Observing modified gravity with redshift surveys

Wide-deep galaxy redshift surveys have the power to yield information on bothH(z) and fg(z)

through measurements of Baryon Acoustic Oscillations (BAO) and redshift-space distortions. In particular, if gravity is not modified and matter is not interacting other than gravitationally, then a detection of the expansion rate is directly linked to a unique prediction of the growth rate. Otherwise galaxy redshift surveys provide a unique and crucial way to make a combined analysis of H(z) and fg(z) to test gravity. As a wide-deep survey, Euclid allows us to measure H(z)

directly from BAO, but also indirectly through the angular diameter distanceDA(z) (and possibly

distance ratios from weak lensing). Most importantly, Euclid survey enables us to measure the cosmic growth history using two independent methods: fg(z) from galaxy clustering, and G(z)

from weak lensing. In the following we discuss the estimation of [H(z), DA(z) and fg(z)] from

galaxy clustering.

From the measure of BAO in the matter power spectrum or in the 2-point correlation function one can infer information on the expansion rate of the universe. In fact, the sound waves imprinted in the CMB can be also detected in the clustering of galaxies, thereby completing an important test of our theory of gravitational structure formation.

The BAO in the radial and tangential directions offer a way to measure the Hubble parameter and angular diameter distance, respectively. In the simplest FLRW universe the basis to define distances is the dimensionless, radial, comoving distance:

χ(z)≡

Z z

0

dz0

E(z0₎. (1.7.22)

The dimensionless version of the Hubble parameter is: E2(z) = Ω(0)_m(1 +z)3+ Ωk(1 +z)2+ (Ωk−Ω(0)m) exp Z z 0 3(1 +w(˜z)) 1 + ˜z d˜z . (1.7.23) The standard cosmological distances are related toχ(z) via

DA(z) = c H0(1 +z) √ −Ωk sinp−Ωkχ(z) (1.7.24) where the luminosity distance,DL(z), is given by the distance duality:

The coupling betweenDA(z) andDL(z) persists in any metric theory of gravity as long as photon

number is conserved (see Section 4.2 for cases in which the duality relation is violated). BAO yield bothDA(z) andH(z) making use of an almost completely linear physics (unlike for example

SN Ia, demanding complex and poorly understood mechanisms of explosions). Furthermore, they provide the chance of constraining the growth rate through the change in the amplitude of the power spectrum.

The characteristic scale of the BAO is set by the sound horizon at decoupling. Consequently, one can attain the angular diameter distance and Hubble parameter separately. This scale along the line of sight (s||(z)) measures H(z) through H(z) = c∆z/s||(z), while the tangential mode

measures the angular diameter distanceDA(z) =s⊥/∆θ(1 +z).

One can then use the power spectrum to derive predictions on the parameter constraining power of the survey (see e.g., [61, 562, 1224, 1231, 418]).

In order to explore the cosmological parameter constraints from a given redshift survey, one needs to specify the measurement uncertainties of the galaxy power spectrum. In general, the statistical error on the measurement of the galaxy power spectrumPg(k) at a given wave-number

bin is [477] _∆P g Pg 2 = 2(2π) 2 Vsurveyk2∆k∆µ 1 + 1 ngPg 2 , (1.7.26)

where ng is the mean number density of galaxies, Vsurvey is the comoving survey volume of the

galaxy survey, andµis the cosine of the angle betweenkand the line-of-sight directionµ=~k·ˆr/k. In general, the observed galaxy power spectrum is different from the true spectrum, and it can be reconstructed approximately assuming a reference cosmology (which we consider to be our fiducial cosmology) as (e.g., [1075])

Pobs(kref⊥, krefk, z) =

DA(z)2refH(z)

DA(z)2H(z)ref

Pg(kref⊥, krefk, z) +Pshot, (1.7.27)

where Pg(kref⊥, krefk, z) =b(z)2 " 1 +β(z) k2 refk k2 ref⊥+kref2 k #2 ×Pmatter(k, z). (1.7.28)

In Eq. (1.7.27), H(z) and DA(z) are the Hubble parameter and the angular diameter distance,

respectively, and the prefactor (DA(z)2refH(z))/(DA(z)2H(z)ref) encapsulates the geometrical dis-

tortions due to the Alcock–Paczynski effect [1075, 115]. Their values in the reference cosmology are distinguished by the subscript ‘ref’, while those in the true cosmology have no subscript. k⊥andkk

are the wave-numbers across and along the line of sight in the true cosmology, and they are related to the wave-numbers calculated assuming the reference cosmology bykref⊥ =k⊥DA(z)/DA(z)ref

and krefk = kkH(z)ref/H(z). Pshot is the unknown white shot noise that remains even after the

conventional shot noise of inverse number density has been subtracted [1075]. In Eq. (1.7.28),b(z) is thelinear biasfactor between galaxy and matter density distributions,fg(z) is the linear growth

rate,7 _{and β(z) = f}

g(z)/b(z) is the linear redshift-space distortion parameter [649]. The linear

matter power spectrumPmatter(k, z) in Eq. (1.7.27) takes the form

Pmatter(k, z) = 8π2_c4_k 0∆2R(k0) 25H4 0Ω2m T2(k) _G(z) G(z= 0) 2_k k0 ns e−k2µ2σ2r_{, (1.7.29)}

whereG(z) is the usualscale independent linear growth-factor in the absence of massive neutrino free-streaming (see Eq. (25) in [451]), whose fiducial value in each redshift bin is computed through numerical integration of the differential equations governing the growth of linear perturbations in

7 _{In presence of massive neutrinosf}

presence of dark energy [779] or employing the approximation of Eq. (1.3.22). T(k) depends on matter and baryon densities8 _{(neglecting dark energy at early times), and is computed in each}

redshift bin using a Boltzmann code like_camb9 _{[743] or}

cmbfast.

In Eq. (1.7.29) a damping factore−k2µ2σ_r2 _{has been added, due to redshift uncertainties, where}

σr = (∂r/∂z)σz, r(z) being the comoving distance [1226, 1075], and assumed that the power

spectrum of primordial curvature perturbations,PR(k), is

∆2_R(k)≡k 3_P R(k) 2π2 = ∆ 2 R(k0) k k0 ns , (1.7.30)

wherek0= 0.002/Mpc, ∆2R(k0)|fid = 2.45×10−9 is the dimensionless amplitude of the primordial

curvature perturbations evaluated at a pivot scalek0, andns is the scalar spectral index [729].

In the limit where the survey volume is much larger than the scale of any features inPobs(k), it

has been shown that the redshift survey Fisher matrix for a given redshift bin can be approximated as [1151] F_ijLSS = Z 1 −1 Z kmax kmin ∂lnPobs(k, µ) ∂pi ∂lnPobs(k, µ) ∂pj Veff(k, µ) 2πk2_dkdµ 2(2π)3 , (1.7.31)

where the derivatives are evaluated at the parameter values pi of the fiducial model, andVeff is

the effective volume of the survey:

Veff(k, µ) = _n gPg(k, µ) ngPg(k, µ) + 1 2 Vsurvey, (1.7.32)

where the comoving number densityng(z) is assumed to be spatially constant. Due to azimuthal

symmetry around the line of sight, the three-dimensional galaxy redshift power spectrumPobs(~k)

depends only onkandµ, i.e., is reduced to two dimensions by symmetry [1075]. The total Fisher matrix can be obtained by summing over the redshift bins.

To minimize nonlinear effects, one should restrict wave-numbers to the quasi-linear regime, e.g., imposing thatkmax is given by requiring that the variance of matter fluctuations in a sphere

of radiusR is, for instance,σ2_{(R) = 0.25 forR =π/(2k}

max). Or one could model the nonlinear

distortions as in [452]. On scales larger than (∼ 100h−1 _{Mpc) where we focus our analysis,}

nonlinear effects can be represented in fact as a displacement field in Lagrangian space modeled by an elliptical Gaussian function. Therefore, following [452, 1076], to model nonlinear effect we multiplyP(k) by the factor

exp ( −k2 " (1−µ2_)Σ2 ⊥ 2 + µ2_Σ2 k 2 #) , (1.7.33)

where Σ⊥ and Σk represent the displacement across and along the line of sight, respectively.

They are related to the growth factor G and to the growth rate fg through Σ⊥ = Σ0G and

Σk = Σ0G(1 +fg). The value of Σ0 is proportional to σ8. For a reference cosmology where

σ8= 0.8 [704], we have Σ0= 11h−1 Mpc.

Finally, we note that when actual data are available, the usual way to measureβ =fg/bis by

fitting the measured galaxy redshift-space correlation functionξ(σ, π) to a model [939]: ξ(σ, π) =

Z ∞

−∞

dv f(v) ˜ξ(σ, π−v/H0), (1.7.34)

8 _{If we assume that neutrinos have a non-vanishing mass, then the transfer function is also redshift-dependent.} 9 _{http://camb.info/}

wheref(v) describes the small-scale random motion (usually modeled by a Gaussian that depends on the galaxy pairwise peculiar velocity dispersion), and ˜ξ(σ, π) is the model accounting for coherent infall velocities:10

˜

ξ(σ, π) =ξ0(s)P0(µ) +ξ2(s)P2(µ) +ξ4(s)P4(µ). (1.7.35)

Pl(µ) are Legendre polynomials; µ = cosθ, where θ denotes the angle between r and π; ξ0(s),

ξ2(s), andξ4(s) depend onβ and the real-space correlation functionξ(r).

The bias between galaxy and matter distributions can be estimated from either galaxy cluster- ing, or weak lensing. To determine bias, we can assume that the galaxy density perturbationδg is

related to the matter density perturbationδm(x) as [494]:

δg =bδm(x) +b2δm2(x)/2. (1.7.36)

Bias can be derived from galaxy clustering by measuring the galaxy bispectrum:

hδgk1δgk2δgk1i = (2π) 3 P(k1)P(k2) J(k1,k2)/b+b2/b2 +cyc.}δD(k1+k2+k3), (1.7.37)

whereJ is a function that depends on the shape of the triangle formed by (k1,k2,k3) inkspace,

but only depends very weakly on cosmology [856, 1204].

In general, bias can be measured from weak lensing through the comparison of the shear-shear and shear-galaxy correlations functions. A combined constraint on bias and the growth factorG(z) can be derived from weak lensing by comparing the cross-correlations of multiple redshift slices.

Of course, if bias is assumed to be linear (b2= 0) and scale independent, or is parametrized in

some simple way, e.g., with a power law scale dependence, then it is possible to estimate it even from linear galaxy clustering alone, as we will see in Section 1.8.3.