency. For $\Delta\theta_{\mathrm{RA}} = 180$ deg, both $g_{+}$ and $g_{\times}$ have tight spread in $|r|$-$\Delta\theta_{\mathrm{RA}}$ relation. This means a more homogeneous and healthy weight for galaxies across different position angles. Therefore, $\Delta\theta_{\mathrm{RA}} = 180$ deg is preferred, given both the inverse-variance-weighted effective trend and the homogeneity of shear uncertainty across position angles.

The reference exposure time of the Medium Tier grism is 4 passes $\times$ 2 dithers per pass $\times$ 190 s per exposure, and the four passes have K-sequence roll angles around (0, 10, 170, 180) degrees [43]. This roll-angle setup also optimizes KL shear uncertainty, as discussed, and we conclude that KL does not require changes to the current reference roll-angle setting.

From our roll angle study simulations, we estimate an overall shape noise level of $\sigma_{\epsilon}^{\mathrm{KL}} = 0.05$ per galaxy for this galaxy sample. This corresponds to a rotational-velocity measurement uncertainty of $\sigma_v \sim25$–$30$ km s$^{-1}$, which is more conservative than assumed in [34]. Studies with the Hubble Space Telescope have already achieved a precision of $\sigma_v = 15$ km s$^{-1}$ using multiple roll angles [84].

Forecasting the new survey strategy

Given our galaxy sample selection in Sect. III and the simulations that motivate the KL shape noise level for the roll-angle strategy of the Medium Tier (Sect. IV), we compute our data vector and covariance matrix as explained in Sect. II.

Figure 4 shows the difference of the simulated input KL $3 \times2$pt data vector (black dots) computed at Planck 2018 cosmology [1] and best-fit cosmologies from recent experiments:

• DESI DR2 BAO+CMB+DESY5 in $w_{0}w_{a}$CDM [9] (blue)

• DES Y6 $3 \times2$pt in $\Lambda$CDM [5] (orange)

• DES Y6 $3 \times2$pt+SN+BAO in $w_{0}w_{a}$CDM [30] (green)

Error bars are obtained as the square root of the diagonal elements of the KL covariance matrix, and the shaded region indicates scale cuts in galaxy clustering and galaxy-

galaxy lensing. We see strong differences in the models on small scales in the cosmic shear part of the data vector. These differences significantly exceed the error bars, indicating that we can best discriminate between models on these scales if we can control systematics, particularly baryonic physics.

We further explore the difference between WL and KL in the shape-noise contribution over the total uncertainty budget in Figure 5. Both scenarios assume the fiducial forecast setup with 10 tomographic bins over the Medium Tier. At the same cosmic variance, KL is less dominated by shape noise than WL. While this opens the exciting opportunity to constrain cosmological parameters and different cosmological models more tightly, it also requires better modeling of baryonic physics and other small-scale cosmic shear uncertainties.

We focus on two aspects of the KL forecasts: (i) updating the KL forecast of [34] with the ROTAC Medium Tier survey configuration (Sect. V A), and (ii) comparing the KL constraining power under different survey tiers and data timelines, including the performance of KL with the Y1 data (Sect. V B). For each forecast, we run a standard $\Lambda$CDM and a $w_{0}w_{a}$CDM scenario. Our choice of priors and the fiducial values are summarized in Table II. Our baseline forecast setting adopts 10 tomographic bins and 20 multipole bins logarithmically spaced between $\ell_{\min} = 20$ and $\ell_{\max} = 4{,}000$.

Weak Lensing and Kinematic Lensing in $\Lambda$CDM and $w_{0}w_{a}$CDM

As shown in Table I, the new survey configuration returns a total number density of the KL sample of 3.9 arcmin$^{-2}$ over a survey area of 2,300 deg$^{2}$. The WL galaxy sample in the new survey design comes out to 41.3 arcmin$^{-2}$, also over the same area. Compared to the previous study [34], we see a reduced number density because the new survey design uses a shallower, wider survey strategy.

Design	Area (deg$^{2}$)	$N_{\mathrm{gal}}$ (arcmin$^{-2}$)
WL	2300	41.3
KL	2300	3.9
KL Deep Tier	19.2	9.65
KL Y1	700	3.9

Similar to the old survey configuration, the new configuration shows a significant improvement in KL over WL (see Figure 6). In $\Lambda$CDM, KL shrinks the 1D marginalized error bar for $\Omega_m$ and $\sigma_8$ by a factor of 2.2 and 1.4, rerespectively, and it improves $\mathrm{FoM}_{\Omega_m\sigma_8}$¹⁴ by a factor of 2.2 (Table III). In $w_0w_a$CDM, we find a 2.1 times improvement in constraining power in $\mathrm{FoM}_{w_0w_a}$ (see Table IV). Overall, we find a slightly less significant improvement of KL over WL compared to the old survey design [34]. This is also partly because this paper treats baryonic physics more realistically, which affects KL more than WL. We further explore the importance of baryonic physics

Dataset	$\Omega_{\mathrm{m}}$	$\sigma_8$	$n_{\mathrm{s}}$	$\Omega_{\mathrm{b}}$	$h_0$	$\mathrm{FoM}_{\Omega_{\mathrm{m}}\sigma_8}$
WL	0.00829	0.00929	0.02826	0.01965	0.09675	22200.46
KL	0.00378	0.00642	0.02689	0.01902	0.09302	49856.78
KL Deep Tier	0.02376	0.03093	0.06179	0.02634	0.13567	3058.13
KL Y1	0.00643	0.00966	0.03187	0.02340	0.11181	24412.32
KL-2 $\times$ 2pt	0.01134	0.01511	0.04047	0.00462	0.04740	10188.61
KL-3 $\times$ 2pt	0.00304	0.00539	0.02233	0.00488	0.03443	71017.35

in Figure 7. KL with perfectly calibrated baryonic systematics gives 5.3 times better constraints in $\mathrm{FoM}_{\Omega_m\sigma_8}$ and 2.5 times better constraints in $\mathrm{FoM}_{w_0w_a}$ compared to KL with baryonic systematics, as they can be controlled today. In comparison, baryonic systematics degrade WL’s constraining power by 2.5 times in $\mathrm{FoM}_{\Omega_m\sigma_8}$ and 1.3 times in $\mathrm{FoM}_{w_0w_a}$. As expected, KL is more sensitive to baryonic feedback than WL because of smaller shape-noise uncertainties on small scales (Figure 5). This opens up a promising opportunity to improve KL constraining power through better modeling and cross-calibration of baryonic physics.

KL performance for different survey tiers

We forecast a 700 deg$^2$ survey at Medium Tier depth, which Roman will have observed approximately by the end of 2028¹⁵. We refer to this dataset as Y1 data. In this section, we forecast the KL performance on Y1 data (KL Y1) and compare with the full Medium Tier. Throughout the discussion, we refer to the full dataset as the Y5 data, since the Roman primary mission is scheduled for five years, although we note that the Medium Tier field will likely have been observed before the five-year timeline.

Dataset	$\Omega_{\mathrm{m}}$	$\sigma_{8}$	$n_{\mathrm{s}}$	$w_{0}$	$w_{\mathrm{a}}$	$\Omega_{\mathrm{b}}$	$h_{0}$	$\mathrm{FoM}_{w_{0}w_{\mathrm{a}}}$
WL	0.02546	0.02534	0.03160	0.25327	0.85267	0.02042	0.09752	10.9526
KL	0.01514	0.01859	0.03029	0.17657	0.67906	0.01987	0.09586	22.5411
KL Deep Tier	0.04499	0.04808	0.06667	0.47433	1.47330	0.02594	0.13793	2.4493
KL Y1	0.02252	0.02642	0.03511	0.26719	1.00195	0.02416	0.11168	10.0753
KL-2 × 2pt	0.03812	0.03315	0.04522	0.35348	1.07038	0.00750	0.06130	7.5095
KL-3 × 2pt	0.01289	0.01501	0.02372	0.15566	0.54830	0.00541	0.03972	34.5773

In addition to the Medium Tier data, the Deep Tier, which will have completed observations by early 2028, will be highly useful for KL. We have estimated the KL sample for the Deep Tier in Sect. III A as the black redshift distribution in Figure 1. For the Y1 data, we assume it shares the same detection sensitivity as the Y5 data. That is, it shares the same redshift distribution and galaxy number density as our estimation in Sect. III A, but covers only 700 deg$^{2}$ as noted in Table I. We adopt the same tomographic bins, multipole bins, and the fiducial values as the Y5 KL forecast.

We compare the KL performance of Deep, Y1, and Y5 and summarize the results in Table III (also see Figure 8) and Table IV. The 1D marginalized error bar of $\Omega_{\mathrm{m}}$ ($\sigma_{8}$) is 0.00378 (0.00642), 0.00643 (0.00966), and 0.02376 (0.03093) for Y5, Y1, and the Deep Tier, respectively, and the corresponding $\mathrm{FoM}_{\Omega_{\mathrm{m}}\sigma_{8}}$ is 49,857, 24,412,

and 3,058. In $w_{0}w_{\mathrm{a}}$CDM, $\mathrm{FoM}_{w_{0}w_{\mathrm{a}}}$ is 22.54, 10.08, and 2.45 for Y5, Y1, and the Deep Tier. Even though the Deep Tier has a much higher galaxy number density, its $\mathrm{FoM}_{\Omega_{\mathrm{m}}\sigma_{8}}$ is 16.3 times lower than the Y5 constraint, and its $w_{0}$–$w_{\mathrm{a}}$ is prior-dominated. The $\mathrm{FoM}_{\Omega_{\mathrm{m}}\sigma_{8}}$ of Y1 is 1.1 times larger than the WL with full Medium Tier data and has comparable $\mathrm{FoM}_{w_{0}w_{\mathrm{a}}}$, while Y1 only has 700 deg$^{2}$ compared to 2,300 deg$^{2}$ of the Y5 data, demonstrating the reduction in shape noise.

PDFstring analysis strategies

In this section, we explore replacing WL with KL in the 3 × 2pt framework. We follow the same fiducial analysis setup as in Sect. V with 10 tomographic bins and 20 multipole bins. In Sect. VIB, we quantify the information gain as we push toward narrower tomographic bins. We also quantify the impact of the Limber approximation in a narrow tomographic scenario.

Limber Regime: Standard PDFstring strategy with a spectroscopic sample

We use the KL sample for both lens and source galaxy samples in our KL-3 × 2pt analysis. This lens-equal-source idea has been discussed to eliminate nuisance parameters, especially those related to redshift uncertainties, and to improve self-calibration of other galaxy sample-related uncertainties [77, 78, 85]. For a KL-3 × 2pt analysis, we note that using the full 3D clustering sample may be more beneficial, but we postpone a corresponding analysis to future work. We adopt a scale cut of $R_{\min}=21\,\mathrm{Mpc}/h$ for the linear galaxy bias modeling following [42, 86]. Table II summarizes the prior ranges and the fiducial values for all parameters.

We first run a baseline KL-3 × 2pt analysis with 10 tomographic bins to demonstrate how the constraining power changes as we include more probes.

In Figure 9, we see a mild increase in constraining power in $\Omega_{\mathrm{m}}$–$\sigma_{8}$ and $w_{0}$–$w_{\mathrm{a}}$ when we combine KL-cosmic shear and KL-2 × 2pt (the combination of galaxy-galaxy lensing and galaxy clustering). KL-2 × 2pt alone gives rather loose constraints on these parameters; however, it breaks degeneracies $\Omega_b$–$h_0$ and $\Omega_m$–$h_0$. $\mathrm{FoM}_{\Omega_m\sigma_8}$ changes by

1.4 times and $\mathrm{FoM}_{w_0w_a}$ increases by 1.5 times, as shown in Table III and IV.

This mild improvement is due to the conservative choice of $R_{\min}$, which again indicates opportunities for increased constraining power through better modeling of small-scale systematics, in this case galaxy bias. In Figure 10, we calculate the data vector signal-to-noise ratio (SNR), defined as

for KL, KL-2 × 2pt, and KL-3 × 2pt with respect to different $R_{\min}$. We see that the KL-2 × 2pt SNR decreases rapidly with increasing $R_{\min}$ and it matches KL-cosmic shear near 8 Mpc/$h$. This analysis motivates significant investment into small-scale modeling of galaxy bias and corresponding priors.

Replacing the KL sample with the main Roman spectroscopic sample for the lens sample to increase number density is another promising avenue to increase constraining power. At lower redshift, the lens sample could be amended by DESI galaxies, which could be combined with the source sample from the Medium Tier. The authors in [40] forecast that the constraining power on the dark energy equation of state can be improved by 192% after including KL shear ratio (KiLeR) in the combined datasets of CMB, BAO, and SN.

Non-Limber Regime: Narrow tomographic bins

Since all galaxies in the KL-3 × 2pt have spectroscopic redshifts, adopting more tomographic bins is an interesting prospect. Such a narrow tomographic-bin analysis would more sensitively probe the time evolution of the dark energy equation of state. We explore this by increasing the number of tomographic bins on a fixed redshift range.

We examine three scenarios for KL-3 × 2pt: 10 (the baseline), 15, and 20 tomographic bins. The simulated analyses show a mild improvement in FoM$_{w_0w_a}$ from 34.58 to 45.93 as we increase the number of tomographic bins from 10 to 20. A similar trend appears in [34] in the cosmic shear-only scenario: increasing the number of tomographic bins does not drastically improve constraining power when the dark energy equation of state evolves slowly over time.

An additional complication of the narrow-bin KL-3 × 2pt concept is the breakdown of the Limber approximation in the clustering part of the data vector. If non-Limber effects are ignored, this can cause significant parameter biases in the inference [47, 87]. We study this scenario in Figure 11, where we analyze a non-Limber-computed data vector with the Limber model. To isolate the effect of model misspecification, we exclude baryonic effects in these analyses, so the bias shift is stronger in Figure 11 than in reality.

We quantify the parameter shift normalized to the Limber posterior in Figure 12. The upper panel shows the Mahalanobis distance of the posterior mean shift in the $w_0w_a$CDM cosmology [47, 88]. The bottom panel shows the bias in each parameter. The model misspecification impacts $b^0_1$ and $\sigma_8$ the most, shifting the 1D marginalized mean by $6.2\sigma$ and $2.7\sigma$ in the 20 tomographic bin case.

Conclusion

In this paper, we perform simulated analyses to forecast KL individually and KL multiprobe strategies with the updated Roman HLWAS survey design. We estimate realistic KL samples using the COSMOS Mock Catalog and derive redshift distributions for the Deep and Medium Tiers of the HLWAS survey. We also consider how telescope roll-angle separation affects KL measurements and explore different scenarios to assess which separation best constrains KL. We compare KL to the traditional photometric WL analysis. Our forecasts include realistic systematic effects, including photometric-redshift and shape-measurement uncertainties, intrinsic alignment, baryonic physics, and galaxy bias.

We find that KL outperforms WL by a factor of $\ge$ two in the FoM for either $\Lambda$CDM or $w_0w_a$CDM. For Year 1 data with 700 deg$^2$ of full depth, KL alone provides constraining power comparable to the Y5 WL forecast. We note that this study does not include the WL Wide Tier. We further find that Roman’s current roll-angle strategy is near-optimal for KL.

We extend our analysis to explore a multiprobe idea by using the KL sample directly as the lens sample to perform a KL-3 × 2pt analysis. Using the KL sample eliminates photometric uncertainties and enables narrow tomographic cross-correlations between lensing and clustering. Joint clustering and lensing analyses using finer tomographic bins probe the redshift evolution of structure growth and geometry of the universe. While this can better constrain time-dependent dark energy models, the improvement depends strongly on the model’s redshift variability; for smooth dark energy evolution, the information gain is not significant. We further show that such an analysis needs to account for non-Limber effects, as they can otherwise bias results in the narrow-tomographic-bin scenario.

Given KL’s low shape noise and corresponding sensitivity to small scales, its performance is more sensitive to baryonic systematics, elevating baryonic systematics to an even greater concern than in standard WL analyses. We quantify the difference in KL constraining power between state-of-the-art baryonic systematics modeling and a scenario where baryonic physics errors are subdominant. We find that there are significant gains possible if baryonic physics models can be improved (either via simulations or self-calibration).

Several possible future extensions of this work are worth noting. First, using the full Roman clustering sample, without downselecting to the KL sample, is an interesting avenue, especially when combining full 3D clustering constraints with KL. This clustering sample can be extended to lower redshift by using DESI or Euclid galaxies. Second, combining WL and KL in a multiprobe analysis is promising, since the WL sample extends to higher redshift. Third, given Roman’s projected lifetime of $\sim22$ years¹⁶, the community could consider extending the current spectroscopic survey to the Roman wide area. While this adds challenges, since the Roman Wide Tier is covered only in H-band [41], it will strongly boost KL constraining power, given that KL is cosmic-variance limited on most scales.

Acknowledgments

This work was supported by NASA Roman WFS 22-ROMAN22-0016. This material is based upon High Performance Computing (HPC) resources supported by the University of Arizona TRIF, UITS, and the Office of Research and Partnerships and maintained by the UArizona Research Technologies department. YHH thanks Haley Bowden for the useful discussion on the WL sample in CPIP DC1.

Impacts of Roll Angle Separation

In this section, we develop the intuition for how the shear-shape degeneracy breaks when 2D grism spectra are included, and how different roll angles impact the performance. Here we show two examples with different intrinsic position angles in Figure 13 ($\theta_{\mathrm{int}} = 0$ deg in the left panel, and $\theta_{\mathrm{int}} = 280$ deg in the right panel). The figures show the partial derivatives of mock observations (broadband images and 2D grism spectra) with respect to different model parameters: $g_{+}$, $g_{\times}$, $\cos i$, and $\theta_{\mathrm{int}}$, with blue (red) meaning negative (positive) partial derivative. The perfect degeneracy between $g_{+}$-$\cos i$ and $g_{\times}$-$\theta_{\mathrm{int}}$ can be visualized immediately from the same patterns in the first column of Figure 13. While the response of 2D grism spectra to $g_{+}$ and $g_{\times}$ is very similar to their image response counterparts, the grism response to $\cos i$ and $\theta_{\mathrm{int}}$ is very different in this case (low inclination $\cos i = 0.9$), and adding the first grism observation helps in breaking the degeneracy. When the second grism spectrum is further included, while its responses to $g_{+}$ and $g_{\times}$ are still very similar to the image counterparts and change slowly at different roll angle separations, the responses to $\cos i$ and $\theta_{\mathrm{int}}$ show richer phenomena as a function of roll angle separation. The roll-angle optimization question then turns into which roll angle provides the most independent $\cos i$ and $\theta_{\mathrm{int}}$ response patterns, and the answer depends on the exact geometry configuration of the galaxy. For the $\theta_{\mathrm{int}} = 0$ deg example, $\Delta\theta_{\mathrm{RA}} = 180$ deg provides the most independent $\cos i$ response pattern while $\Delta\theta_{\mathrm{RA}} = 130$ deg provides the most independent $\theta_{\mathrm{int}}$ response pattern among the three roll angle examples. The $\theta_{\mathrm{int}} = 280$ deg example has a different preference.

References