Introduction
The recent DESI data analysis [1 , 2 ] provided tantalizing hints for dynamical dark energy, i.e. dark energy with a variable equation of state (EoS). The equation of state alone, however, does not capture the entire phenomenology of dark energy, and models based on scalar fields, somewhat analogous to inflationary scalar fields, have been proposed and examined [3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ] . In order to mimic the DESI best fit evolving EoS w ( z ) w(z) w ( z ) w < − 1 w < -1 w < − 1 w > − 1 w > -1 w > − 1 [15 ] . Interestingly, non-minimally coupled scalar field models that introduce an interaction of the scalar field with matter seem to provide a good fit to current cosmological data while naturally producing an effective phantom crossing without introducing ghost or gradient instabilities [16 , 17 ] . Such models, however, automatically generate a fifth force acting on baryonic matter, that is tightly constrained by laboratory experiments and Solar System tests [18 , 19 ] (except if new ingredients are added to screen the fifth force on small scales [20 , 21 , 22 ] ). An elegant solution to evade those constraints while still solving the “phantom crossing” problem, is to couple only dark matter to the scalar field, as shown in [23 , 24 , 25 , 26 , 27 ] .
Perhaps the simplest and one of the most widely studied models of this type is a dark matter-dark energy interaction, namely the coupled dark energy model. In its basic incarnation, this model depends on a single constant coupling Q d m Q_{\mathrm{dm}} Q dm [28 , 29 ] , which for a point source of mass M M M
Ψ ^ ( r ) = − G M r ( 1 + 2 Q d m 2 e − m r ) . (1) \hat{\Psi}(r)=-\frac{GM}{r}\left(1+2Q_{\mathrm{dm}}^{2}e^{-mr}\right). \tag*{(1)} Ψ ^ ( r ) = − r GM ( 1 + 2 Q dm 2 e − m r ) . (1) Here, M M M m m m ϕ \phi ϕ Q d m Q_{\mathrm{dm}} Q dm [30 , 31 , 32 , 33 ] . Analyses including DESI data have also found hints of a nonzero coupling, although their significance depends on the datasets and assumptions adopted [24 , 25 , 34 , 35 , 36 , 37 ] . If this evidence is confirmed by additional data, e.g. Euclid, the dark-dark interaction could become a new standard cosmological model.
The coupled dark energy model contains a startling prediction: that the weak equivalence principle is broken at cosmological scales. That is, baryons and dark matter fall differently in a gravitational potential: while baryons remain uncoupled and feel standard gravity, following the geodesics of the metric, dark matter feels an extra interaction mediated by the scalar field and therefore deviates from the geodesics of the metric g μ ν g_{\mu\nu} g μν ϕ \phi ϕ [24 , 38 ] (and in principle also with space, but this would occur only beyond linear theory).
Recently, a direct way to test the EP in large scale structure has been identified [39 , 40 , 41 , 42 ] . The main idea is to employ the gravitational redshift effect to measure the gravitational potential and compare this with the effective potential Ψ ~ \tilde{\Psi} Ψ ~ [43 , 44 , 45 , 46 ] , which can be extracted from the two-point correlation function or the power spectrum by cross-correlating two (or more) sub-populations of galaxies with different biases [47 , 48 , 49 , 50 , 51 , 52 ] . This technique, multi-tracer with relativistic corrections, has been exploited in Ref. [41 ] to provide the first forecasts of EP-violation in present and future surveys, in a way that is independent of (i.e., marginalized over) the power spectrum shape, the background cosmology, and the perturbation growth rate.
This paper is devoted to forecasting the performance of this cosmological EP test in the context of coupled dark energy (CDE) and its generalization to doubly coupled dark energy (DCDE). In the latter model, the scalar field is conformally coupled to dark matter and nonminimally coupled to gravity. We consider the observable E P E_P E P [41 ] , which provides a null test of the EP and can be measured in a model-independent way from galaxy power spectra. We express this quantity in the CDE and DCDE models, obtaining a general form that helps understand the roles of the various sectors of the theory’s Lagrangian. The model reduces to CDE when the nonminimal gravitational coupling is switched off. We then perform Fisher forecasts on these couplings for DESI, Euclid and SKA2 through a projection from E P E_P E P k k k E P E_P E P
Evolution equations in a doubly coupled dark energy model
In this section, we derive the evolution equations for the DCDE model, where dark matter and the metric are conformally coupled to a scalar field ϕ \phi ϕ f 1 f_1 f 1 f 2 f_2 f 2 f 1 , f 2 f_1, f_2 f 1 , f 2 [28 , 53 ] . The corresponding action of a scalar-tensor theory, where a dynamical scalar field ϕ \phi ϕ [54 ] ) and the dark matter sector (cf. Ref. [55 ] ), is given by:
S = ∫ d 4 x − g [ 1 2 κ f 1 ( ϕ ) R + L ϕ ( g μ ν ) + L b ( g μ ν , ψ b ) + L d m ( f 2 ( ϕ ) g μ ν , ψ D M ) ] , (2) S = \int d^4x\sqrt{-g}\left[\frac{1}{2\kappa}f_1(\phi)R + \mathcal{L}_{\phi}(g_{\mu\nu}) + \mathcal{L}_{b}(g_{\mu\nu},\psi_b) + \mathcal{L}_{\mathrm{dm}}(f_2(\phi)g_{\mu\nu},\psi_{\mathrm{DM}})\right], \tag*{(2)} S = ∫ d 4 x − g [ 2 κ 1 f 1 ( ϕ ) R + L ϕ ( g μν ) + L b ( g μν , ψ b ) + L dm ( f 2 ( ϕ ) g μν , ψ DM ) ] , (2) where ψ b \psi_b ψ b ψ d m \psi_{\mathrm{dm}} ψ dm κ = 8 π G \kappa= 8\pi G κ = 8 π G
L ϕ ( g μ ν ) = − 1 2 ( 1 − 6 Q 2 ) f 1 ( ϕ ) ∇ μ ϕ ∇ μ ϕ − V ( ϕ ) , (3) \mathcal{L}_{\phi}(g_{\mu\nu}) = -\frac{1}{2}(1-6Q^2)f_1(\phi)\nabla_{\mu}\phi\nabla^{\mu}\phi- V(\phi), \tag*{(3)} L ϕ ( g μν ) = − 2 1 ( 1 − 6 Q 2 ) f 1 ( ϕ ) ∇ μ ϕ ∇ μ ϕ − V ( ϕ ) , (3) where Q Q Q (3) is chosen so that Q Q Q f 1 f_1 f 1 f 1 f_1 f 1 f 2 f_2 f 2 Q Q Q f 2 ≠ c o n s t f_2 \ne\mathrm{const} f 2 = const
Varying the action (2) with respect to the metric g μ ν g_{\mu\nu} g μν ϕ \phi ϕ
f 1 G μ ν + g μ ν □ f 1 − ∇ μ ∇ ν f 1 = κ ( T μ ν d m + T μ ν b + T μ ν ϕ ) , (4) f_1G_{\mu\nu} + g_{\mu\nu}\Box f_1 - \nabla_{\mu}\nabla_{\nu}f_1 = \kappa\left(T^{\mathrm{dm}}_{\mu\nu} + T^{b}_{\mu\nu} + T^{\phi}_{\mu\nu}\right), \tag*{(4)} f 1 G μν + g μν □ f 1 − ∇ μ ∇ ν f 1 = κ ( T μν dm + T μν b + T μν ϕ ) , (4) Figure 1. Schematic illustration of interactions between the Ricci scalar R R R L d m / L b L_{\rm dm}/L_{\rm b} L dm / L b ϕ \phi ϕ Q ( ϕ ) Q(\phi) Q ( ϕ ) Q d m ( ϕ ) Q_{\rm dm}(\phi) Q dm ( ϕ )
□ ϕ + 1 2 f 1 , ϕ f 1 ∇ μ ϕ ∇ μ ϕ = 1 ( 1 − 6 Q 2 ) f 1 ( V , ϕ − 1 2 κ f 1 , ϕ R − 1 2 f 2 , ϕ f 2 T d m ) (5) \Box\phi+\frac{1}{2}\frac{f_{1,\phi}}{f_1}\nabla_\mu\phi\nabla^\mu\phi=\frac{1}{(1-6Q^2)f_1}\left(V_{,\phi}-\frac{1}{2\kappa}f_{1,\phi}R-\frac{1}{2}\frac{f_{2,\phi}}{f_2}T^{\rm dm}\right) \tag*{(5)} □ ϕ + 2 1 f 1 f 1 , ϕ ∇ μ ϕ ∇ μ ϕ = ( 1 − 6 Q 2 ) f 1 1 ( V , ϕ − 2 κ 1 f 1 , ϕ R − 2 1 f 2 f 2 , ϕ T dm ) (5) where G μ ν = R μ ν − 1 2 R g μ ν G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} G μν = R μν − 2 1 R g μν □ ϕ ≡ ∇ μ ∇ μ ϕ \Box\phi\equiv\nabla_\mu\nabla^\mu\phi □ ϕ ≡ ∇ μ ∇ μ ϕ ϕ \phi ϕ { b , d m } \{\mathrm{b},\mathrm{dm}\} { b , dm } T d m T^{\rm dm} T dm f i , ϕ = d f i / d ϕ f_{i,\phi}=df_i/d\phi f i , ϕ = d f i / d ϕ V , ϕ = d V / d ϕ V_{,\phi}=dV/d\phi V , ϕ = d V / d ϕ
We consider a flat, linearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) metric in the conformal Newtonian gauge, where the two scalar degrees of freedom are characterized by the gravitational potentials Ψ \Psi Ψ Φ \Phi Φ
d s 2 = a 2 [ − ( 1 + 2 Ψ ) d η 2 + ( 1 − 2 Φ ) d x i d x i ] (6) ds^2=a^2\left[-(1+2\Psi)d\eta^2+(1-2\Phi)dx^i dx_i\right] \tag*{(6)} d s 2 = a 2 [ − ( 1 + 2Ψ ) d η 2 + ( 1 − 2Φ ) d x i d x i ] (6) where a a a η \eta η (6) and assuming pressureless dark matter and baryons, one can obtain the Friedmann equation,
3 H 2 f 1 = κ [ ρ d m + ρ b + 1 2 ( 1 − 6 Q 2 ) f 1 H 2 ϕ ′ 2 + V ( ϕ ) ] − 3 H 2 f 1 , ϕ ϕ ′ (7) 3H^2f_1=\kappa\left[\rho_{\rm dm}+\rho_{\rm b}+\frac{1}{2}(1-6Q^2)f_1H^2\phi'^2+V(\phi)\right]-3H^2f_{1,\phi}\phi' \tag*{(7)} 3 H 2 f 1 = κ [ ρ dm + ρ b + 2 1 ( 1 − 6 Q 2 ) f 1 H 2 ϕ ′2 + V ( ϕ ) ] − 3 H 2 f 1 , ϕ ϕ ′ (7) where a prime denotes differentiation with respect to the e e e N = ln a N=\ln a N = ln a ρ ϕ = 1 2 ( 1 − 6 Q 2 ) f 1 ( ϕ ) H 2 ϕ ′ 2 + V ( ϕ ) \rho_\phi=\frac{1}{2}(1-6Q^2)f_1(\phi)H^2\phi'^2+V(\phi) ρ ϕ = 2 1 ( 1 − 6 Q 2 ) f 1 ( ϕ ) H 2 ϕ ′2 + V ( ϕ )
ρ D E , e f f = 1 2 ( 1 − 6 Q 2 ) f 1 ( ϕ ) H 2 ϕ ′ 2 + V ( ϕ ) − 3 H 2 f 1 , ϕ ϕ ′ κ (8) \rho_{\rm DE,eff}=\frac{1}{2}(1-6Q^2)f_1(\phi)H^2\phi'^2+V(\phi)-\frac{3H^2f_{1,\phi}\phi'}{\kappa} \tag*{(8)} ρ DE , eff = 2 1 ( 1 − 6 Q 2 ) f 1 ( ϕ ) H 2 ϕ ′2 + V ( ϕ ) − κ 3 H 2 f 1 , ϕ ϕ ′ (8) such that
3 H 2 f 1 = κ ( ρ d m + ρ b + ρ D E , e f f ) (9) 3H^2f_1=\kappa\left(\rho_{\rm dm}+\rho_{\rm b}+\rho_{\rm DE,eff}\right) \tag*{(9)} 3 H 2 f 1 = κ ( ρ dm + ρ b + ρ DE , eff ) (9) and one can define
Ω ~ d m = κ ρ d m 3 H 2 f 1 ; Ω ~ b = κ ρ b 3 H 2 f 1 ; Ω ~ D E , e f f = κ ρ D E , e f f 3 H 2 f 1 , (10) \widetilde{\Omega}_{\rm dm}=\frac{\kappa\rho_{\rm dm}}{3H^2f_1};\quad\widetilde{\Omega}_{\rm b}=\frac{\kappa\rho_{\rm b}}{3H^2f_1};\quad\widetilde{\Omega}_{\rm DE,eff}=\frac{\kappa\rho_{\rm DE,eff}}{3H^2f_1}, \tag*{(10)} Ω dm = 3 H 2 f 1 κ ρ dm ; Ω b = 3 H 2 f 1 κ ρ b ; Ω DE , eff = 3 H 2 f 1 κ ρ DE , eff , (10) to ensure that Ω ~ d m + Ω ~ b + Ω ~ D E , e f f = 1 \widetilde{\Omega}_{\rm dm}+\widetilde{\Omega}_{\rm b}+\widetilde{\Omega}_{\rm DE,eff}=1 Ω dm + Ω b + Ω DE , eff = 1 [56 ] in which these density fractions are defined in a similar way. The background Klein-Gordon equation is given by
ϕ ′ ′ + ( 3 + H ′ H ) ϕ ′ + 1 2 f 1 , ϕ f 1 ϕ ′ 2 = − 1 ( 1 − 6 Q 2 ) f 1 H 2 [ V , ϕ − 3 f 1 , ϕ κ H 2 ( 2 + H ′ H ) + f 2 , ϕ 2 f 2 ρ d m ] (11) \phi''+\left(3+\frac{H'}{H}\right)\phi'+\frac{1}{2}\frac{f_{1,\phi}}{f_1}\phi'^2=-\frac{1}{(1-6Q^2)f_1H^2}\left[V_{,\phi}-\frac{3f_{1,\phi}}{\kappa}H^2\left(2+\frac{H'}{H}\right)+\frac{f_{2,\phi}}{2f_2}\rho_{\rm dm}\right] \tag*{(11)} ϕ ′′ + ( 3 + H H ′ ) ϕ ′ + 2 1 f 1 f 1 , ϕ ϕ ′2 = − ( 1 − 6 Q 2 ) f 1 H 2 1 [ V , ϕ − κ 3 f 1 , ϕ H 2 ( 2 + H H ′ ) + 2 f 2 f 2 , ϕ ρ dm ] (11) Defining Q d m ( ϕ ) ≡ f 2 , ϕ 2 f 2 Q_{\rm dm}(\phi)\equiv\frac{f_{2,\phi}}{2f_2} Q dm ( ϕ ) ≡ 2 f 2 f 2 , ϕ (2) :
∇ ν T μ ν b = 0 (12) \nabla^\nu T^{\rm b}_{\mu\nu}=0 \tag*{(12)} ∇ ν T μν b = 0 (12) ∇ ν T μ ν d m = Q d m T d m ∇ μ ϕ (13) \nabla^\nu T^{\rm dm}_{\mu\nu}=Q_{\rm dm}T^{\rm dm}\nabla_\mu\phi \tag*{(13)} ∇ ν T μν dm = Q dm T dm ∇ μ ϕ (13) ∇ ν T μ ν D E , e f f = − Q d m T d m ∇ μ ϕ + 1 κ G μ ν ∇ ν f 1 (14) \nabla^\nu T^{\rm DE,eff}_{\mu\nu}=-Q_{\rm dm}T^{\rm dm}\nabla_\mu\phi+\frac{1}{\kappa}G_{\mu\nu}\nabla^\nu f_1 \tag*{(14)} ∇ ν T μν DE , eff = − Q dm T dm ∇ μ ϕ + κ 1 G μν ∇ ν f 1 (14) The time components of the above conservation equations at the background level then lead to
ρ b ′ + 3 ρ b = 0 , (15) \rho'_b+3\rho_b=0, \tag*{(15)} ρ b ′ + 3 ρ b = 0 , (15) ρ d m ′ + 3 ρ d m = Q d m ρ d m ϕ ′ , (16) \rho'_{\rm dm}+3\rho_{\rm dm}=Q_{\rm dm}\rho_{\rm dm}\phi', \tag*{(16)} ρ dm ′ + 3 ρ dm = Q dm ρ dm ϕ ′ , (16) ρ D E , e f f ′ + 3 ( ρ D E , e f f + p D E , e f f ) = − Q d m ρ d m ϕ ′ + 3 H 2 κ f 1 , ϕ ϕ ′ . (17) \rho'_{\rm DE,eff}+3(\rho_{\rm DE,eff}+p_{\rm DE,eff})=-Q_{\rm dm}\rho_{\rm dm}\phi'+\frac{3\mathcal{H}^2}{\kappa}f_{1,\phi}\phi'. \tag*{(17)} ρ DE , eff ′ + 3 ( ρ DE , eff + p DE , eff ) = − Q dm ρ dm ϕ ′ + κ 3 H 2 f 1 , ϕ ϕ ′ . (17) Solving Eq. (16) , we have
ρ d m ( N ) = m d m ( N ) n d m ( N ) = ρ d m , 0 e − 3 N e ∫ ϕ 0 ϕ Q d m ( ϕ ^ ) d ϕ ^ , (18) \rho_{\rm dm}(N)=m_{\rm dm}(N)n_{\rm dm}(N)=\rho_{{\rm dm},0}e^{-3N}e^{\int_{\phi_0}^{\phi}Q_{\rm dm}(\hat{\phi})d\hat{\phi}}, \tag*{(18)} ρ dm ( N ) = m dm ( N ) n dm ( N ) = ρ dm , 0 e − 3 N e ∫ ϕ 0 ϕ Q dm ( ϕ ^ ) d ϕ ^ , (18) where a subscript ”0” denotes the quantities at the current epoch. Assuming a conserved total number of dark matter particles, the particle mass varies due to the presence of the coupling Q d m Q_{\rm dm} Q dm m d m ( N ) = m d m , 0 e ∫ ϕ 0 ϕ Q d m ( ϕ ^ ) d ϕ ^ m_{\rm dm}(N)=m_{{\rm dm},0}e^{\int_{\phi_0}^{\phi}Q_{\rm dm}(\hat{\phi})d\hat{\phi}} m dm ( N ) = m dm , 0 e ∫ ϕ 0 ϕ Q dm ( ϕ ^ ) d ϕ ^
At the perturbation level, the perturbed scalar field ϕ ( p ) \phi_{\rm(p)} ϕ ( p ) ρ m , ( p ) \rho_{\rm m,(p)} ρ m , ( p ) u m , ( p ) μ u^\mu_{\rm m,(p)} u m , ( p ) μ
ϕ ( p ) = ϕ + φ ; ρ m , ( p ) = ρ m ( 1 + δ m ) ; u m , ( p ) μ = 1 a ( 1 − Ψ , v m i ) (19) \phi_{\rm(p)}=\phi+\varphi\ ;\quad\rho_{\rm m,(p)}=\rho_{\rm m}(1+\delta_{\rm m})\ ;\quad u^\mu_{\rm m,(p)}=\frac{1}{a}(1-\Psi,v^i_{\rm m}) \tag*{(19)} ϕ ( p ) = ϕ + φ ; ρ m , ( p ) = ρ m ( 1 + δ m ) ; u m , ( p ) μ = a 1 ( 1 − Ψ , v m i ) (19) with a subscript (p) denoting the perturbed quantities, v i = d x i / d η v^i=dx^i/d\eta v i = d x i / d η x i x^i x i m = { b , d m } m=\{\mathrm{b},\mathrm{dm}\} m = { b , dm } (6) into the matter conservation equations (12) and (13) , the continuity and Euler equations in Fourier space3
δ b ′ = − θ b + 3 Φ ′ , (20) \delta'_b=-\theta_b+3\Phi', \tag*{(20)} δ b ′ = − θ b + 3 Φ ′ , (20) θ b ′ + θ b ( 1 + H ′ H ) − Ψ λ 2 = 0 , (21) \theta'_b+\theta_b\left(1+\frac{\mathcal{H}'}{\mathcal{H}}\right)-\frac{\Psi}{\lambda^2}=0, \tag*{(21)} θ b ′ + θ b ( 1 + H H ′ ) − λ 2 Ψ = 0 , (21) δ d m ′ + θ d m − 3 Φ ′ = Q d m φ ′ + Q d m , ϕ ϕ ′ φ , (22) \delta'_{\rm dm}+\theta_{\rm dm}-3\Phi'=Q_{\rm dm}\varphi'+Q_{{\rm dm},\phi}\phi'\varphi, \tag*{(22)} δ dm ′ + θ dm − 3 Φ ′ = Q dm φ ′ + Q dm , ϕ ϕ ′ φ , (22) θ d m ′ + θ d m ( 1 + H ′ H + Q d m ϕ ′ ) − Q d m φ + Ψ λ 2 = 0 , (23) \theta'_{\rm dm}+\theta_{\rm dm}\left(1+\frac{\mathcal{H}'}{\mathcal{H}}+Q_{\rm dm}\phi'\right)-\frac{Q_{\rm dm}\varphi+\Psi}{\lambda^2}=0, \tag*{(23)} θ dm ′ + θ dm ( 1 + H H ′ + Q dm ϕ ′ ) − λ 2 Q dm φ + Ψ = 0 , (23) where θ m \theta_m θ m θ m = i k i v m i / H \theta_m=ik_iv^i_m/\mathcal{H} θ m = i k i v m i / H k k k H = a H \mathcal{H}=aH H = a H λ = H / k \lambda=\mathcal{H}/k λ = H / k Q d m , ϕ ≡ d Q d m / d ϕ Q_{{\rm dm},\phi}\equiv dQ_{\rm dm}/d\phi Q dm , ϕ ≡ d Q dm / d ϕ
Φ = − λ 2 2 f 1 [ 2 V ^ Ψ + φ V ^ , ϕ − 3 φ f 1 , ϕ ϕ ϕ ′ + 3 ( δ b Ω b + δ d m Ω d m ) + Φ ′ ( f 1 + 3 ϕ ′ f 1 , ϕ ) − 3 f 1 , ϕ ( φ + φ ′ ) + 6 Ψ ( Ω b + Ω d m ) + κ ( 1 − 6 Q 2 ) ( f 1 ϕ ′ φ ′ + f 1 , ϕ ϕ ′ φ 2 2 ) ] + φ f 1 , ϕ 2 f 1 , (24) \Phi=-\frac{\lambda^2}{2f_1}\left[2\hat{V}\Psi+\varphi\hat{V}_{,\phi}-3\varphi f_{1,\phi\phi}\phi'+3(\delta_b\Omega_b+\delta_{\rm dm}\Omega_{\rm dm})+\Phi'(f_1+3\phi'f_{1,\phi})-3f_{1,\phi}(\varphi+\varphi')+6\Psi(\Omega_b+\Omega_{\rm dm})+\kappa(1-6Q^2)\left(f_1\phi'\varphi'+\frac{f_{1,\phi}\phi'\varphi^2}{2}\right)\right]+\varphi\frac{f_{1,\phi}}{2f_1}, \tag*{(24)} Φ = − 2 f 1 λ 2 [ 2 V ^ Ψ + φ V ^ , ϕ − 3 φ f 1 , ϕϕ ϕ ′ + 3 ( δ b Ω b + δ dm Ω dm ) + Φ ′ ( f 1 + 3 ϕ ′ f 1 , ϕ ) − 3 f 1 , ϕ ( φ + φ ′ ) + 6Ψ ( Ω b + Ω dm ) + κ ( 1 − 6 Q 2 ) ( f 1 ϕ ′ φ ′ + 2 f 1 , ϕ ϕ ′ φ 2 ) ] + φ 2 f 1 f 1 , ϕ , (24) Ψ = Φ − φ f 1 , ϕ f 1 , (25) \Psi=\Phi-\varphi\frac{f_{1,\phi}}{f_1}, \tag*{(25)} Ψ = Φ − φ f 1 f 1 , ϕ , (25) with V ^ = V a 2 / H 2 \hat{V}=Va^2/\mathcal{H}^2 V ^ = V a 2 / H 2 κ = c = 1 \kappa=c=1 κ = c = 1 Ω m ( N ) = κ ρ m / 3 H 2 = f 1 Ω ~ d m \Omega_m(N)=\kappa\rho_m/3\mathcal{H}^2=f_1\widetilde{\Omega}_{\rm dm} Ω m ( N ) = κ ρ m /3 H 2 = f 1 Ω dm Q d m Q_{\rm dm} Q dm Q = − f 1 , ϕ 2 f 1 Q=-\frac{f_{1,\phi}}{2f_1} Q = − 2 f 1 f 1 , ϕ Q d m = f 2 , ϕ 2 f 2 Q_{\rm dm}=\frac{f_{2,\phi}}{2f_2} Q dm = 2 f 2 f 2 , ϕ (1) with a constant coupling. Then the general solutions are f 1 ( ϕ ) = f 1 , 0 e − 2 Q ( ϕ − ϕ 0 ) f_1(\phi)=f_{1,0}e^{-2Q(\phi-\phi_0)} f 1 ( ϕ ) = f 1 , 0 e − 2 Q ( ϕ − ϕ 0 ) f 2 ( ϕ ) = f 2 , 0 e 2 Q d m ( ϕ − ϕ 0 ) f_2(\phi)=f_{2,0}e^{2Q_{\rm dm}(\phi-\phi_0)} f 2 ( ϕ ) = f 2 , 0 e 2 Q dm ( ϕ − ϕ 0 ) ϕ 0 \phi_0 ϕ 0 f i , 0 ≡ f i ( ϕ 0 ) f_{i,0}\equiv f_i(\phi_0) f i , 0 ≡ f i ( ϕ 0 ) f 1 , 0 = f 2 , 0 = 1 f_{1,0}=f_{2,0}=1 f 1 , 0 = f 2 , 0 = 1 ϕ 0 = 0 \phi_0=0 ϕ 0 = 0 f 1 = e − 2 Q ϕ f_1=e^{-2Q\phi} f 1 = e − 2 Qϕ f 2 = e 2 Q d m ϕ f_2=e^{2Q_{\rm dm}\phi} f 2 = e 2 Q dm ϕ
On sub-horizon scales (λ ≪ 1 \lambda\ll1 λ ≪ 1 [57 ] , by implying that spatial derivatives dominate over the time derivatives of slowly evolving perturbations, i.e. X ∼ X ′ ∼ X ′ ′ X\sim X'\sim X'' X ∼ X ′ ∼ X ′′
f ~ ( k , t ) = ∫ d 3 x f ( x , t ) e − i k ⋅ x ; f ( x , t ) = ∫ d 3 k ( 2 π ) 3 f ~ ( k , t ) e i k ⋅ x . \tilde{f}(\mathbf{k},t)=\int d^3x f(\mathbf{x},t)e^{-i\mathbf{k}\cdot\mathbf{x}};\quad f(\mathbf{x},t)=\int\frac{d^3k}{(2\pi)^3}\tilde{f}(\mathbf{k},t)e^{i\mathbf{k}\cdot\mathbf{x}}. f ~ ( k , t ) = ∫ d 3 x f ( x , t ) e − i k ⋅ x ; f ( x , t ) = ∫ ( 2 π ) 3 d 3 k f ~ ( k , t ) e i k ⋅ x . all perturbation quantities X ∈ { δ , θ , Ψ , Φ } X \in\{\delta,\theta,\Psi,\Phi\} X ∈ { δ , θ , Ψ , Φ } X ′ X' X ′ X ′ ′ ∼ X ≪ λ − 2 X X'' \sim X \ll\lambda^{-2}X X ′′ ∼ X ≪ λ − 2 X 4
( 1 − 6 Q 2 + a 2 m ϕ 2 k 2 ) φ = − 3 λ 2 Q d m Ω ~ d m δ d m − Q ( 2 Ψ − 4 Φ ) (26) \left(1-6Q^2+\frac{a^2m_\phi^2}{k^2}\right)\varphi=-3\lambda^2Q_{\rm dm}\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}-Q(2\Psi-4\Phi) \tag*{(26)} ( 1 − 6 Q 2 + k 2 a 2 m ϕ 2 ) φ = − 3 λ 2 Q dm Ω dm δ dm − Q ( 2Ψ − 4Φ ) (26) where m ϕ 2 = ( V , ϕ ϕ + 2 Q V , ϕ + 2 Q Q d m ρ d m ) / f 1 ( ϕ ) m_\phi^2=(V_{,\phi\phi}+2QV_{,\phi}+2QQ_{\rm dm}\rho_{\rm dm})/f_1(\phi) m ϕ 2 = ( V , ϕϕ + 2 Q V , ϕ + 2 Q Q dm ρ dm ) / f 1 ( ϕ ) (24) reduces to
Φ = − 3 λ 2 2 ( δ b Ω ~ b + δ d m Ω ~ d m ) − Q φ (27) \Phi=-\frac{3\lambda^2}{2}(\delta_b\widetilde{\Omega}_b+\delta_{\rm dm}\widetilde{\Omega}_{\rm dm})-Q\varphi \tag*{(27)} Φ = − 2 3 λ 2 ( δ b Ω b + δ dm Ω dm ) − Qφ (27) and Eq. (26) can be brought into the convenient form,
φ = − 3 λ 2 Y ( k ) [ ( Q + Q d m ) Ω ~ d m δ d m + Q Ω ~ b δ b ] ; Y ( k ) = k 2 k 2 + a 2 m ϕ 2 . (28) \varphi=-3\lambda^2Y(k)\left[(Q+Q_{\rm dm})\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}+Q\widetilde{\Omega}_b\delta_b\right]\ ;\quad Y(k)=\frac{k^2}{k^2+a^2m_\phi^2}. \tag*{(28)} φ = − 3 λ 2 Y ( k ) [ ( Q + Q dm ) Ω dm δ dm + Q Ω b δ b ] ; Y ( k ) = k 2 + a 2 m ϕ 2 k 2 . (28) Thus, Eq. (25) can be re-written for the scalar field Ψ \Psi Ψ
Ψ = − 3 λ 2 2 [ ( 2 Q Y ( k ) ( Q + Q d m ) + 1 ) Ω ~ d m δ d m + ( 2 Q 2 Y ( k ) + 1 ) Ω ~ b δ b ] . (29) \Psi=-\frac{3\lambda^2}{2}\left[(2QY(k)(Q+Q_{\rm dm})+1)\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}+(2Q^2Y(k)+1)\widetilde{\Omega}_b\delta_b\right]. \tag*{(29)} Ψ = − 2 3 λ 2 [ ( 2 Q Y ( k ) ( Q + Q dm ) + 1 ) Ω dm δ dm + ( 2 Q 2 Y ( k ) + 1 ) Ω b δ b ] . (29) These are the Poisson equations for the perturbation quantities φ \varphi φ Ψ \Psi Ψ
To determine the growth of matter perturbations at deep subhorizon scales, the continuity equations (20) and (22) can be simplified to δ b ′ = − θ b \delta'_b=-\theta_b δ b ′ = − θ b δ d m ′ = − θ d m \delta'_{\rm dm}=-\theta_{\rm dm} δ dm ′ = − θ dm Ψ , φ ∼ λ 2 δ m \Psi,\varphi\sim\lambda^2\delta_m Ψ , φ ∼ λ 2 δ m (20) and (22) and replacing θ b , θ d m \theta_b,\theta_{\rm dm} θ b , θ dm
δ b ′ ′ + δ b ′ ( 1 + H ′ H ) + Ψ λ 2 = 0 (30) \delta_b''+\delta_b'\left(1+\frac{\mathcal{H}'}{\mathcal{H}}\right)+\frac{\Psi}{\lambda^2}=0 \tag*{(30)} δ b ′′ + δ b ′ ( 1 + H H ′ ) + λ 2 Ψ = 0 (30) δ d m ′ ′ + δ d m ′ ( 1 + H ′ H + Q d m ϕ ′ ) + Ψ + Q d m φ λ 2 = 0. (31) \delta_{\rm dm}''+\delta_{\rm dm}'\left(1+\frac{\mathcal{H}'}{\mathcal{H}}+Q_{\rm dm}\phi'\right)+\frac{\Psi+Q_{\rm dm}\varphi}{\lambda^2}=0. \tag*{(31)} δ dm ′′ + δ dm ′ ( 1 + H H ′ + Q dm ϕ ′ ) + λ 2 Ψ + Q dm φ = 0. (31) Plugging in the gravitational potentials we have
δ b ′ ′ + ( 1 + H ′ H ) δ b ′ − 3 2 [ ( 1 + 2 Q Y ( Q + Q d m ) ) Ω ~ d m δ d m + ( 1 + 2 Q 2 Y ) Ω ~ b δ b ] = 0 (32) \delta_b''+\left(1+\frac{\mathcal{H}'}{\mathcal{H}}\right)\delta_b'-\frac{3}{2}\left[(1+2QY(Q+Q_{\rm dm}))\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}+(1+2Q^2Y)\widetilde{\Omega}_b\delta_b\right]=0 \tag*{(32)} δ b ′′ + ( 1 + H H ′ ) δ b ′ − 2 3 [ ( 1 + 2 Q Y ( Q + Q dm )) Ω dm δ dm + ( 1 + 2 Q 2 Y ) Ω b δ b ] = 0 (32) δ d m ′ ′ + ( 1 + H ′ H + Q d m ϕ ′ ) δ d m ′ − 3 2 { [ 1 + 2 Y ( Q + Q d m ) 2 ] Ω ~ d m δ d m + [ 1 + 2 Q Y ( Q + Q d m ) ] Ω ~ b δ b } = 0. (33) \delta_{\rm dm}''+\left(1+\frac{\mathcal{H}'}{\mathcal{H}}+Q_{\rm dm}\phi'\right)\delta_{\rm dm}'-\frac{3}{2}\left\{[1+2Y(Q+Q_{\rm dm})^2]\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}+[1+2QY(Q+Q_{\rm dm})]\widetilde{\Omega}_b\delta_b\right\}=0. \tag*{(33)} δ dm ′′ + ( 1 + H H ′ + Q dm ϕ ′ ) δ dm ′ − 2 3 { [ 1 + 2 Y ( Q + Q dm ) 2 ] Ω dm δ dm + [ 1 + 2 Q Y ( Q + Q dm )] Ω b δ b } = 0. (33) As expected, for Q = 0 Q=0 Q = 0 [32 , 38 ] .
The effective potential acting on dark matter test particles mentioned in the introduction is simply Ψ ^ d m ≡ Ψ + Q d m φ \widehat{\Psi}_{\rm dm}\equiv\Psi+Q_{\rm dm}\varphi Ψ dm ≡ Ψ + Q dm φ Ψ ^ b ≡ Ψ \widehat{\Psi}_b\equiv\Psi Ψ b ≡ Ψ Y ( k ) Y(k) Y ( k ) 2 ( Q + Q d m ) 2 e − m ( φ ) r 2(Q+Q_{\rm dm})^2e^{-m(\varphi)r} 2 ( Q + Q dm ) 2 e − m ( φ ) r 2 Q 2 e − m ( φ ) r 2Q^2e^{-m(\varphi)r} 2 Q 2 e − m ( φ ) r Q + Q d m Q+Q_{\rm dm} Q + Q dm Q Q Q
Testing the Weak Equivalence Principle
In this section, we derive a combination of quantities in the model described in Sec. II that can be employed to test the EP with galaxy clustering, as shown in Refs. [41 , 42 ] .
For a generic model, the Poisson equation for the gravitational potential Ψ \Psi Ψ
Ψ = − 3 2 ∑ m μ G , m ( k , z ) λ 2 Ω m δ m (34) \Psi=-\frac{3}{2}\sum_m\mu_{G,m}(k,z)\lambda^2\Omega_m\delta_m \tag*{(34)} Ψ = − 2 3 m ∑ μ G , m ( k , z ) λ 2 Ω m δ m (34) where the summation runs over the matter species m = { d m , b } \mathrm{m} = \{\mathrm{dm}, \mathrm{b}\} m = { dm , b } μ G , m \mu_{G,\mathrm{m}} μ G , m μ G , m ≠ 1 \mu_{G,\mathrm{m}} \ne1 μ G , m = 1 (29) can be used to identify that,
μ G , d m ( k , z ) = 1 f 1 [ Y ( k ) ( 2 Q 2 + 2 Q d m Q ) + 1 ] , (35) \mu_{G,\mathrm{dm}}(k,z) = \frac{1}{f_1}\left[Y(k)(2Q^2+2Q_{\mathrm{dm}}Q)+1\right], \tag*{(35)} μ G , dm ( k , z ) = f 1 1 [ Y ( k ) ( 2 Q 2 + 2 Q dm Q ) + 1 ] , (35) μ G , b ( k , z ) = 1 f 1 [ Y ( k ) 2 Q 2 + 1 ] . (36) \mu_{G,\mathrm{b}}(k,z) = \frac{1}{f_1}\left[Y(k)2Q^2+1\right]. \tag*{(36)} μ G , b ( k , z ) = f 1 1 [ Y ( k ) 2 Q 2 + 1 ] . (36) In Ref. [41 ] , it was shown that a parameter, denoted as E P E_P E P E P E_P E P E P E_P E P
Proposed in [39 ] , the most general cosmological Euler equation for dark matter in the modified gravity scenario can be expressed by introducing two phenomenological free functions (see, e.g., Refs [41 , 42 , 58 ] ), that become nonzero when deviating from the standard Λ \Lambda Λ
θ d m ′ + θ d m [ 1 + H ′ H + Θ d m ( k , z ) ] − [ 1 + Γ d m ( k , z ) ] λ − 2 Ψ = 0. (37) \theta'_{\mathrm{dm}}+\theta_{\mathrm{dm}}\left[1+\frac{\mathcal{H}'}{\mathcal{H}}+\Theta_{\mathrm{dm}}(k,z)\right]-\left[1+\Gamma_{\mathrm{dm}}(k,z)\right]\lambda^{-2}\Psi=0. \tag*{(37)} θ dm ′ + θ dm [ 1 + H H ′ + Θ dm ( k , z ) ] − [ 1 + Γ dm ( k , z ) ] λ − 2 Ψ = 0. (37) Here, the functions Θ d m , Γ d m \Theta_{\mathrm{dm}},\Gamma_{\mathrm{dm}} Θ dm , Γ dm (23) one can find that [28 ]
Θ d m = Q d m ϕ ′ ; Γ d m = Q d m ϕ Ψ . (38) \Theta_{\mathrm{dm}}=Q_{\mathrm{dm}}\phi';\qquad\Gamma_{\mathrm{dm}}=\frac{Q_{\mathrm{dm}}\phi}{\Psi}. \tag*{(38)} Θ dm = Q dm ϕ ′ ; Γ dm = Ψ Q dm ϕ . (38) As expected, Θ \Theta Θ Γ \Gamma Γ Q d m → 0 Q_{\mathrm{dm}}\to0 Q dm → 0 ϕ \phi ϕ Ψ \Psi Ψ (28) and (29) , Γ d m \Gamma_{\mathrm{dm}} Γ dm
Γ d m = 2 Q d m Y ( k ) ( Q + Q d m ) Ω d m + Q Ω b b [ 2 Q Y ( k ) ( Q + Q d m ) + 1 ] Ω d m + [ 2 Q 2 Y ( k ) + 1 ] Ω b b . (39) \Gamma_{\mathrm{dm}}=2Q_{\mathrm{dm}}Y(k)\frac{(Q+Q_{\mathrm{dm}})\Omega_{\mathrm{dm}}+Q\Omega_{\mathrm{b}}b}{\left[2QY(k)(Q+Q_{\mathrm{dm}})+1\right]\Omega_{\mathrm{dm}}+\left[2Q^2Y(k)+1\right]\Omega_{\mathrm{b}}b}. \tag*{(39)} Γ dm = 2 Q dm Y ( k ) [ 2 Q Y ( k ) ( Q + Q dm ) + 1 ] Ω dm + [ 2 Q 2 Y ( k ) + 1 ] Ω b b ( Q + Q dm ) Ω dm + Q Ω b b . (39) with the time-dependent bias function b ( z ) b(z) b ( z ) δ b = b ( z ) δ d m \delta_{\mathrm{b}}=b(z)\delta_{\mathrm{dm}} δ b = b ( z ) δ dm (30) and (31) . Notice that this bias is exclusively due to the EP violation and does not include fluidodynamical effects that are expected to be important at non-linear scales. Referring only to the dark matter component, E P E_P E P [41 ] in the following way,
E P = 1 + Θ d m − 3 Ω d m μ G , d m Γ d m 2 f g , (40) E_P=1+\Theta_{\mathrm{dm}}-\frac{3\Omega_{\mathrm{dm}}\mu_{G,\mathrm{dm}}\Gamma_{\mathrm{dm}}}{2f_g}, \tag*{(40)} E P = 1 + Θ dm − 2 f g 3 Ω dm μ G , dm Γ dm , (40) where f g ≡ δ d m ′ / δ d m f_g\equiv\delta'_{\mathrm{dm}}/\delta_{\mathrm{dm}} f g ≡ δ dm ′ / δ dm Θ d m \Theta_{\mathrm{dm}} Θ dm Γ d m \Gamma_{\mathrm{dm}} Γ dm E P E_P E P E P ≠ 1 E_P\ne1 E P = 1 (35) , (38) and (39) , E P E_P E P
E P = 1 + Q d m ϕ ′ − 3 Ω d m Q d m Y ( k ) 2 f g f 1 [ 2 ( Q + Q d m ) Ω d m + 2 Q Ω b b ] [ 2 Q Y ( k ) ( Q + Q d m ) + 1 ] [ 2 Q Y ( k ) ( Q + Q d m ) + 1 ] Ω d m + [ 2 Q 2 Y ( k ) + 1 ] Ω b b , (41) E_P=1+Q_{\mathrm{dm}}\phi'-\frac{3\Omega_{\mathrm{dm}}Q_{\mathrm{dm}}Y(k)}{2f_gf_1}\frac{\left[2(Q+Q_{\mathrm{dm}})\Omega_{\mathrm{dm}}+2Q\Omega_{\mathrm{b}}b\right]\left[2QY(k)(Q+Q_{\mathrm{dm}})+1\right]}{\left[2QY(k)(Q+Q_{\mathrm{dm}})+1\right]\Omega_{\mathrm{dm}}+\left[2Q^2Y(k)+1\right]\Omega_{\mathrm{b}}b}, \tag*{(41)} E P = 1 + Q dm ϕ ′ − 2 f g f 1 3 Ω dm Q dm Y ( k ) [ 2 Q Y ( k ) ( Q + Q dm ) + 1 ] Ω dm + [ 2 Q 2 Y ( k ) + 1 ] Ω b b [ 2 ( Q + Q dm ) Ω dm + 2 Q Ω b b ] [ 2 Q Y ( k ) ( Q + Q dm ) + 1 ] , (41) where we recall that f 1 = e − 2 Q ϕ f_1=e^{-2Q\phi} f 1 = e − 2 Qϕ E P E_P E P
E P = 1 + Q d m ϕ ′ − 3 Q d m Y ( k ) Ω d m f 1 Q + Q d m f g = 1 + Q d m ϕ ′ − 3 Q d m Ω ~ d m Y ( k ) Q + Q d m f g . (42) \begin{aligned}
E_P=1+Q_{\mathrm{dm}}\phi'-3Q_{\mathrm{dm}}Y(k)\frac{\Omega_{\mathrm{dm}}}{f_1}\frac{Q+Q_{\mathrm{dm}}}{f_g} \\
&=1+Q_{\mathrm{dm}}\phi'-3Q_{\mathrm{dm}}\widetilde{\Omega}_{\mathrm{dm}}Y(k)\frac{Q+Q_{\mathrm{dm}}}{f_g}. \tag*{(42)}
\end{aligned} E P = 1 + Q dm ϕ ′ − 3 Q dm Y ( k ) f 1 Ω dm f g Q + Q dm = 1 + Q dm ϕ ′ − 3 Q dm Ω dm Y ( k ) f g Q + Q dm . (42) Recall that Ω ~ d m = Ω d m / f 1 \widetilde{\Omega}_{\rm dm}=\Omega_{\rm dm}/f_1 Ω dm = Ω dm / f 1 E P → 1 E_P\to1 E P → 1 Q d m → 0 Q_{\rm dm}\to0 Q dm → 0
x ( N ) = ϕ ′ 6 , y ( N ) = 1 H V 3 f 1 . (43) x(N)=\frac{\phi'}{\sqrt{6}},\qquad y(N)=\frac{1}{H}\sqrt{\frac{V}{3f_1}}. \tag*{(43)} x ( N ) = 6 ϕ ′ , y ( N ) = H 1 3 f 1 V . (43) Neglecting baryons, the effective dark energy and dark matter density parameters can then be expressed as
Ω ~ D E , e f f = ρ D E , e f f 3 H 2 f 1 = ( 1 − 6 Q 2 ) x 2 + y 2 + 2 6 Q x , (44) \widetilde{\Omega}_{{\rm DE,eff}}=\frac{\rho_{{\rm DE,eff}}}{3H^2f_1}=(1-6Q^2)x^2+y^2+2\sqrt{6}Qx, \tag*{(44)} Ω DE , eff = 3 H 2 f 1 ρ DE , eff = ( 1 − 6 Q 2 ) x 2 + y 2 + 2 6 Q x , (44) Ω ~ d m = 1 − Ω ~ D E , e f f = 1 − ( 1 − 6 Q 2 ) x 2 − y 2 − 2 6 Q x , (45) \widetilde{\Omega}_{\rm dm}=1-\widetilde{\Omega}_{{\rm DE,eff}}=1-(1-6Q^2)x^2-y^2-2\sqrt{6}Qx, \tag*{(45)} Ω dm = 1 − Ω DE , eff = 1 − ( 1 − 6 Q 2 ) x 2 − y 2 − 2 6 Q x , (45) and E P E_P E P
E P = 1 + 6 Q d m x − 3 Q d m [ 1 − ( 1 − 6 Q 2 ) x 2 − y 2 − 2 6 Q x ] Y ( k ) Q + Q d m f g . (46) E_P=1+\sqrt{6}Q_{\rm dm}x-3Q_{\rm dm}\left[1-(1-6Q^2)x^2-y^2-2\sqrt{6}Qx\right]Y(k)\frac{Q+Q_{\rm dm}}{f_g}. \tag*{(46)} E P = 1 + 6 Q dm x − 3 Q dm [ 1 − ( 1 − 6 Q 2 ) x 2 − y 2 − 2 6 Q x ] Y ( k ) f g Q + Q dm . (46) We now want to investigate how strong the couplings Q d m Q_{\rm dm} Q dm Q Q Q Q d m Q_{\rm dm} Q dm Q Q Q Q d m Q_{\rm dm} Q dm
The first scenario is interesting since it corresponds to the simplest non-trivial coupled model, in which f 1 = 1 f_1=1 f 1 = 1 Q = 0 Q=0 Q = 0 Ω d m = Ω ~ d m \Omega_{\rm dm}=\widetilde{\Omega}_{\rm dm} Ω dm = Ω dm a 2 m ϕ 2 ≪ k 2 a^2m_\phi^2\ll k^2 a 2 m ϕ 2 ≪ k 2 Y → 1 Y\to1 Y → 1 [24 , 25 , 34 , 35 , 36 , 37 ] ). Under these assumptions, E P E_P E P k k k
E P = 1 + Q d m ϕ ′ − 3 Ω d m Q d m 2 f g = 1 + 6 Q d m x − 3 ( 1 − x 2 − y 2 ) Q d m 2 f g . (47) \begin{aligned}
E_P&=1+Q_{\rm dm}\phi'-3\frac{\Omega_{\rm dm}Q_{\rm dm}^2}{f_g}\\
&=1+\sqrt{6}Q_{\rm dm}x-3\frac{(1-x^2-y^2)Q_{\rm dm}^2}{f_g}. \tag*{(47)}
\end{aligned} E P = 1 + Q dm ϕ ′ − 3 f g Ω dm Q dm 2 = 1 + 6 Q dm x − 3 f g ( 1 − x 2 − y 2 ) Q dm 2 . (47) Since we expect the behavior of the model to be close to that of Λ \Lambda Λ Λ \Lambda Λ f g ≈ Ω d m γ f_g\approx\Omega_{\rm dm}^{\gamma} f g ≈ Ω dm γ Ω d m \Omega_{\rm dm} Ω dm γ = 0.545 \gamma=0.545 γ = 0.545
By further introducing the dimensionless coupling{^5} β d m = − 6 2 Q d m \beta_{\rm dm}=-\frac{\sqrt{6}}{2}Q_{\rm dm} β dm = − 2 6 Q dm E P E_P E P
E P ( N ) = 1 − 2 β d m x − 2 β d m 2 ( 1 − x 2 − y 2 ) 1 − γ . (48) E_P(N)=1-2\beta_{\rm dm}x-2\beta_{\rm dm}^2(1-x^2-y^2)^{1-\gamma}. \tag*{(48)} E P ( N ) = 1 − 2 β dm x − 2 β dm 2 ( 1 − x 2 − y 2 ) 1 − γ . (48) The evolution of x x x y y y x x x y y y [59 ] for a linear potential V ≈ V 0 − V 0 μ ( ϕ − ϕ 0 ) V\approx V_0-V_0\mu(\phi-\phi_0) V ≈ V 0 − V 0 μ ( ϕ − ϕ 0 ) μ \mu μ ϕ 0 = 0 \phi_0=0 ϕ 0 = 0 V ′ ( N ) = − V 0 μ ϕ ′ = − V 0 μ 6 x V'(N)=-V_0\mu\phi'=-V_0\mu\sqrt{6}x V ′ ( N ) = − V 0 μ ϕ ′ = − V 0 μ 6 x
x ( N ) = 1 6 e 3 N E ~ ( N ) { 4 6 β d m [ e 3 N i E ~ ( N i ) + Ω m 0 Ω Λ 0 ln ( e 3 N i / 2 ( E ~ ( N i ) − Ω Λ 0 ) e 3 N / 2 ( E ~ ( N ) − Ω Λ 0 ) ) ] + μ [ Ω m 0 Ω Λ 0 ln ( e 3 N / 2 ( E ~ ( N ) − Ω Λ 0 ) e 3 N i / 2 ( E ~ ( N i ) − Ω Λ 0 ) ) + e 3 N E ~ ( N ) − e 3 N i E ~ ( N i ) ] } . (49) x(N)=\frac{1}{\sqrt{6}e^{3N}\widetilde{E}(N)}\left\{\frac{4}{\sqrt{6}}\beta_{\rm dm}\left[e^{3N_i}\widetilde{E}(N_i)+\frac{\Omega_{m0}}{\sqrt{\Omega_{\Lambda0}}}\ln\left(\frac{e^{3N_i/2}(\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}})}{e^{3N/2}(\widetilde{E}(N)-\sqrt{\Omega_{\Lambda0}})}\right)\right]+\mu\left[\frac{\Omega_{m0}}{\sqrt{\Omega_{\Lambda0}}}\ln\left(\frac{e^{3N/2}(\widetilde{E}(N)-\sqrt{\Omega_{\Lambda0}})}{e^{3N_i/2}(\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}})}\right)+e^{3N}\widetilde{E}(N)-e^{3N_i}\widetilde{E}(N_i)\right]\right\}. \tag*{(49)} x ( N ) = 6 e 3 N E ( N ) 1 { 6 4 β dm [ e 3 N i E ( N i ) + Ω Λ0 Ω m 0 ln ( e 3 N /2 ( E ( N ) − Ω Λ0 ) e 3 N i /2 ( E ( N i ) − Ω Λ0 ) ) ] + μ [ Ω Λ0 Ω m 0 ln ( e 3 N i /2 ( E ( N i ) − Ω Λ0 ) e 3 N /2 ( E ( N ) − Ω Λ0 ) ) + e 3 N E ( N ) − e 3 N i E ( N i ) ] } . (49) V ( N ) = V 0 { 1 − 4 6 β d m N μ + μ ( 8 6 β d m − 2 μ 3 Ω Λ 0 ) [ Ω Λ 0 Ω m 0 e 3 N i E ~ ( N i ) ( E ~ ( N ) − 1 ) − ln ( e 3 N i / 2 ( E ~ ( N i ) − Ω Λ 0 ) 1 − Ω Λ 0 ) + E ~ ( N ) ln ( e 3 N i / 2 ( E ~ ( N i ) − Ω Λ 0 ) e 3 N / 2 ( E ~ ( N ) − Ω Λ 0 ) ) ] } . (50) V(N)=V_0\left\{1-\frac{4}{\sqrt{6}}\beta_{\rm dm}N\mu+\mu\left(\frac{\frac{8}{\sqrt{6}}\beta_{\rm dm}-2\mu}{3\sqrt{\Omega_{\Lambda0}}}\right)\left[\frac{\sqrt{\Omega_{\Lambda0}}}{\Omega_{m0}}e^{3N_i}\widetilde{E}(N_i)\left(\widetilde{E}(N)-1\right)-\ln\left(\frac{e^{3N_i/2}(\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}})}{1-\sqrt{\Omega_{\Lambda0}}}\right)+\widetilde{E}(N)\ln\left(\frac{e^{3N_i/2}(\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}})}{e^{3N/2}(\widetilde{E}(N)-\sqrt{\Omega_{\Lambda0}})}\right)\right]\right\}. \tag*{(50)} V ( N ) = V 0 { 1 − 6 4 β dm N μ + μ ( 3 Ω Λ0 6 8 β dm − 2 μ ) [ Ω m 0 Ω Λ0 e 3 N i E ( N i ) ( E ( N ) − 1 ) − ln ( 1 − Ω Λ0 e 3 N i /2 ( E ( N i ) − Ω Λ0 ) ) + E ( N ) ln ( e 3 N /2 ( E ( N ) − Ω Λ0 ) e 3 N i /2 ( E ( N i ) − Ω Λ0 ) ) ] } . (50) where N i N_i N i N i = − 3 N_i=-3 N i = − 3 Ω Λ 0 = 1 − Ω m 0 \Omega_{\Lambda0}=1-\Omega_{m0} Ω Λ0 = 1 − Ω m 0 V 0 = 3 H 0 2 Ω Λ 0 V_0=3H_0^2\Omega_{\Lambda0} V 0 = 3 H 0 2 Ω Λ0 E ~ ( N ) = Ω m 0 e − 3 N + 1 − Ω m 0 \widetilde{E}(N)=\sqrt{\Omega_{m0}e^{-3N}+1-\Omega_{m0}} E ( N ) = Ω m 0 e − 3 N + 1 − Ω m 0 x x x ϕ \phi ϕ x i ( N = − 3 ) = 2 3 β d m x_i(N=-3)=\frac{2}{3}\beta_{\rm dm} x i ( N = − 3 ) = 3 2 β dm
Next, for the doubly coupled dark energy, where both Q Q Q Q d m Q_{\rm dm} Q dm β = − 6 2 Q \beta=-\frac{\sqrt{6}}{2}Q β = − 2 6 Q Y → 1 Y\to1 Y → 1 f g ≈ Ω ~ E P γ f_g\approx\widetilde{\Omega}_{\rm EP}^{\gamma} f g ≈ Ω EP γ E P E_{\rm P} E P
E P = 1 − 2 β d m x − 2 β d m ( β + β d m ) [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] 1 − γ . (51) E_{\rm P}=1-2\beta_{\rm dm}x-2\beta_{\rm dm}(\beta+\beta_{\rm dm})\left[1-(1-4\beta^2)x^2-y^2+4\beta x\right]^{1-\gamma}. \tag*{(51)} E P = 1 − 2 β dm x − 2 β dm ( β + β dm ) [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] 1 − γ . (51) The quantities x x x V V V
x ( N ) = e 3 N i E ~ ( N i ) e 3 N E ~ ( N ) 2 ( β + β d m ) 3 − 4 β ( β + β d m ) + 1 3 ( 1 − 4 β 2 ) [ 4 β + 3 2 μ ] [ 1 − e 3 N i E ~ ( N i ) e 3 N E ~ ( N ) ] + Ω m 0 3 ( 1 − 4 β 2 ) 1 − Ω m 0 e 3 N E ~ ( N ) [ 2 β d m − 2 β − 3 2 μ ] × ln [ e 3 N i / 2 ( E ~ ( N i ) − 1 − Ω m 0 ) e 3 N / 2 ( E ~ ( N ) − 1 − Ω m 0 ) ] . (52) \begin{aligned}
x(N)={}&\frac{e^{3N_i}\widetilde{E}(N_i)}{e^{3N}\widetilde{E}(N)}
\frac{2(\beta+\beta_{\rm dm})}{3-4\beta(\beta+\beta_{\rm dm})}
+\frac{1}{3(1-4\beta^2)}
\left[4\beta+\sqrt{\frac{3}{2}}\mu\right]
\left[1-\frac{e^{3N_i}\widetilde{E}(N_i)}{e^{3N}\widetilde{E}(N)}\right]\\
&+\frac{\Omega_{m0}}{3(1-4\beta^2)\sqrt{1-\Omega_{m0}e^{3N}\widetilde{E}(N)}}
\left[2\beta_{\rm dm}-2\beta-\sqrt{\frac{3}{2}}\mu\right]\\
&\qquad\times\ln\left[
\frac{e^{3N_i/2}\left(\widetilde{E}(N_i)-\sqrt{1-\Omega_{m0}}\right)}
{e^{3N/2}\left(\widetilde{E}(N)-\sqrt{1-\Omega_{m0}}\right)}
\right]. \tag*{(52)}
\end{aligned} x ( N ) = e 3 N E ( N ) e 3 N i E ( N i ) 3 − 4 β ( β + β dm ) 2 ( β + β dm ) + 3 ( 1 − 4 β 2 ) 1 [ 4 β + 2 3 μ ] [ 1 − e 3 N E ( N ) e 3 N i E ( N i ) ] + 3 ( 1 − 4 β 2 ) 1 − Ω m 0 e 3 N E ( N ) Ω m 0 [ 2 β dm − 2 β − 2 3 μ ] × ln e 3 N /2 ( E ( N ) − 1 − Ω m 0 ) e 3 N i /2 ( E ( N i ) − 1 − Ω m 0 ) . (52) V ( N ) = V 0 { 1 − 2 6 μ ( β + β d m ) 3 ( 1 − 4 β 2 ) N + 2 6 μ e 3 N i E ~ ( N i ) 3 Ω m 0 [ x ( N i ) − 4 β + 3 2 μ 3 ( 1 − 4 β 2 ) ] [ E ~ ( N ) − 1 ] + 2 6 μ [ 2 β d m − 2 β − 3 2 μ ] 9 ( 1 − 4 β 2 ) Ω Λ 0 [ E ~ ( N ) ln ( e 3 N i / 2 [ E ~ ( N i ) − Ω Λ 0 ] e 3 N / 2 [ E ~ ( N ) − Ω Λ 0 ] ) − ln ( e 3 N i / 2 [ E ~ ( N i ) − Ω Λ 0 ] 1 − Ω Λ 0 ) ] } . (53) \begin{aligned}
V(N)=V_0\Bigg\{1
&-\frac{2\sqrt{6}\mu(\beta+\beta_{\rm dm})}{3(1-4\beta^2)}N
+\frac{2\sqrt{6}\mu e^{3N_i}\widetilde{E}(N_i)}{3\Omega_{m0}}
\left[x(N_i)-\frac{4\beta+\sqrt{\frac{3}{2}}\mu}{3(1-4\beta^2)}\right]
\left[\widetilde{E}(N)-1\right]\\
&+\frac{2\sqrt{6}\mu\left[2\beta_{\rm dm}-2\beta-\sqrt{\frac{3}{2}}\mu\right]}
{9(1-4\beta^2)\sqrt{\Omega_{\Lambda0}}}
\Bigg[
\widetilde{E}(N)\ln\left(
\frac{e^{3N_i/2}\left[\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}}\right]}
{e^{3N/2}\left[\widetilde{E}(N)-\sqrt{\Omega_{\Lambda0}}\right]}
\right)\\
&\qquad\qquad\qquad\qquad
-\ln\left(
\frac{e^{3N_i/2}\left[\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}}\right]}
{1-\sqrt{\Omega_{\Lambda0}}}
\right)
\Bigg]\Bigg\}. \tag*{(53)}
\end{aligned} V ( N ) = V 0 { 1 − 3 ( 1 − 4 β 2 ) 2 6 μ ( β + β dm ) N + 3 Ω m 0 2 6 μ e 3 N i E ( N i ) x ( N i ) − 3 ( 1 − 4 β 2 ) 4 β + 2 3 μ [ E ( N ) − 1 ] + 9 ( 1 − 4 β 2 ) Ω Λ0 2 6 μ [ 2 β dm − 2 β − 2 3 μ ] [ E ( N ) ln e 3 N /2 [ E ( N ) − Ω Λ0 ] e 3 N i /2 [ E ( N i ) − Ω Λ0 ] − ln 1 − Ω Λ0 e 3 N i /2 [ E ( N i ) − Ω Λ0 ] ] } . (53) The derivation of Eqs. (52) and (53) is provided in App. § 2.2 . Moreover, the phase-space system for the doubly coupled model including baryons is summarized in App. § 2.3 . The expressions Eqs. (48) –(53) will be implemented in our forecasts. Next, we describe our forecasting methodology in Sec. IV and provide the forecast results for the constant couplings β \beta β β d m \beta_{\rm dm} β dm
Fisher forecast for the couplings
In Ref. [41 ] , it has been shown that the auto- and cross-matter linear power spectra for two galaxy populations (with the subscripts B, F, for ”bright” and ”faint”, respectively) up to order λ = H / k \lambda=\mathcal{H}/k λ = H / k
P F F = ( 1 + β F μ 2 ) 2 β B β F S g 2 B P . (54) P_{\rm FF}=(1+\beta_{\rm F}\mu^2)^2\frac{\beta_{\rm B}}{\beta_{\rm F}}S_g^2BP. \tag*{(54)} P FF = ( 1 + β F μ 2 ) 2 β F β B S g 2 B P . (54) P B B = ( 1 + β B μ 2 ) 2 β F β B S g 2 B P . (55) P_{\rm BB}=(1+\beta_{\rm B}\mu^2)^2\frac{\beta_{\rm F}}{\beta_{\rm B}}S_g^2BP. \tag*{(55)} P BB = ( 1 + β B μ 2 ) 2 β B β F S g 2 B P . (55) P F B = [ ( 1 + β F μ 2 ) ( 1 + β B μ 2 ) + i λ μ ( τ 1 + μ 2 τ 2 ) ] S g 2 B P . (56) P_{\rm FB}=\left[(1+\beta_{\rm F}\mu^2)(1+\beta_{\rm B}\mu^2)+i\lambda\mu(\tau_1+\mu^2\tau_2)\right]S_g^2BP. \tag*{(56)} P FB = [ ( 1 + β F μ 2 ) ( 1 + β B μ 2 ) + iλ μ ( τ 1 + μ 2 τ 2 ) ] S g 2 B P . (56) in which μ \mu μ β B , F = f g / b B , F \beta_{\rm B,F}=f_g/b_{\rm B,F} β B , F = f g / b B , F b B , F b_{\rm B,F} b B , F P P P z = 0 z=0 z = 0 B ≡ b B ( z ) b F ( z ) G 2 ( z ) B\equiv b_{\rm B}(z)b_{\rm F}(z)G^2(z) B ≡ b B ( z ) b F ( z ) G 2 ( z ) G ≡ δ m ( z ) / δ m 0 G\equiv\delta_m(z)/\delta_{m0} G ≡ δ m ( z ) / δ m 0
S g 2 ( k , μ , z ) = e − 1 2 k 2 μ 2 σ g 2 . (57) S_g^2(k,\mu,z)=e^{-\frac{1}{2}k^2\mu^2\sigma_g^2}. \tag*{(57)} S g 2 ( k , μ , z ) = e − 2 1 k 2 μ 2 σ g 2 . (57) in which the damping strengths σ g \sigma_g σ g τ 1 \tau_1 τ 1 τ 2 \tau_2 τ 2
τ 1 = β B α B − β F α F − E P ( β B − β F ) , (58) \tau_1=\beta_{\rm B}\alpha_{\rm B}-\beta_{\rm F}\alpha_{\rm F}-E_{\rm P}(\beta_{\rm B}-\beta_{\rm F}), \tag*{(58)} τ 1 = β B α B − β F α F − E P ( β B − β F ) , (58) τ 2 = β B β F ( α B − α F ) . (59) \tau_2=\beta_{\rm B}\beta_{\rm F}(\alpha_{\rm B}-\alpha_{\rm F}). \tag*{(59)} τ 2 = β B β F ( α B − α F ) . (59) where we have defined for each galaxy population
α ≡ 1 − 5 s + 5 s − 2 H r − H ′ H + f e v o l . (60) \alpha\equiv1 - 5s + \frac{5s - 2}{\mathcal{H}r} - \frac{\mathcal{H}'}{\mathcal{H}} + f^{\mathrm{evol}}. \tag*{(60)} α ≡ 1 − 5 s + H r 5 s − 2 − H H ′ + f evol . (60) Here, s s s f e v o l f^{\mathrm{evol}} f evol f e v o l = 0 f^{\mathrm{evol}} = 0 f evol = 0 [41 , 42 ] . Moreover, we assume in the following that α B , F \alpha_{B,F} α B , F [60 , 61 ] . By grouping the power spectra Eqs. (54) –(56) into polynomials of μ a λ b \mu^a\lambda^b μ a λ b τ 1 \tau_1 τ 1 (58) , we see then explicitly that E P E_P E P
The spectra in Eqs. (54) –(56) encode also the Alcock-Paczynski (AP) effect through k k k μ \mu μ
k = α A P k r ; μ = μ r h α A P ; α A P = 1 d μ r 2 ( h 2 d 2 − 1 ) + 1 , (61) k = \alpha_{\mathrm{AP}} k_r; \qquad\mu= \frac{\mu_r h}{\alpha_{\mathrm{AP}}}; \qquad\alpha_{\mathrm{AP}} = \frac{1}{d}\sqrt{\mu_r^2(h^2d^2-1)+1}, \tag*{(61)} k = α AP k r ; μ = α AP μ r h ; α AP = d 1 μ r 2 ( h 2 d 2 − 1 ) + 1 , (61) with k r k_r k r μ r \mu_r μ r h ≡ E / E r h \equiv E/E_r h ≡ E / E r d ≡ L A / L A r d \equiv L_A/L_{Ar} d ≡ L A / L A r E ( z ) ≡ H ( z ) / H 0 E(z) \equiv H(z)/H_0 E ( z ) ≡ H ( z ) / H 0 L A ( z ) ≡ H 0 D A ( z ) L_A(z) \equiv H_0D_A(z) L A ( z ) ≡ H 0 D A ( z ) d d d h d ≡ h d h_d \equiv hd h d ≡ h d [41 ] , a forecast analysis has been performed for a set of parameters { h d , d , B , σ g , β B , β F , E P } \{h_d,d,B,\sigma_g,\beta_B,\beta_F,E_P\} { h d , d , B , σ g , β B , β F , E P } z 1 z_1 z 1 B ( z ) B(z) B ( z ) k k k k ∈ ( 0.01 − 0.12 ) h / M p c k \in(0.01-0.12)\,h/\mathrm{Mpc} k ∈ ( 0.01 − 0.12 ) h / Mpc Δ k = 0.01 h / M p c \Delta k = 0.01\,h/\mathrm{Mpc} Δ k = 0.01 h / Mpc P ( k ) P(k) P ( k ) [41 ] , we assume uninformative flat priors for all parameters except d d d [62 ] and App. § 2.5 ), while fixing B ( z 1 ) = 1 B(z_1) = 1 B ( z 1 ) = 1 P ( k , z 1 ) P(k,z_1) P ( k , z 1 ) B B B
Here, we follow the methodology of Refs. [41 , 42 ] , with only slightly different survey specifications (see § 1.4 ), to obtain constraints on { h d , d , B , σ g , β B , β F , E P } \{h_d,d,B,\sigma_g,\beta_B,\beta_F,E_P\} { h d , d , B , σ g , β B , β F , E P } { β d m , Ω m 0 } \{\beta_{\mathrm{dm}},\Omega_{m0}\} { β dm , Ω m 0 } { β , β d m , Ω m 0 } \{\beta,\beta_{\mathrm{dm}},\Omega_{m0}\} { β , β dm , Ω m 0 } β \beta β β d m \beta_{\mathrm{dm}} β dm E P ( z ) E_P(z) E P ( z ) (48) and (51) ) and the expansion rate E ( z ) E(z) E ( z ) (120) and (78) ), with the latter modifying h d h_d h d d d d { E P , h d , d } \{E_P,h_d,d\} { E P , h d , d } 6 § 2.4 .
Survey specifications and fiducials
Here we consider three upcoming galaxy surveys and forecast how well they can constrain the coupling strengths: the DESI Bright Galaxy Sample (BGS) [63 ] , a Euclid-like stage-IV Hα \alpha α [64 ] , and the SKA Phase 2 HI spectroscopic galaxy survey [65 ] . In addition, we compute the fiducial matter power spectrum at the lowest redshift bin, background quantities, and linear growth rate assuming a standard flat Λ \Lambda Λ [66 ] . To forecast the couplings, we also need to specify the initial time N i N_i N i β \beta β β d m \beta_{\mathrm{dm}} β dm Ω m 0 \Omega_{m0} Ω m 0 x x x y y y [59 ] , we adopt N i = − 3 N_i=-3 N i = − 3 x i x_i x i ϕ \phi ϕ § 2.2 ). Finally, we take the fiducial values Ω m 0 = 0.315 \Omega_{m0}=0.315 Ω m 0 = 0.315 β = 0.1 \beta=0.1 β = 0.1 β d m = 0.2 \beta_{\mathrm{dm}}=0.2 β dm = 0.2 V 0 = 3 ( 1 − Ω m 0 ) ≈ 2.055 V_0=3(1-\Omega_{m0})\approx2.055 V 0 = 3 ( 1 − Ω m 0 ) ≈ 2.055 μ = 0.145 \mu=0.145 μ = 0.145 [24 , 26 , 32 ] ). These analyses, however, assume a constant coupling throughout the entire cosmic evolution, while here we only consider late-time data.
For each survey, we split the full galaxy sample into two populations (bright and faint) within each redshift bin. This is done by choosing a flux threshold that divides the sample equally between the two populations. As in [41 ] , we choose the fiducial biases to be b g , B = b g + 0.5 b_{g,B}=b_g+0.5 b g , B = b g + 0.5 b g , F = b g − 0.5 b_{g,F}=b_g-0.5 b g , F = b g − 0.5 b g b_g b g [61 ] .
Table II. Survey specifications for a Euclid-like spectroscopic survey, covering the redshift range 0.9 ≤ z ≤ 1.8 0.9 \le z \le1.8 0.9 ≤ z ≤ 1.8 z = 0.9 , 1.1 , 1.3 , 1.5 , 1.8 z = 0.9, 1.1, 1.3, 1.5, 1.8 z = 0.9 , 1.1 , 1.3 , 1.5 , 1.8 [60 , 67 ] .
The remaining survey specifications for DESI BGS, Euclid, and SKA2 are identical to those in Ref. [42 ] . We list them explicitly in Table I –Table III . These include the redshift bins, survey volumes, galaxy number densities, and average galaxy biases, as well as the treatment of galaxy redshift uncertainties (associated with σ g \sigma_g σ g
Table I. Survey specifications for DESI BGS covering the redshift range 0 ≤ z ≤ 0.5 0 \le z \le0.5 0 ≤ z ≤ 0.5 Δ z = 0.1 \Delta z = 0.1 Δ z = 0.1 n g n_g n g 10 − 3 ( h / M p c ) 3 10^{-3}(h/\mathrm{Mpc})^3 1 0 − 3 ( h / Mpc ) 3 h = 0.67 h = 0.67 h = 0.67 V V V ( G p c / h ) 3 (\mathrm{Gpc}/h)^3 ( Gpc / h ) 3 b g , B , b g , F = b g ± 0.5 b_{g,B},b_{g,F} = b_g \pm0.5 b g , B , b g , F = b g ± 0.5 n g n_g n g s B , s F s_B,s_F s B , s F [60 ] . The observed absolute magnitude limit is chosen to be m c = 19.5 m_c = 19.5 m c = 19.5 f e v o l f_{\mathrm{evol}} f evol
Table III. SKA2 survey specifications covering the redshift range 0.2 ≤ z ≤ 0.8 0.2 \le z \le0.8 0.2 ≤ z ≤ 0.8 Δ z = 0.1 \Delta z = 0.1 Δ z = 0.1 [60 , 65 ] .
The forecasting results presented in the next section are based on the aforementioned survey specifications.
Results and discussion
We present our forecasting results in Tables IV–VII for DESI, Euclid, SKA2, and, just to illustrate the effect of the number density, a survey mimicking SKA2 but with ten times as many galaxies. All the results shown in these tables are absolute errors obtained after marginalising over all other parameters. No priors are taken into account except those on d d d { h d , d , E P } \{h_d,d,E_P\} { h d , d , E P } E P E_P E P E E E h d h_d h d d d d
Table V. Same as Table. Table IV, but for the Euclid spectroscopic survey.
general as far as the cosmological model is concerned. By doing so, we expect our results to be more robust than those obtained from analyses carried out assuming specific models, at the price of possibly weaker constraints. Moreover, these constraints are fully obtained from the late-time Universe, in contrast to tighter constraints obtained including CMB measurements and assuming constant couplings throughout the evolution.
Among the three surveys, SKA2 could provide the tightest constraints on h d h_d h d d d d E P E_{\mathrm{P}} E P [41 , 42 ] , thanks to its large number of detected galaxies. Projecting these constraints onto the CDE model, we obtain relative 1 σ 1\sigma 1 σ β d m \beta_{\mathrm{dm}} β dm Ω m 0 \Omega_{\mathrm{m0}} Ω m0 β d m \beta_{\mathrm{dm}} β dm (48) , the uncertainty of β d m \beta_{\mathrm{dm}} β dm E P E_{\mathrm{P}} E P Ω m 0 \Omega_{\mathrm{m0}} Ω m0
σ β d m ≈ ∣ 1 ∂ E P / ∂ β d m ∣ f i d σ E P . (62) \sigma_{\beta_{\mathrm{dm}}} \approx\left|\frac{1}{\partial E_{\mathrm{P}}/\partial\beta_{\mathrm{dm}}}\right|_{\mathrm{fid}} \sigma_{E_{\mathrm{P}}}. \tag*{(62)} σ β dm ≈ ∂ E P / ∂ β dm 1 fid σ E P . (62) For SKA2, ∣ ∂ E P / ∂ β d m ∣ f i d ∼ 0.9 − 1.12 |\partial E_{\mathrm{P}}/\partial\beta_{\mathrm{dm}}|_{\mathrm{fid}} \sim0.9 - 1.12 ∣ ∂ E P / ∂ β dm ∣ fid ∼ 0.9 − 1.12 1 σ 1\sigma 1 σ β d m \beta_{\mathrm{dm}} β dm E P E_{\mathrm{P}} E P σ β d m ≈ 0.044 \sigma_{\beta_{\mathrm{dm}}} \approx0.044 σ β dm ≈ 0.044 1 σ 1\sigma 1 σ σ β d m = 0.044 \sigma_{\beta_{\mathrm{dm}}} = 0.044 σ β dm = 0.044 Table VI , despite neglecting the dependence of h d h_d h d d d d β d m \beta_{\mathrm{dm}} β dm Ω m 0 \Omega_{\mathrm{m0}} Ω m0 σ β d m = 0.034 \sigma_{\beta_{\mathrm{dm}}} = 0.034 σ β dm = 0.034 E P E_{\mathrm{P}} E P β d m \beta_{\mathrm{dm}} β dm β d m \beta_{\mathrm{dm}} β dm
Table VI. Same as Table. Table IV, but for the SKA2.
model independence. For instance, we explored the case in which the power spectrum is fixed at fiducial, while leaving the remaining parameters free to vary. In this case, the relative 1 σ 1\sigma 1 σ β d m \beta_{\rm dm} β dm
On the other hand, β d m \beta_{\rm dm} β dm E E E Ω m 0 \Omega_{\rm m0} Ω m0 h d h_d h d d d d β d m \beta_{\rm dm} β dm Ω m 0 \Omega_{\rm m0} Ω m0 − 0.93 -0.93 − 0.93 − 0.99 -0.99 − 0.99 − 0.65 -0.65 − 0.65 Λ \Lambda Λ − 0.0094 -0.0094 − 0.0094 0.0019 0.0019 0.0019 0.056 0.056 0.056 1 σ 1\sigma 1 σ β d m \beta_{\rm dm} β dm Ω m 0 \Omega_{\rm m0} Ω m0
We also present in Figure 2 the two-dimensional confidence contours between β d m \beta_{\rm dm} β dm Ω m 0 \Omega_{\rm m0} Ω m0 ( c 1 , c 2 ) (c_1,c_2) ( c 1 , c 2 ) C = c 1 β d m + c 2 Ω m 0 C=c_1\beta_{\rm dm}+c_2\Omega_{\rm m0} C = c 1 β dm + c 2 Ω m0 ( c 1 , c 2 ) ≃ ( 0.27 , 0.96 ) , ( 0.21 , 0.98 ) , ( 0.17 , 0.99 ) (c_1,c_2)\simeq(0.27,0.96),(0.21,0.98),(0.17,0.99) ( c 1 , c 2 ) ≃ ( 0.27 , 0.96 ) , ( 0.21 , 0.98 ) , ( 0.17 , 0.99 ) 1 σ 1\sigma 1 σ C C C Figure 2 .
Figure 2. Forecast constraints on β d m \beta_{\rm dm} β dm Ω m 0 \Omega_{\rm m0} Ω m0 1 σ 1\sigma 1 σ 2 σ 2\sigma 2 σ
For the DCDE model, one can further constrain the universal coupling β \beta β E P E_P E P β d m \beta_{\rm dm} β dm β \beta β β d m \beta_{\rm dm} β dm Ω m 0 \Omega_{\rm m0} Ω m0 β \beta β β d m \beta_{\rm dm} β dm β \beta β
We also explored the effect of increasing the galaxy number density. For instance, increasing the SKA2 galaxy number density by a factor of ten could substantially improve the projected constraints, as listed in Table VII. For CDE, the 1 σ 1\sigma 1 σ σ β d m = 0.0081 \sigma_{\beta_{\rm dm}}=0.0081 σ β dm = 0.0081 σ Ω m 0 = 0.003 \sigma_{\Omega_{\rm m0}}=0.003 σ Ω m0 = 0.003 1 σ 1\sigma 1 σ σ β = 0.21 \sigma_\beta= 0.21 σ β = 0.21 σ β d m = 0.12 \sigma_{\beta_{\rm dm}} = 0.12 σ β dm = 0.12 σ Ω m 0 = 0.038 \sigma_{\Omega_{\rm m0}} = 0.038 σ Ω m0 = 0.038 E P E_{\rm P} E P z = 0.45 z = 0.45 z = 0.45
Lastly, this methodology can be extended to other EP-violating models or to alternative shapes of the scalar-field potential. Once the corresponding E P E_{\rm P} E P E E E
Conclusion
In this paper, we studied model-independent cosmological tests of the weak equivalence principle in the context of a doubly coupled dark energy model, in which a scalar field couples simultaneously to gravity (associated with β \beta β β d m \beta_{\rm dm} β dm β \beta β E P E_{\rm P} E P E P E_{\rm P} E P β d m \beta_{\rm dm} β dm E P E_{\rm P} E P
We then investigated whether these couplings, assumed here to be constant during the relevant late-time epoch, could be detected by next-generation cosmological surveys, namely DESI, Euclid and SKA2. To preserve robustness, we projected the model-independent constraints on { E P , h d , d } \{E_{\rm P}, h_{\rm d}, d\} { E P , h d , d } [41 , 42 ] using two different galaxy populations, onto the CDE and DCDE model parameters. This could give us, for the first time, constraints on β d m \beta_{\rm dm} β dm
To summarise our results, our forecasts reveal the potential of EP measurements from SKA2 to constrain β d m \beta_{\rm dm} β dm E P E_{\rm P} E P
Our results suggest several directions for further investigation. First, one could explore how the constraints depend on more complex shapes of the scalar-field potential. This requires to solve for x x x y y y E P E_{\rm P} E P [41 ] . One could also investigate whether extending the analysis to more than two galaxy populations provides further improvements. Finally, adding more cosmological probes could help reduce the degeneracy between β \beta β β d m \beta_{\rm dm} β dm
Acknowledgments
The authors thank C. Bonvin and S. Castello for useful comments. LA acknowledges support by DFG under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and under Project 554679582 ”GeoGrav: Cosmological Geometry and Gravity with nonlinear physics” and by Tamkeen under the NYUAD Research Institute grant CG014.
Yukawa corrections to the effective gravitational potentials
From Eqs. (28) , (29) and (31) , the effective potential for dark matter particles Ψ ^ d m ≡ Ψ + Q d m φ \hat{\Psi}_{\rm dm} \equiv\Psi+ Q_{\rm dm}\varphi Ψ ^ dm ≡ Ψ + Q dm φ
Ψ ^ d m = Ψ + Q d m φ = − 3 λ 2 2 [ ( 2 Q Y ( k ) ( Q + Q d m ) + 1 ) δ ~ d m δ d m + ( 2 Q 2 Y ( k ) + 1 ) δ ~ b δ b ] − 3 λ 2 2 [ 2 Q d m Y ( k ) ( Q + Q d m ) δ ~ d m δ d m + 2 Y ( k ) Q Q d m δ ~ b δ b ] = − 3 λ 2 2 { [ 1 + 2 Y ( Q + Q d m ) 2 ] δ ~ d m δ d m + [ 1 + 2 Y Q ( Q + Q d m ) ] δ ~ b δ b } . (63) \begin{aligned}
\hat{\Psi}_{\rm dm} &= \Psi+ Q_{\rm dm}\varphi\\
&= -\frac{3\lambda^2}{2}\left[(2QY(k)(Q+Q_{\rm dm})+1)\tilde{\delta}_{\rm dm}\delta_{\rm dm}+(2Q^2Y(k)+1)\tilde{\delta}_{b}\delta_b\right] \\
&\quad-\frac{3\lambda^2}{2}\left[2Q_{\rm dm}Y(k)(Q+Q_{\rm dm})\tilde{\delta}_{\rm dm}\delta_{\rm dm}+2Y(k)QQ_{\rm dm}\tilde{\delta}_{b}\delta_b\right] \\
&= -\frac{3\lambda^2}{2}\left\{\left[1+2Y(Q+Q_{\rm dm})^2\right]\tilde{\delta}_{\rm dm}\delta_{\rm dm}+\left[1+2YQ(Q+Q_{\rm dm})\right]\tilde{\delta}_{b}\delta_b\right\}. \tag*{(63)}
\end{aligned} Ψ ^ dm = Ψ + Q dm φ = − 2 3 λ 2 [ ( 2 Q Y ( k ) ( Q + Q dm ) + 1 ) δ ~ dm δ dm + ( 2 Q 2 Y ( k ) + 1 ) δ ~ b δ b ] − 2 3 λ 2 [ 2 Q dm Y ( k ) ( Q + Q dm ) δ ~ dm δ dm + 2 Y ( k ) Q Q dm δ ~ b δ b ] = − 2 3 λ 2 { [ 1 + 2 Y ( Q + Q dm ) 2 ] δ ~ dm δ dm + [ 1 + 2 Y Q ( Q + Q dm ) ] δ ~ b δ b } . (63) Following Ref. [28 ] , the density contrast for a dark matter particle in the DCDE model, located in empty space at the origin, is given by
Ω ~ d m δ d m = ρ ˉ d m ρ c r i t ρ d m − ρ ˉ d m ρ ˉ d m = δ ρ d m ρ c r i t = δ ρ d m 3 f 1 H 2 . (64) \widetilde{\Omega}_{\rm dm}\delta_{\rm dm}=\frac{\bar{\rho}_{\rm dm}}{\rho_{\rm crit}}\frac{\rho_{\rm dm}-\bar{\rho}_{\rm dm}}{\bar{\rho}_{\rm dm}}=\frac{\delta\rho_{\rm dm}}{\rho_{\rm crit}}=\frac{\delta\rho_{\rm dm}}{3f_1H^2}. \tag*{(64)} Ω dm δ dm = ρ crit ρ ˉ dm ρ ˉ dm ρ dm − ρ ˉ dm = ρ crit δ ρ dm = 3 f 1 H 2 δ ρ dm . (64) On the other hand we know that
m d m ( a ) = m d m , 0 e Q d m [ ϕ ( a ) − ϕ 0 ] = a 3 ∫ d 3 x δ ρ d m ( x ) . (65) m_{\rm dm}(a)=m_{{\rm dm},0}e^{Q_{\rm dm}[\phi(a)-\phi_0]}=a^3\int d^3x\delta\rho_{\rm dm}(\boldsymbol{x}). \tag*{(65)} m dm ( a ) = m dm , 0 e Q dm [ ϕ ( a ) − ϕ 0 ] = a 3 ∫ d 3 x δ ρ dm ( x ) . (65) Solving this gives,
δ ρ d m ( x ) = m d m a 3 δ D ( 3 ) ( x ) , (66) \delta\rho_{\rm dm}(\boldsymbol{x})=\frac{m_{\rm dm}}{a^3}\delta_D^{(3)}(\boldsymbol{x}), \tag*{(66)} δ ρ dm ( x ) = a 3 m dm δ D ( 3 ) ( x ) , (66) with δ D ( 3 ) ( x ) \delta_D^{(3)}(\boldsymbol{x}) δ D ( 3 ) ( x ) ∫ d 3 x δ D ( 3 ) ( x ) = 1 \int d^3x\delta_D^{(3)}(\boldsymbol{x})=1 ∫ d 3 x δ D ( 3 ) ( x ) = 1
δ ρ d m ( k ) = m d m a 3 . (67) \delta\rho_{\rm dm}(\boldsymbol{k})=\frac{m_{\rm dm}}{a^3}. \tag*{(67)} δ ρ dm ( k ) = a 3 m dm . (67) Plugging back into Eq. (64)
3 2 λ 2 Ω ~ d m δ d m = 3 2 λ 2 δ ρ d m 3 f 1 H 2 = m d m 2 f 1 k 2 a , (68) \frac{3}{2}\lambda^2\widetilde{\Omega}_{\rm dm}\delta_{\rm dm}=\frac{3}{2}\lambda^2\frac{\delta\rho_{\rm dm}}{3f_1H^2}=\frac{m_{\rm dm}}{2f_1k^2a}, \tag*{(68)} 2 3 λ 2 Ω dm δ dm = 2 3 λ 2 3 f 1 H 2 δ ρ dm = 2 f 1 k 2 a m dm , (68) and Ψ ^ d m ← d m \widehat{\Psi}_{{\rm dm}\leftarrow{\rm dm}} Ψ dm ← dm
Ψ ^ d m ← d m ( k ) = − m d m 2 a f 1 [ 1 k 2 + 2 ( Q + Q d m ) 2 k 2 + a 2 m ϕ 2 ] , (69) \widehat{\Psi}_{{\rm dm}\leftarrow{\rm dm}}(\boldsymbol{k})=-\frac{m_{\rm dm}}{2af_1}\left[\frac{1}{k^2}+\frac{2(Q+Q_{\rm dm})^2}{k^2+a^2m_\phi^2}\right], \tag*{(69)} Ψ dm ← dm ( k ) = − 2 a f 1 m dm [ k 2 1 + k 2 + a 2 m ϕ 2 2 ( Q + Q dm ) 2 ] , (69) and transforming to real space this gives (with r = a x \boldsymbol{r}=a\boldsymbol{x} r = a x
Ψ ^ d m ← d m ( r ) = − G m d m f 1 r [ 1 + 2 ( Q + Q d m ) 2 e − m ϕ r ] . (70) \widehat{\Psi}_{{\rm dm}\leftarrow{\rm dm}}(r)=-\frac{Gm_{\rm dm}}{f_1r}\left[1+2(Q+Q_{\rm dm})^2e^{-m_\phi r}\right]. \tag*{(70)} Ψ dm ← dm ( r ) = − f 1 r G m dm [ 1 + 2 ( Q + Q dm ) 2 e − m ϕ r ] . (70) Likewise, for a baryon particle located at the origin, its effective potential acting on a dark matter test particle is given by
Ψ ^ d m ← b ( r ) = − G m b f 1 r [ 1 + 2 Q ( Q + Q d m ) e − m ϕ r ] . (71) \widehat{\Psi}_{{\rm dm}\leftarrow{\rm b}}(r)=-\frac{Gm_{\rm b}}{f_1r}\left[1+2Q(Q+Q_{\rm dm})e^{-m_\phi r}\right]. \tag*{(71)} Ψ dm ← b ( r ) = − f 1 r G m b [ 1 + 2 Q ( Q + Q dm ) e − m ϕ r ] . (71) On the other hand, for a baryon test particle the effective potential is simply Ψ ^ b = Ψ \widehat{\Psi}_{\rm b}=\Psi Ψ b = Ψ (30) . Consider a single baryon particle as the source, then the effective potential is
Ψ ^ b ← b ( k ) = − m b 2 a f 1 [ 1 k 2 + 2 Q 2 k 2 + a 2 m ϕ 2 ] , (72) \widehat{\Psi}_{{\rm b}\leftarrow{\rm b}}(\boldsymbol{k})=-\frac{m_{\rm b}}{2af_1}\left[\frac{1}{k^2}+\frac{2Q^2}{k^2+a^2m_\phi^2}\right], \tag*{(72)} Ψ b ← b ( k ) = − 2 a f 1 m b [ k 2 1 + k 2 + a 2 m ϕ 2 2 Q 2 ] , (72) and transforming to real space this gives
Ψ ^ b ← b ( r ) = − G m b f 1 r [ 1 + 2 Q 2 e − m ϕ r ] . (73) \widehat{\Psi}_{{\rm b}\leftarrow{\rm b}}(r)=-\frac{Gm_{\rm b}}{f_1r}\left[1+2Q^2e^{-m_\phi r}\right]. \tag*{(73)} Ψ b ← b ( r ) = − f 1 r G m b [ 1 + 2 Q 2 e − m ϕ r ] . (73) Approximate analytical solutions for x ( N ) = ϕ ′ 6 x(N)=\frac{\phi'}{\sqrt{6}} x ( N ) = 6 ϕ ′ y ( N ) = 1 H V 3 f 1 y(N)=\frac{1}{H}\sqrt{\frac{V}{3f_1}} y ( N ) = H 1 3 f 1 V
It has been shown in [59 ] that one can find approximate analytical solutions for x x x y y y β d m \beta_{\rm dm} β dm [59 ] (see their Fig. 3) indicates that assuming a background close to Λ \Lambda Λ x x x y y y β \beta β β d m \beta_{\rm dm} β dm
We first derive the expansion rate E E E [59 ] , taking the time derivative of Eq. (9) and plug in Eqs. (16) and (17) , we have
H ′ H = − 3 2 ( 1 + p D E , e f f 3 f 1 H 2 ) (74) \frac{H'}{H}=-\frac{3}{2}\left(1+\frac{p_{\mathrm{DE,eff}}}{3f_1H^2}\right) \tag*{(74)} H H ′ = − 2 3 ( 1 + 3 f 1 H 2 p DE , eff ) (74) where p D E , e f f p_{\mathrm{DE,eff}} p DE , eff κ = 1 \kappa=1 κ = 1
T μ ν D E , e f f = T μ ν ϕ + ∇ μ ∇ ν f 1 − g μ ν □ f 1 (75) T_{\mu\nu}^{\mathrm{DE,eff}}=T_{\mu\nu}^{\phi}+\nabla_\mu\nabla_\nu f_1-g_{\mu\nu}\Box f_1 \tag*{(75)} T μν DE , eff = T μν ϕ + ∇ μ ∇ ν f 1 − g μν □ f 1 (75) and therefore
p D E , e f f = 1 2 ( 1 − 6 Q 2 ) f 1 H 2 ϕ ′ 2 − V + H 2 [ f 1 ′ ′ + ( 2 + H ′ H ) f 1 ′ ] (76) p_{\mathrm{DE,eff}}=\frac{1}{2}(1-6Q^2)f_1H^2\phi'^2-V+H^2\left[f_1''+\left(2+\frac{H'}{H}\right)f_1'\right] \tag*{(76)} p DE , eff = 2 1 ( 1 − 6 Q 2 ) f 1 H 2 ϕ ′2 − V + H 2 [ f 1 ′′ + ( 2 + H H ′ ) f 1 ′ ] (76) Expressing the pressure in terms of x x x y y y f 1 ′ ′ f_1'' f 1 ′′ H ′ / H H'/H H ′ / H
p D E , e f f 3 f 1 H 2 = 4 3 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 4 3 β 2 ) x 2 − y 2 − 4 3 β x ] + 4 3 β β d m [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] − 2 6 3 β V , ϕ V y 2 (77) \frac{p_{\mathrm{DE,eff}}}{3f_1H^2}=\frac{4}{3}\beta^2+(1-4\beta^2)\left[\left(1-\frac{4}{3}\beta^2\right)x^2-y^2-\frac{4}{3}\beta x\right]+\frac{4}{3}\beta\beta_{\rm dm}\left[1-(1-4\beta^2)x^2-y^2+4\beta x\right]-\frac{2\sqrt{6}}{3}\beta\frac{V_{,\phi}}{V}y^2 \tag*{(77)} 3 f 1 H 2 p DE , eff = 3 4 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 3 4 β 2 ) x 2 − y 2 − 3 4 β x ] + 3 4 β β dm [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] − 3 2 6 β V V , ϕ y 2 (77) It is easy to find that p D E , e f f 3 f 1 H 2 = x 2 − y 2 \frac{p_{\mathrm{DE,eff}}}{3f_1H^2}=x^2-y^2 3 f 1 H 2 p DE , eff = x 2 − y 2 β = 0 \beta=0 β = 0 E E E (74) :
E ( N ) = exp [ − 3 2 ∫ 0 N ( 1 + p D E , e f f ( N ′ ) 3 f 1 ( N ′ ) H 2 ( N ′ ) ) d N ′ ] (78) E(N)=\exp\left[-\frac{3}{2}\int_0^N\left(1+\frac{p_{\mathrm{DE,eff}}(N')}{3f_1(N')H^2(N')}\right)\mathrm{d}N'\right] \tag*{(78)} E ( N ) = exp [ − 2 3 ∫ 0 N ( 1 + 3 f 1 ( N ′ ) H 2 ( N ′ ) p DE , eff ( N ′ ) ) d N ′ ] (78) by plugging in Eq. (77) . We emphasize that this integral is an exact solution of E E E x ( N ) x(N) x ( N ) y ( N ) y(N) y ( N ) ϕ ′ = 6 x \phi'=\sqrt{6}x ϕ ′ = 6 x − 2 6 β = Q = − f 1 , ϕ f 1 -\frac{2}{\sqrt{6}}\beta=Q=-\frac{f_{1,\phi}}{f_1} − 6 2 β = Q = − f 1 f 1 , ϕ − 2 6 β d m = Q d m = f 2 , ϕ f 2 -\frac{2}{\sqrt{6}}\beta_{\rm dm}=Q_{\rm dm}=\frac{f_{2,\phi}}{f_2} − 6 2 β dm = Q dm = f 2 f 2 , ϕ V = V 0 − V 0 μ ϕ V=V_0-V_0\mu\phi V = V 0 − V 0 μ ϕ (11) becomes
x ′ + ( 3 + H ′ H ) x + 2 β x 2 = 1 1 − 4 β 2 [ V 0 μ 6 f 1 H 2 + 2 β ( 2 + H ′ H ) + β d m Ω ~ d m ] (79) x'+\left(3+\frac{H'}{H}\right)x+2\beta x^2=\frac{1}{1-4\beta^2}\left[\frac{V_0\mu}{\sqrt{6}f_1H^2}+2\beta\left(2+\frac{H'}{H}\right)+\beta_{\rm dm}\widetilde{\Omega}_{\rm dm}\right] \tag*{(79)} x ′ + ( 3 + H H ′ ) x + 2 β x 2 = 1 − 4 β 2 1 [ 6 f 1 H 2 V 0 μ + 2 β ( 2 + H H ′ ) + β dm Ω dm ] (79) which is still exact. Assuming the background is close to Λ \Lambda Λ β , β d m \beta,\beta_{\rm dm} β , β dm w D E , e f f = p D E , e f f / ρ D E , e f f ≈ − 1 w_{\mathrm{DE,eff}}=p_{\mathrm{DE,eff}}/\rho_{\mathrm{DE,eff}}\approx-1 w DE , eff = p DE , eff / ρ DE , eff ≈ − 1 f 1 , f 2 ≈ 1 f_1,f_2\approx1 f 1 , f 2 ≈ 1 V 0 ≈ 3 H 0 2 ( 1 − Ω m 0 ) V_0\approx3H_0^2(1-\Omega_{\rm m0}) V 0 ≈ 3 H 0 2 ( 1 − Ω m0 )
Ω ~ d m ≈ Ω m 0 Ω m 0 + ( 1 − Ω m 0 ) e 3 N ; H ′ H ≈ − 3 2 Ω ~ d m ≈ − 3 Ω m 0 2 [ Ω m 0 + ( 1 − Ω m 0 ) e 3 N ] ; H 2 ≈ H 0 2 ( Ω m 0 e − 3 N + 1 − Ω m 0 ) (80) \widetilde{\Omega}_{\rm dm}\approx\frac{\Omega_{\rm m0}}{\Omega_{\rm m0}+(1-\Omega_{\rm m0})e^{3N}};\qquad\frac{H'}{H}\approx-\frac{3}{2}\widetilde{\Omega}_{\rm dm}\approx-\frac{3\Omega_{\rm m0}}{2[\Omega_{\rm m0}+(1-\Omega_{\rm m0})e^{3N}]};\qquad H^2\approx H_0^2(\Omega_{\rm m0}e^{-3N}+1-\Omega_{\rm m0}) \tag*{(80)} Ω dm ≈ Ω m0 + ( 1 − Ω m0 ) e 3 N Ω m0 ; H H ′ ≈ − 2 3 Ω dm ≈ − 2 [ Ω m0 + ( 1 − Ω m0 ) e 3 N ] 3 Ω m0 ; H 2 ≈ H 0 2 ( Ω m0 e − 3 N + 1 − Ω m0 ) (80) In addition, within the low-redshift range considered in this paper, we can assume that the kinetic energy of the scalar field is much smaller than its potential energy, allowing us to neglect the term quadratic in x x x (79) , which now can be reduced to
x ′ + ( 3 − 3 Ω m 0 e − 3 N 2 E ~ 2 ) x = 1 1 − 4 β 2 [ V 0 μ 6 H 0 2 E ~ 2 + 4 β + ( β d m − 3 β ) Ω m 0 e − 3 N E ~ 2 ] (81) x'+\left(3-\frac{3\Omega_{\rm m0}e^{-3N}}{2\widetilde{E}^2}\right)x=\frac{1}{1-4\beta^2}\left[\frac{V_0\mu}{\sqrt{6}H_0^2\widetilde{E}^2}+4\beta+(\beta_{\rm dm}-3\beta)\frac{\Omega_{\rm m0}e^{-3N}}{\widetilde{E}^2}\right] \tag*{(81)} x ′ + ( 3 − 2 E 2 3 Ω m0 e − 3 N ) x = 1 − 4 β 2 1 [ 6 H 0 2 E 2 V 0 μ + 4 β + ( β dm − 3 β ) E 2 Ω m0 e − 3 N ] (81) recall that E ~ \widetilde{E} E E ~ ( N ) = Ω m 0 e − 3 N + 1 − Ω m 0 \widetilde{E}(N)=\sqrt{\Omega_{\rm m0}e^{-3N}+1-\Omega_{\rm m0}} E ( N ) = Ω m0 e − 3 N + 1 − Ω m0 (81) has analytical solutions
x ( N ) = e 3 N i E ~ ( N i ) e 3 N E ~ ( N ) x i ( N i ) + 1 3 ( 1 − 4 β 2 ) [ 4 β + 3 2 μ ] [ 1 − e 3 N i E ~ ( N i ) e 3 N E ~ ( N ) ] + Ω m 0 3 ( 1 − 4 β 2 ) 1 − Ω m 0 e 3 N E ~ ( N ) [ 2 β d m − 2 β − 3 2 μ ] × ln [ e 3 N i / 2 ( E ~ ( N i ) − 1 − Ω m 0 ) e 3 N / 2 ( E ~ ( N ) − 1 − Ω m 0 ) ] . (82) \begin{aligned}
x(N)={}&\frac{e^{3N_i}\widetilde{E}(N_i)}{e^{3N}\widetilde{E}(N)}x_i(N_i)+\frac{1}{3(1-4\beta^2)}\left[4\beta+\sqrt{\frac{3}{2}}\mu\right]\left[1-\frac{e^{3N_i}\widetilde{E}(N_i)}{e^{3N}\widetilde{E}(N)}\right]\\
&+\frac{\Omega_{\rm m0}}{3(1-4\beta^2)\sqrt{1-\Omega_{\rm m0}e^{3N}\widetilde{E}(N)}}\left[2\beta_{\rm dm}-2\beta-\sqrt{\frac{3}{2}}\mu\right]\\
&\times\ln\left[\frac{e^{3N_i/2}\left(\widetilde{E}(N_i)-\sqrt{1-\Omega_{\rm m0}}\right)}{e^{3N/2}\left(\widetilde{E}(N)-\sqrt{1-\Omega_{\rm m0}}\right)}\right]. \tag*{(82)}
\end{aligned} x ( N ) = e 3 N E ( N ) e 3 N i E ( N i ) x i ( N i ) + 3 ( 1 − 4 β 2 ) 1 [ 4 β + 2 3 μ ] [ 1 − e 3 N E ( N ) e 3 N i E ( N i ) ] + 3 ( 1 − 4 β 2 ) 1 − Ω m0 e 3 N E ( N ) Ω m0 [ 2 β dm − 2 β − 2 3 μ ] × ln e 3 N /2 ( E ( N ) − 1 − Ω m0 ) e 3 N i /2 ( E ( N i ) − 1 − Ω m0 ) . (82) Using V ′ ( N ) = − V 0 μ ϕ ′ = − V 0 μ 6 x V'(N)=-V_0\mu\phi'=-V_0\mu\sqrt{6}x V ′ ( N ) = − V 0 μ ϕ ′ = − V 0 μ 6 x
V ( N ) = V 0 { 1 − 2 6 μ ( β + β d m ) 3 ( 1 − 4 β 2 ) N + 2 6 μ e 3 N i E ~ ( N i ) 3 Ω m 0 [ x ( N i ) − 4 β + 3 2 μ 3 ( 1 − 4 β 2 ) ] [ E ~ ( N ) − 1 ] + 2 6 μ [ 2 β d m − 2 β − 3 2 μ ] 9 ( 1 − 4 β 2 ) Ω Λ 0 [ E ~ ( N ) ln ( e 3 N i / 2 [ E ~ ( N i ) − Ω Λ 0 ] e 3 N / 2 [ E ~ ( N ) − Ω Λ 0 ] ) − ln ( e 3 N i / 2 [ E ~ ( N i ) − Ω Λ 0 ] 1 − Ω Λ 0 ) ] } . (83) \begin{aligned}
V(N)=V_0\Biggl\{1&-\frac{2\sqrt{6}\mu(\beta+\beta_{\rm dm})}{3(1-4\beta^2)}N+\frac{2\sqrt{6}\mu e^{3N_i}\widetilde{E}(N_i)}{3\Omega_{m0}}\left[x(N_i)-\frac{4\beta+\sqrt{\frac{3}{2}}\mu}{3(1-4\beta^2)}\right]\left[\widetilde{E}(N)-1\right]\\
&+\frac{2\sqrt{6}\mu\left[2\beta_{\rm dm}-2\beta-\sqrt{\frac{3}{2}}\mu\right]}{9(1-4\beta^2)\sqrt{\Omega_{\Lambda0}}}\Biggl[\widetilde{E}(N)\ln\left(\frac{e^{3N_i/2}\left[\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}}\right]}{e^{3N/2}\left[\widetilde{E}(N)-\sqrt{\Omega_{\Lambda0}}\right]}\right)\\
&\qquad-\ln\left(\frac{e^{3N_i/2}\left[\widetilde{E}(N_i)-\sqrt{\Omega_{\Lambda0}}\right]}{1-\sqrt{\Omega_{\Lambda0}}}\right)\Biggr]\Biggr\}. \tag*{(83)}
\end{aligned} V ( N ) = V 0 { 1 − 3 ( 1 − 4 β 2 ) 2 6 μ ( β + β dm ) N + 3 Ω m 0 2 6 μ e 3 N i E ( N i ) x ( N i ) − 3 ( 1 − 4 β 2 ) 4 β + 2 3 μ [ E ( N ) − 1 ] + 9 ( 1 − 4 β 2 ) Ω Λ0 2 6 μ [ 2 β dm − 2 β − 2 3 μ ] [ E ( N ) ln e 3 N /2 [ E ( N ) − Ω Λ0 ] e 3 N i /2 [ E ( N i ) − Ω Λ0 ] − ln 1 − Ω Λ0 e 3 N i /2 [ E ( N i ) − Ω Λ0 ] ] } . (83) One can verify that, in the β = 0 \beta=0 β = 0 (82) and (B10) reduce to Eqs. (49) and (50) , as expected. The final step is to find the initial condition x i ≡ x ( N i ) x_i\equiv x(N_i) x i ≡ x ( N i ) (82) . We can obtain from the fixed point that satisfies the conditions x ′ = 0 x'=0 x ′ = 0 y = 0 y=0 y = 0 ϕ \phi ϕ (45) reads
Ω ~ d m ( N i ) = ρ d m ( N i ) 3 f 1 ( N i ) H 2 ( N i ) = 1 − ( 1 − 6 Q 2 ) x i 2 − 2 6 Q x i = 1 − ( 1 − 4 β 2 ) x i 2 + 4 β x i , (84) \widetilde{\Omega}_{\rm dm}(N_i)=\frac{\rho_{\rm dm}(N_i)}{3f_1(N_i)H^2(N_i)}=1-\left(1-6Q^2\right)x_i^2-2\sqrt{6}Qx_i=1-\left(1-4\beta^2\right)x_i^2+4\beta x_i, \tag*{(84)} Ω dm ( N i ) = 3 f 1 ( N i ) H 2 ( N i ) ρ dm ( N i ) = 1 − ( 1 − 6 Q 2 ) x i 2 − 2 6 Q x i = 1 − ( 1 − 4 β 2 ) x i 2 + 4 β x i , (84) Note that Eq. (84) follows from the exact constraint in the DCDE case. We no longer assume a background close to Λ \Lambda Λ
Ω ~ d m ′ Ω ~ d m = ρ d m ′ ρ d m − f 1 ′ f 1 − 2 H ′ H = 0 , (85) \frac{\widetilde{\Omega}'_{\rm dm}}{\widetilde{\Omega}_{\rm dm}}=\frac{\rho'_{\rm dm}}{\rho_{\rm dm}}-\frac{f'_1}{f_1}-2\frac{H'}{H}=0, \tag*{(85)} Ω dm Ω dm ′ = ρ dm ρ dm ′ − f 1 f 1 ′ − 2 H H ′ = 0 , (85) it is easy to find that
ρ d m ′ ρ d m = Q d m ϕ ′ − 3 = − 2 β d m x i − 3 ; f 1 ′ f 1 = f 1 , ϕ ϕ ′ f 1 = − 2 Q ϕ ′ = 4 β x i , (86) \frac{\rho'_{\rm dm}}{\rho_{\rm dm}}=Q_{\rm dm}\phi'-3=-2\beta_{\rm dm}x_i-3;\qquad\frac{f'_1}{f_1}=\frac{f_{1,\phi}\phi'}{f_1}=-2Q\phi'=4\beta x_i, \tag*{(86)} ρ dm ρ dm ′ = Q dm ϕ ′ − 3 = − 2 β dm x i − 3 ; f 1 f 1 ′ = f 1 f 1 , ϕ ϕ ′ = − 2 Q ϕ ′ = 4 β x i , (86) and therefore
H ′ H = − 3 2 − ( β d m + 2 β ) x i . (87) \frac{H'}{H}=-\frac{3}{2}-(\beta_{\rm dm}+2\beta)x_i. \tag*{(87)} H H ′ = − 2 3 − ( β dm + 2 β ) x i . (87) Substituting Eqs. (84) and (B14) into Eq. (79) , we find
x i = 2 ( β + β d m ) 3 − 4 β ( β + β d m ) . (88) x_i=\frac{2(\beta+\beta_{\rm dm})}{3-4\beta(\beta+\beta_{\rm dm})}. \tag*{(88)} x i = 3 − 4 β ( β + β dm ) 2 ( β + β dm ) . (88) Setting β = 0 \beta=0 β = 0 x i = 2 β d m / 3 x_i=2\beta_{\rm dm}/3 x i = 2 β dm /3 β d m = 0 \beta_{\rm dm}=0 β dm = 0
Ω ~ D E , e f f = ( 1 − 4 β 2 ) x i 2 − 4 β x i = ( 1 − 4 β 2 ) [ 2 ( β + β d m ) 3 − 4 β ( β + β d m ) ] 2 − 8 β ( β + β d m ) 3 − 4 β ( β + β d m ) . (89) \widetilde{\Omega}_{\rm DE,eff}=(1-4\beta^2)x_i^2-4\beta x_i=(1-4\beta^2)\left[\frac{2(\beta+\beta_{\rm dm})}{3-4\beta(\beta+\beta_{\rm dm})}\right]^2-\frac{8\beta(\beta+\beta_{\rm dm})}{3-4\beta(\beta+\beta_{\rm dm})}. \tag*{(89)} Ω DE , eff = ( 1 − 4 β 2 ) x i 2 − 4 β x i = ( 1 − 4 β 2 ) [ 3 − 4 β ( β + β dm ) 2 ( β + β dm ) ] 2 − 3 − 4 β ( β + β dm ) 8 β ( β + β dm ) . (89) We also discuss here a particular case. From Eq. (87) , we see that, in this regime, we can identify the effective EoS
w e f f ≡ − 1 − 2 3 H ′ H = 2 3 ( β d m + 2 β ) x i . (90) w_{\rm eff}\equiv-1-\frac{2}{3}\frac{H'}{H}=\frac{2}{3}(\beta_{\rm dm}+2\beta)x_i. \tag*{(90)} w eff ≡ − 1 − 3 2 H H ′ = 3 2 ( β dm + 2 β ) x i . (90) Interestingly, for β d m = − 2 β \beta_{\rm dm}=-2\beta β dm = − 2 β w e f f = 0 w_{\rm eff}=0 w eff = 0 x i ≠ 0 x_i\ne0 x i = 0 (31) (again neglecting baryons)
δ d m ′ ′ + ( 1 + H ′ H − 2 β d m x ) δ d m ′ − 3 2 [ 1 + 4 3 Y ( k ) ( β + β d m ) 2 ] [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] δ d m = 0 , (91) \delta''_{\rm dm}+\left(1+\frac{H'}{H}-2\beta_{\rm dm}x\right)\delta'_{\rm dm}-\frac{3}{2}\left[1+\frac{4}{3}Y(k)(\beta+\beta_{\rm dm})^2\right]\left[1-(1-4\beta^2)x^2-y^2+4\beta x\right]\delta_{\rm dm}=0, \tag*{(91)} δ dm ′′ + ( 1 + H H ′ − 2 β dm x ) δ dm ′ − 2 3 [ 1 + 3 4 Y ( k ) ( β + β dm ) 2 ] [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x ] δ dm = 0 , (91) and plugging in this particular case with y = 0 y=0 y = 0 Y ( k ) = 1 Y(k)=1 Y ( k ) = 1 β d m = − 2 β \beta_{\rm dm}=-2\beta β dm = − 2 β x = x i x=x_i x = x i
δ d m ′ ′ + ( 1 2 − 8 β 2 3 + 4 β 2 ) δ d m ′ − 9 − 4 β 2 2 ( 3 + 4 β 2 ) δ d m = 0. (92) \delta''_{\rm dm}+\left(\frac{1}{2}-\frac{8\beta^2}{3+4\beta^2}\right)\delta'_{\rm dm}-\frac{9-4\beta^2}{2(3+4\beta^2)}\delta_{\rm dm}=0. \tag*{(92)} δ dm ′′ + ( 2 1 − 3 + 4 β 2 8 β 2 ) δ dm ′ − 2 ( 3 + 4 β 2 ) 9 − 4 β 2 δ dm = 0. (92) For β 2 < 9 / 4 \beta^2 < 9/4 β 2 < 9/4 δ d m ∝ a \delta_{\rm dm} \propto a δ dm ∝ a f g = 1 f_g = 1 f g = 1 Λ \Lambda Λ E P E_P E P
E P = 9 ( 1 − 4 β 2 ) ( 3 + 4 β 2 ) 2 (93) E_P = \frac{9(1-4\beta^2)}{(3+4\beta^2)^2} \tag*{(93)} E P = ( 3 + 4 β 2 ) 2 9 ( 1 − 4 β 2 ) (93) which could still deviate from unity. For this particular case, then, while the background and the linearly perturbed behavior are indistinguishable from uncoupled dark energy, the deviation from the equivalence principle can still be detected.
Phase-space system for the doubly coupled model including baryons
In this appendix we rewrite the background equations in terms of the variables x x x y y y Ω ~ d m \widetilde{\Omega}_{\rm dm} Ω dm κ = 1 \kappa= 1 κ = 1
1 = ρ d m 3 f 1 H 2 + ρ b 3 f 1 H 2 + 1 2 ( 1 − 6 Q 2 ) ϕ ′ 2 3 + V ( ϕ ) 3 f 1 H 2 − f 1 , ϕ ϕ ′ f 1 = Ω ~ d m + Ω ~ b + ( 1 − 6 Q 2 ) x 2 + y 2 − 4 β x . (94) \begin{aligned}
1 &= \frac{\rho_{\rm dm}}{3f_1H^2}+\frac{\rho_b}{3f_1H^2}+\frac{1}{2}(1-6Q^2)\frac{\phi'^2}{3}+\frac{V(\phi)}{3f_1H^2}-\frac{f_{1,\phi}\phi'}{f_1} \\
&= \widetilde{\Omega}_{\rm dm}+\widetilde{\Omega}_b+(1-6Q^2)x^2+y^2-4\beta x . \tag*{(94)}
\end{aligned} 1 = 3 f 1 H 2 ρ dm + 3 f 1 H 2 ρ b + 2 1 ( 1 − 6 Q 2 ) 3 ϕ ′2 + 3 f 1 H 2 V ( ϕ ) − f 1 f 1 , ϕ ϕ ′ = Ω dm + Ω b + ( 1 − 6 Q 2 ) x 2 + y 2 − 4 β x . (94) Introducing a third variable z = 1 H ρ b 3 f 1 z=\frac{1}{H}\sqrt{\frac{\rho_b}{3f_1}} z = H 1 3 f 1 ρ b
Ω ~ d m = 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 . (95) \widetilde{\Omega}_{\rm dm}=1-(1-4\beta^2)x^2-y^2+4\beta x-z^2 . \tag*{(95)} Ω dm = 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 . (95) From
H ′ H = − 3 2 ( 1 + p D E , e f f 3 f 1 H 2 ) , (96) \frac{H'}{H}=-\frac{3}{2}\left(1+\frac{p_{\rm DE,eff}}{3f_1H^2}\right), \tag*{(96)} H H ′ = − 2 3 ( 1 + 3 f 1 H 2 p DE , eff ) , (96) we include baryons and get
− 2 3 H ′ H = 1 + 4 3 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 4 3 β 2 ) x 2 − y 2 − 4 3 β x ] + 4 3 β β d m [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 ] − 2 6 3 β V , ϕ V y 2 . (97) -\frac{2}{3}\frac{H'}{H}=1+\frac{4}{3}\beta^2+(1-4\beta^2)\left[\left(1-\frac{4}{3}\beta^2\right)x^2-y^2-\frac{4}{3}\beta x\right]+\frac{4}{3}\beta\beta_{\rm dm}\left[1-(1-4\beta^2)x^2-y^2+4\beta x-z^2\right]-\frac{2\sqrt{6}}{3}\beta\frac{V_{,\phi}}{V}y^2 . \tag*{(97)} − 3 2 H H ′ = 1 + 3 4 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 3 4 β 2 ) x 2 − y 2 − 3 4 β x ] + 3 4 β β dm [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 ] − 3 2 6 β V V , ϕ y 2 . (97) Putting now V = V 0 e − μ ϕ V=V_0e^{-\mu\phi} V = V 0 e − μ ϕ x , y , z x,y,z x , y , z
− 2 3 H ′ H = 1 + 4 3 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 4 3 β 2 ) x 2 − y 2 − 4 3 β x ] + 4 3 β β d m [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 ] + 2 6 3 β μ y 2 . (98) -\frac{2}{3}\frac{H'}{H}=1+\frac{4}{3}\beta^2+(1-4\beta^2)\left[\left(1-\frac{4}{3}\beta^2\right)x^2-y^2-\frac{4}{3}\beta x\right]+\frac{4}{3}\beta\beta_{\rm dm}\left[1-(1-4\beta^2)x^2-y^2+4\beta x-z^2\right]+\frac{2\sqrt{6}}{3}\beta\mu y^2 . \tag*{(98)} − 3 2 H H ′ = 1 + 3 4 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 3 4 β 2 ) x 2 − y 2 − 3 4 β x ] + 3 4 β β dm [ 1 − ( 1 − 4 β 2 ) x 2 − y 2 + 4 β x − z 2 ] + 3 2 6 β μ y 2 . (98) Continuing with the Klein-Gordon equation:
ϕ ′ ′ + ( 3 + H ′ H ) ϕ ′ + 1 2 f 1 , ϕ f 1 ϕ ′ 2 = − 1 ( 1 − 4 β 2 ) f 1 H 2 [ V , ϕ − 3 f 1 , ϕ f 1 H 2 ( 2 + H ′ H ) + f 2 , ϕ 2 f 2 ρ d m ] , (99) \phi''+\left(3+\frac{H'}{H}\right)\phi'+\frac{1}{2}\frac{f_{1,\phi}}{f_1}\phi'^2=-\frac{1}{(1-4\beta^2)f_1H^2}\left[V_{,\phi}-\frac{3f_{1,\phi}}{f_1}H^2\left(2+\frac{H'}{H}\right)+\frac{f_{2,\phi}}{2f_2}\rho_{\rm dm}\right], \tag*{(99)} ϕ ′′ + ( 3 + H H ′ ) ϕ ′ + 2 1 f 1 f 1 , ϕ ϕ ′2 = − ( 1 − 4 β 2 ) f 1 H 2 1 [ V , ϕ − f 1 3 f 1 , ϕ H 2 ( 2 + H H ′ ) + 2 f 2 f 2 , ϕ ρ dm ] , (99) which can be written as
x ′ + ( 3 + H ′ H ) x + 2 β x 2 = 3 μ y 2 ( 1 − 4 β 2 ) 6 + 2 1 − 4 β 2 β ( 2 + H ′ H ) + β d m Ω ~ d m 1 − 4 β 2 . (100) x'+\left(3+\frac{H'}{H}\right)x+2\beta x^2=\frac{3\mu y^2}{(1-4\beta^2)\sqrt{6}}+\frac{2}{1-4\beta^2}\beta\left(2+\frac{H'}{H}\right)+\beta_{\rm dm}\frac{\widetilde{\Omega}_{\rm dm}}{1-4\beta^2}. \tag*{(100)} x ′ + ( 3 + H H ′ ) x + 2 β x 2 = ( 1 − 4 β 2 ) 6 3 μ y 2 + 1 − 4 β 2 2 β ( 2 + H H ′ ) + β dm 1 − 4 β 2 Ω dm . (100) Then, from the definition of y y y z z z
y ′ y = − H ′ H + 1 2 V ′ V − 1 2 f 1 ′ f 1 = − H ′ H − 1 2 μ 6 x − 2 β x . (101) \begin{aligned}
\frac{y'}{y}=-\frac{H'}{H}+\frac{1}{2}\frac{V'}{V}-\frac{1}{2}\frac{f_1'}{f_1} \\
&=-\frac{H'}{H}-\frac{1}{2}\mu\sqrt{6}x-2\beta x . \tag*{(101)}
\end{aligned} y y ′ = − H H ′ + 2 1 V V ′ − 2 1 f 1 f 1 ′ = − H H ′ − 2 1 μ 6 x − 2 β x . (101) z ′ z = − 3 2 − H ′ H − 2 β x . (102) \frac{z'}{z}=-\frac{3}{2}-\frac{H'}{H}-2\beta x . \tag*{(102)} z z ′ = − 2 3 − H H ′ − 2 β x . (102) So the final system of autonomous equations is:
x ′ = − ( 3 + H ′ H ) x − 2 β x 2 + 6 μ y 2 2 ( 1 − 4 β 2 ) + 2 1 − 4 β 2 β ( 2 + H ′ H ) + β d m Ω ~ d m 1 − 4 β 2 , (103) x'=-\left(3+\frac{H'}{H}\right)x-2\beta x^2+\frac{\sqrt{6}\mu y^2}{2(1-4\beta^2)}+\frac{2}{1-4\beta^2}\beta\left(2+\frac{H'}{H}\right)+\beta_{\rm dm}\frac{\widetilde{\Omega}_{\rm dm}}{1-4\beta^2}, \tag*{(103)} x ′ = − ( 3 + H H ′ ) x − 2 β x 2 + 2 ( 1 − 4 β 2 ) 6 μ y 2 + 1 − 4 β 2 2 β ( 2 + H H ′ ) + β dm 1 − 4 β 2 Ω dm , (103) y ′ = y ( − H ′ H − 1 2 μ 6 x − 2 β x ) , (104) y'=y\left(-\frac{H'}{H}-\frac{1}{2}\mu\sqrt{6}x-2\beta x\right), \tag*{(104)} y ′ = y ( − H H ′ − 2 1 μ 6 x − 2 β x ) , (104) z ′ = z ( − 3 2 − H ′ H − 2 β x ) . (105) z'=z\left(-\frac{3}{2}-\frac{H'}{H}-2\beta x\right). \tag*{(105)} z ′ = z ( − 2 3 − H H ′ − 2 β x ) . (105) For β = 0 \beta=0 β = 0
x ′ = − ( 3 + H ′ H ) x + 6 μ y 2 2 + β d m ( 1 − x 2 − y 2 − z 2 ) , (106) x'=-\left(3+\frac{H'}{H}\right)x+\frac{\sqrt{6}\mu y^2}{2}+\beta_{\rm dm}\left(1-x^2-y^2-z^2\right), \tag*{(106)} x ′ = − ( 3 + H H ′ ) x + 2 6 μ y 2 + β dm ( 1 − x 2 − y 2 − z 2 ) , (106) y ′ = y ( − H ′ H − 6 2 μ x ) , (107) y'=y\left(-\frac{H'}{H}-\frac{\sqrt{6}}{2}\mu x\right), \tag*{(107)} y ′ = y ( − H H ′ − 2 6 μx ) , (107) z ′ = z ( − 3 2 − H ′ H ) . (108) z'=z\left(-\frac{3}{2}-\frac{H'}{H}\right). \tag*{(108)} z ′ = z ( − 2 3 − H H ′ ) . (108) which coincides with the usual equations of CDE, with
H ′ H = − 3 2 ( 1 + x 2 − y 2 ) (109) \frac{H'}{H}=-\frac{3}{2}\left(1+x^2-y^2\right) \tag*{(109)} H H ′ = − 2 3 ( 1 + x 2 − y 2 ) (109) Now we find the critical points of the system Eqs. Eqs. (103) , (C11) for y = 0 y=0 y = 0 z = 0 z=0 z = 0
x ′ = − ( 3 + H ′ H ) x − 2 β x 2 + 2 β 1 − 4 β 2 ( 2 + H ′ H ) + β d m 1 − ( 1 − 4 β 2 ) x 2 + 4 β x 1 − 4 β 2 , (110) x'=-\left(3+\frac{H'}{H}\right)x-2\beta x^2+\frac{2\beta}{1-4\beta^2}\left(2+\frac{H'}{H}\right)+\beta_{\rm dm}\frac{1-(1-4\beta^2)x^2+4\beta x}{1-4\beta^2}, \tag*{(110)} x ′ = − ( 3 + H H ′ ) x − 2 β x 2 + 1 − 4 β 2 2 β ( 2 + H H ′ ) + β dm 1 − 4 β 2 1 − ( 1 − 4 β 2 ) x 2 + 4 β x , (110) y ′ = 0 , (111) y'=0, \tag*{(111)} y ′ = 0 , (111) z ′ = 0 , (112) z'=0, \tag*{(112)} z ′ = 0 , (112) with
− 2 3 H ′ H = 1 + 4 3 β 2 + ( 1 − 4 β 2 ) [ ( 1 − 4 3 β 2 ) x 2 − 4 3 β x + 4 3 β β d m [ 1 − ( 1 − 4 β 2 ) x 2 + 4 β x ] . (113) -\frac{2}{3}\frac{H'}{H}=1+\frac{4}{3}\beta^2+(1-4\beta^2)[(1-\frac{4}{3}\beta^2)x^2-\frac{4}{3}\beta x+\frac{4}{3}\beta\beta_{\rm dm}[1-(1-4\beta^2)x^2+4\beta x]. \tag*{(113)} − 3 2 H H ′ = 1 + 3 4 β 2 + ( 1 − 4 β 2 ) [( 1 − 3 4 β 2 ) x 2 − 3 4 β x + 3 4 β β dm [ 1 − ( 1 − 4 β 2 ) x 2 + 4 β x ] . (113) Then we can simplify the x ′ x' x ′
x ′ = 1 2 ( ( 4 β 2 − 1 ) x 2 + 4 β x + 1 ) ( 2 β + β d m ( 4 β x + 2 ) + ( 4 β 2 − 3 ) x ) . (114) x'=\frac{1}{2}\left((4\beta^2-1)x^2+4\beta x+1\right)\left(2\beta+\beta_{\rm dm}(4\beta x+2)+(4\beta^2-3)x\right). \tag*{(114)} x ′ = 2 1 ( ( 4 β 2 − 1 ) x 2 + 4 β x + 1 ) ( 2 β + β dm ( 4 β x + 2 ) + ( 4 β 2 − 3 ) x ) . (114) The y = 0 y=0 y = 0 x ′ = 0 x'=0 x ′ = 0
x ± = 1 ± 1 − 2 β , (115) x_{\pm}=\frac{1}{\pm1-2\beta}, \tag*{(115)} x ± = ± 1 − 2 β 1 , (115) x i = 2 ( β + β d m ) 3 − 4 β ( β + β d m ) . (116) x_i=\frac{2(\beta+\beta_{\rm dm})}{3-4\beta(\beta+\beta_{\rm dm})}. \tag*{(116)} x i = 3 − 4 β ( β + β dm ) 2 ( β + β dm ) . (116) Since these critical points exist for y = 0 y=0 y = 0 x ± x_{\pm} x ± Ω ~ d m = 0 \widetilde{\Omega}_{\rm dm}=0 Ω dm = 0
Details on the Fisher Matrix Projection
In this appendix, we present details on the Fisher matrix projection. More specifically, we project the model-independent constraints on the parameters { E P , h d , d } \{E_{\rm P},h_d,d\} { E P , h d , d } [41 , 42 ] (with each parameter left free to vary at each redshift bin), onto a set of constant parameters: { β , β d m , Ω m 0 } \{\beta,\beta_{\rm dm},\Omega_{\rm m0}\} { β , β dm , Ω m0 } β = 0 \beta=0 β = 0 F o l d \mathcal{F}_{\rm old} F old { E P , h d , d } \{E_P,h_d,d\} { E P , h d , d } 3 × 3 3\times3 3 × 3 F n e w \mathcal{F}_{\rm new} F new { β , β d m , Ω m 0 } \{\beta,\beta_{\rm dm},\Omega_{\rm m0}\} { β , β dm , Ω m0 }
F n e w = J T F o l d J , (117) \mathcal{F}_{\rm new}=J^{T}\mathcal{F}_{\rm old}J, \tag*{(117)} F new = J T F old J , (117) with J J J J T J^T J T E P ( N i ) ≡ E P , i E_P(N_i)\equiv E_{P,i} E P ( N i ) ≡ E P , i h d ( N i ) ≡ h d , i h_d(N_i)\equiv h_{d,i} h d ( N i ) ≡ h d , i d ( N i ) ≡ d i d(N_i)\equiv d_i d ( N i ) ≡ d i E P ( N ) E_P(N) E P ( N ) (48) and (51) , while those for d ( N ) d(N) d ( N ) h d ( N ) h_d(N) h d ( N )
d ( N ) = L A ( N ) L A r ( N ) = ∫ N 0 e − N ′ d N ′ E ( N ′ ) ∫ N 0 e − N ′ d N ′ E r ( N ′ ) , (118) d(N)=\frac{L_A(N)}{L_{A_r}(N)}=\frac{\int_N^0\frac{e^{-N'}\,dN'}{E(N')}}{\int_N^0\frac{e^{-N'}\,dN'}{E_r(N')}}, \tag*{(118)} d ( N ) = L A r ( N ) L A ( N ) = ∫ N 0 E r ( N ′ ) e − N ′ d N ′ ∫ N 0 E ( N ′ ) e − N ′ d N ′ , (118) h d ( N ) = h ( N ) d ( N ) = E ( N ) E r ( N ) L A ( N ) L A r ( N ) = E ( N ) ∫ N 0 e − N ′ d N ′ E ( N ′ ) E r ( N ) ∫ N 0 e − N ′ d N ′ E r ( N ′ ) , (119) h_d(N)=h(N)d(N)=\frac{E(N)}{E_r(N)}\frac{L_A(N)}{L_{A_r}(N)}=\frac{E(N)\int_N^0\frac{e^{-N'}\,dN'}{E(N')}}{E_r(N)\int_N^0\frac{e^{-N'}\,dN'}{E_r(N')}}, \tag*{(119)} h d ( N ) = h ( N ) d ( N ) = E r ( N ) E ( N ) L A r ( N ) L A ( N ) = E r ( N ) ∫ N 0 E r ( N ′ ) e − N ′ d N ′ E ( N ) ∫ N 0 E ( N ′ ) e − N ′ d N ′ , (119) where E r E_r E r Λ \Lambda Λ E E E β \beta β β d m \beta_{\rm dm} β dm x x x y y y (52) and (53) , with the exact solution given by Eq. (78) . Moreover, in the CDE case, E ( N ) E(N) E ( N )
E ( N ) = e − 3 / 2 ∫ 0 N ( x ( N ′ ) 2 − y ( N ′ ) 2 + 1 ) d N ′ , (120) E(N)=e^{-3/2\int_0^N(x(N')^2-y(N')^2+1)dN'}, \tag*{(120)} E ( N ) = e − 3/2 ∫ 0 N ( x ( N ′ ) 2 − y ( N ′ ) 2 + 1 ) d N ′ , (120) with the x x x y y y (49) and (50) . This expansion is well justified, as the relative difference from E r ( N ) E_r(N) E r ( N ) 0.4 % 0.4\% 0.4% Table VIII .
Table VIII. Comparison of the reference expansion rate E r E_r E r E E E
The Jacobian elements J ( i ) J^{(i)} J ( i ) η i \eta_i η i
J ( i ) = ( ∂ h d , i ∂ β ∂ h d , i ∂ β d m ∂ h d , i ∂ Ω m 0 ∂ d i ∂ β ∂ d i ∂ β d m ∂ d i ∂ Ω m 0 ∂ E P , i ∂ β ∂ E P , i ∂ β d m ∂ E P , i ∂ Ω m 0 ) , (121) J^{(i)}=
\begin{pmatrix}
\frac{\partial h_{d,i}}{\partial\beta} & \frac{\partial h_{d,i}}{\partial\beta_{\rm dm}} & \frac{\partial h_{d,i}}{\partial\Omega_{\rm m0}}\\
\frac{\partial d_i}{\partial\beta} & \frac{\partial d_i}{\partial\beta_{\rm dm}} & \frac{\partial d_i}{\partial\Omega_{\rm m0}}\\
\frac{\partial E_{P,i}}{\partial\beta} & \frac{\partial E_{P,i}}{\partial\beta_{\rm dm}} & \frac{\partial E_{P,i}}{\partial\Omega_{\rm m0}}
\end{pmatrix}, \tag*{(121)} J ( i ) = ∂ β ∂ h d , i ∂ β ∂ d i ∂ β ∂ E P , i ∂ β dm ∂ h d , i ∂ β dm ∂ d i ∂ β dm ∂ E P , i ∂ Ω m0 ∂ h d , i ∂ Ω m0 ∂ d i ∂ Ω m0 ∂ E P , i , (121) with each entry evaluated at the fiducial values, e.g.:
∂ E P ( N i ) ∂ β ∣ f i d = ∂ ∂ β [ E P ( N i ; β , β d m , Ω m 0 ) ] ∣ N = N i ; β = β f i d , β d m = β d m , f i d , Ω m 0 = Ω m 0 , f i d . (122) \left.\frac{\partial E_P(N_i)}{\partial\beta}\right|_{\rm fid}
=
\left.\frac{\partial}{\partial\beta}\left[E_P(N_i;\beta,\beta_{\rm dm},\Omega_{\rm m0})\right]\right|_{N=N_i;\,\beta=\beta_{\rm fid},\,\beta_{\rm dm}=\beta_{\rm dm,fid},\,\Omega_{\rm m0}=\Omega_{\rm m0,fid}}. \tag*{(122)} ∂ β ∂ E P ( N i ) fid = ∂ β ∂ [ E P ( N i ; β , β dm , Ω m0 ) ] N = N i ; β = β fid , β dm = β dm , fid , Ω m0 = Ω m0 , fid . (122) In this paper we set: β f i d = 0.1 \beta_{\rm fid}=0.1 β fid = 0.1 β d m , f i d = 0.2 \beta_{\rm dm,fid}=0.2 β dm , fid = 0.2 Ω m 0 , f i d = 0.315 \Omega_{\rm m0,fid}=0.315 Ω m0 , fid = 0.315
For the full Jacobian J J J N N N
J = ( J ( 1 ) J ( 2 ) ⋮ J ( N ) ) . (123) J=
\begin{pmatrix}
J^{(1)}\\
J^{(2)}\\
\vdots\\
J^{(N)}
\end{pmatrix}. \tag*{(123)} J = J ( 1 ) J ( 2 ) ⋮ J ( N ) . (123) Estimates of the relative error on the luminosity distance
We give now a rough estimate of the current uncertainty on the luminosity distance d d d N N N δ m \delta m δ m d d d
σ d d = ln 10 5 δ m N . (124) \frac{\sigma_d}{d} = \frac{\ln10}{5}\frac{\delta m}{\sqrt{N}}. \tag*{(124)} d σ d = 5 ln 10 N δ m . (124) Using the current Pantheon+ catalog [62 ] , we show the resulting uncertainty on d d d Table IX
Table IX. Pantheon+ supernova luminosity-distance uncertainties.
These results are just a rough estimate, since we assumed a constant δ m \delta m δ m
References 
