1.7 Observational properties of dark energy and modified gravity

Both scalar field dark-energy models and modifications of gravity can in principle lead to any desired expansion history H (z), or equivalently any evolution of the effective dark-energy equation of state parameter w (z ). For canonical scalar fields, this can be achieved by selecting the appropriate potential V (φ) along the evolution of the scalar field φ (t), as was done, e.g., in [102]. For modified gravity models, the same procedure can be followed for example for f (R) type models [e.g. 736]. The evolution history on its own can thus not tell us very much about the physical nature of the mechanism behind the accelerated expansion (although of course a clear measurement showing that w ⁄= − 1 would be a sensational discovery). A smoking gun for modifications of gravity can thus only appear at perturbation level.

In the next subsections we explore how dark energy or modified gravity effects can be detected through weak lensing and redshift surveys.

1.7.1 General remarks

Quite generally, cosmological observations fall into two categories: geometrical probes and structure formation probes. While the former provide a measurement of the Hubble function, the latter are a test of the gravitational theory in an almost Newtonian limit on subhorizon scales. Furthermore, possible effects on the geodesics of test particles need to be derived: naturally, photons follow null-geodesics while massive particles, which constitute the cosmic large-scale structure, move along geodesics for non-relativistic particles.

In some special cases, modified gravity models predict a strong deviation from the standard Friedmann equation as in, e.g., DGP, (1.4.74View Equation). While the Friedmann equation is not know explicitly in more general models of massive gravity (cascading gravity or hard mass gravity), similar modifications are expected to arise and provide characteristic features, [see, e.g., 11Jump To The Next Citation Point, 478]) that could distinguish these models from other scenarios of modified gravity or with additional dynamical degrees of freedom.

In general however the most interesting signatures of modified gravity models are to be found in the perturbation sector. For instance, in DGP, growth functions differ from those in dark-energy models by a few percent for identical Hubble functions, and for that reason, an observation of both the Hubble and the growth function gives a handle on constraining the gravitational theory, [592]. The growth function can be estimated both through weak lensing and through galaxy clustering and redshift distortions.

Concerning the interactions of light with the cosmic large-scale structure, one sees a modified coupling in general models and a difference between the metric potentials. These effects are present in the anisotropy pattern of the CMB, as shown in [792], where smaller fluctuations were found on large angular scales, which can possibly alleviate the tension between the CMB and the ΛCDM model on small multipoles where the CMB spectrum acquires smaller amplitudes due to the ISW-effect on the last-scattering surface, but provides a worse fit to supernova data. An interesting effect inexplicable in GR is the anticorrelation between the CMB temperature and the density of galaxies at high redshift due to a sign change in the integrated Sachs–Wolfe effect. Interestingly, this behavior is very common in modified gravity theories.

A very powerful probe of structure growth is of course weak lensing, but to evaluate the lensing effect it is important to understand the nonlinear structure formation dynamics as a good part of the total signal is generated by small structures. Only recently has it been possible to perform structure formation simulations in modified gravity models, although still without a mechanism in which GR is recovered on very small scales, necessary to be in accordance with local tests of gravity.

In contrast, the number density of collapsed objects relies only little on nonlinear physics and can be used to investigate modified gravity cosmologies. One needs to solve the dynamical equations for a spherically symmetric matter distribution. Modified gravity theories show the feature of lowering the collapse threshold for density fluctuations in the large-scale structure, leading to a higher comoving number density of galaxies and clusters of galaxies. This probe is degenerate with respect to dark-energy cosmologies, which generically give the same trends.

1.7.2 Observing modified gravity with weak lensing

The magnification matrix is a 2 × 2 matrix that relates the true shape of a galaxy to its image. It contains two distinct parts: the convergence, defined as the trace of the matrix, modifies the size of the image, whereas the shear, defined as the symmetric traceless part, distorts the shape of the image. At small scales the shear and the convergence are not independent. They satisfy a consistency relation, and they contain therefore the same information on matter density perturbations. More precisely, the shear and the convergence are both related to the sum of the two Bardeen potentials, Φ + Ψ, integrated along the photon trajectory. At large scales however, this consistency relation does not hold anymore. Various relativistic effects contribute to the convergence, see [150Jump To The Next Citation Point]. Some of these effects are generated along the photon trajectory, whereas others are due to the perturbations of the galaxies redshift. These relativistic effects provide independent information on the two Bardeen potentials, breaking their degeneracy. The convergence is therefore a useful quantity that can increase the discriminatory power of weak lensing.

The convergence can be measured through its effect on the galaxy number density, see e.g. [175Jump To The Next Citation Point]. The standard method extracts the magnification from correlations of distant quasars with foreground clusters, see [804Jump To The Next Citation Point, 657Jump To The Next Citation Point]. Recently, [977Jump To The Next Citation Point, 978Jump To The Next Citation Point] designed a new method that permits to accurately measure auto-correlations of the magnification, as a function of the galaxies redshift. This method potentially allows measurements of the relativistic effects in the convergence. Magnification matrix

We are interested in computing the magnification matrix π’Ÿab in a perturbed Friedmann universe. The magnification matrix relates the true shape of a galaxy to its image, and describes therefore the deformations encountered by a light bundle along its trajectory. π’Ÿ ab can be computed by solving Sachs equation, see [775], that governs propagation of light in a generic geometry. The convergence κ and the shear γ ≡ γ1 + iγ2 are then defined respectively as the trace and the symmetric traceless part of π’Ÿab

( ) --χS--- 1 − κ − γ1 − γ2 π’Ÿab = 1 + z − γ2 1 − κ + γ1 . (1.7.1 ) S
Here zS is the redshift of the source and χS is a time coordinate related to conformal time ηS through χS = ηO − ηS.

We consider a spatially flat (K = 0) Friedmann universe with scalar perturbations. We start from the usual longitudinal (or Newtonian) gauge where the metric is given by

μ ν 2 [ 2 i j] gμνdx dx = a − (1 + 2Ψ )dη + (1 − 2Φ)δijdx dx . (1.7.2 )
We compute π’Ÿab at linear order in Φ and Ψ and then we extract the shear and the convergence. We find, see [150, 125]
∫ 1- χS χS-−-χ- 2 γ = 2 dχ χ χS /∂ (Φ + Ψ ), (1.7.3 ) ∫0χS -- ∫ χS κ = 1- dχ χS-−-χ-/∂/∂(Φ + Ψ) + ΦS − dχ(Φ + Ψ) (1.7.4 ) 2 0 χ χS 0 χS ( 1 ) ( ∫ χS ) + ------ − 1 ΨS + n ⋅ vS − dχ(ΦΛ™+ ΨΛ™) , β„‹S χS 0
where n is the direction of observation and vS is the peculiar velocity of the source. Here we are making use of the angular spin raising / ∂ and lowering /- ∂ operators (see e.g., [560] for a review of the properties of these operators) defined as
s − s -- − s s /∂sX ≡ − sin πœƒ(∂πœƒ + icscπœƒ ∂φ)(sin πœƒ)sX, /∂sX ≡ − sin πœƒ(∂πœƒ − icscπœƒ∂φ )(sin πœƒ)sX, (1.7.5 )
where X s is an arbitrary field of spin s and πœƒ and φ are spherical coordinates.

Eq. (1.7.3View Equation) and the first term in Eq. (1.7.4View Equation) are the standard contributions of the shear and the convergence, but expressed here with the full-sky transverse operators

( ) 1 2 1 2 1 2i ( ) χ2/∂ = χ2- ∂ πœƒ − cot πœƒ∂πœƒ −---2-∂ φ + χ2-sinπœƒ ∂πœƒ∂φ − cotπœƒ ∂πœƒ , (1.7.6 ) ( sin πœƒ ) 1-//- 1-- 2 --1--- χ2∂∂ = χ2 ∂ πœƒ + cot πœƒ∂πœƒ + sin2 πœƒ∂φ . (1.7.7 )
In the flat-sky approximation, where πœƒ is very small, 1 -- χ2/∂/∂ reduces to the 2D Laplacian 2 2 ∂x + ∂y and one recovers the standard expression for the convergence. Similarly, the real part of -12/∂2 χ that corresponds to γ1 reduces to 2 2 ∂y − ∂x and the imaginary part that corresponds to γ2 becomes ∂x∂y.

The other terms in Eq. (1.7.4View Equation) are relativistic corrections to the convergence, that are negligible at small scales but may become relevant at large scales. The terms in the first line are intrinsic corrections, generated respectively by the curvature perturbation at the source position and the Shapiro time-delay. The terms in the second line are due to the fact that we measure the convergence at a fixed redshift of the source zS rather that at a fixed conformal time ηS. Since in a perturbed universe, the observable redshift is itself a perturbed quantity, this transformation generates additional contribution to the convergence. Those are respectively the Sachs–Wolfe contribution, the Doppler contribution and the integrated Sachs–Wolfe contribution. Note that we have neglected the contributions at the observer position since they only give rise to a monopole or dipole term. The dominant correction to the convergence is due to the Doppler term. Therefore in the following we are interested in comparing its amplitude with the amplitude of the standard contribution. To that end we define κst and κvel as

∫ χS χS--−-χ -- κst = dχ 2χ χ /∂ /∂(Φ + Ψ ), (1.7.8 ) (0 S) κ = --1---− 1 n ⋅ v . (1.7.9 ) vel β„‹S χS S Observable quantities

The convergence is not directly observable. However it can be measured through the modifications that it induces on the galaxy number density. Let us introduce the magnification

μ = --1---≃ 1 + 2κ, when |κ |,|γ| β‰ͺ 1. (1.7.10 ) detπ’Ÿ
The magnification modifies the size of a source: dΩO = μd ΩS, where dΩS is the true angular size of the source and d ΩO is the solid angle measured by the observer, i.e. the size of the image. The magnification has therefore an impact on the observed galaxy number density. Let us call ¯n(f)df the number of unlensed galaxies per unit solid angle, at a redshift zS, and with a flux in the range [f, f + df ]. The magnification μ modifies the flux measured by the observer, since it modifies the observed galaxy surface. It affects also the solid angle of observation and hence the number of galaxies per unit of solid angle. These two effects combine to give a galaxy number overdensity, see [175, 804Jump To The Next Citation Point]
( ) δμ = n-(f) −-¯n(f) ≃ 1 + 2 α − 1 (κst + κvel). (1.7.11 ) g ¯n(f )
Here ′ α ≡ − N (> fc)fcβˆ•N (fc), where N (> fc) is the number of galaxies brighter than fc and fc is the flux limit adopted. Hence α is an observable quantity, see e.g. [977Jump To The Next Citation Point, 804Jump To The Next Citation Point]. Recent measurements of the galaxy number overdensity δμg are reported in [804, 657]. The challenge in those measurements is to eliminate intrinsic clustering of galaxies, which induces an overdensity δcgl much larger than δμg. One possibility to separate these two effects is to correlate galaxy number overdensities at widely separated redshifts. One can then measure μ cl ⟨δg(zS)δg (zS′)⟩, where zS is the redshift of the sources and zS′ < zS is the redshift of the lenses. Another possibility, proposed by [977, 978Jump To The Next Citation Point], is to use the unique dependence of δμ g on galaxy flux (i.e., on α) to disentangle δμ g from δcl g. This method, combined with precise measurements of the galaxies redshift, allows to measure auto-correlations of δμg, i.e., ⟨δμg(zS)δμg(zS′)⟩, either for zS ⁄= zS′ or for zS = zS′. The velocity contribution, κvel, has only an effect on ⟨δgμ(zS )δμg(zS′)⟩. The correlations between δcgl (zS′) and vS are indeed completely negligible and hence the source peculiar velocity does not affect μ cl ′ ⟨δg(zS)δg (zS )⟩. In the following we study in detail the contribution of peculiar motion to μ μ ⟨δg(zS)δg(zS)⟩.

The two components of the convergence κst and κvel (and consequently the galaxy number overdensity) are functions of redshift zS and direction of observation n. We can therefore determine the angular power spectrum

μ μ ′ ∑ 2β„“-+-1 ′ ⟨δg(zS,n)δg(zS,n )⟩ = 4π C β„“(zS)Pβ„“(n ⋅ n ). (1.7.12 ) β„“
The angular power spectrum C β„“(zS ) contains two contributions, generated respectively by ⟨κstκst⟩ and ⟨κvelκvel⟩. The cross-term ⟨κvelκst⟩ is negligible since κst contains only Fourier modes with a wave vector k⊥ perpendicular to the line of sight (see Eq. (1.7.8View Equation)), whereas κvel selects modes with wave vector along the line of sight (Eq. (1.7.9View Equation)).

So far the derivation has been completely generic. Eqs. (1.7.3View Equation) and (1.7.4View Equation) are valid in any theory of gravity whose metric can be written as in Eq. (1.7.2View Equation). To evaluate the angular power spectrum we now have to be more specific. In the following we assume GR, with no anisotropic stress such that Φ = Ψ. We use the Fourier transform convention

∫ --1--- 3 ikx v (x,χ ) = (2 π)3 d kv (k,χ)e . (1.7.13 )
The continuity equation, see e.g., [317Jump To The Next Citation Point], allows us to express the peculiar velocity as
GΛ™(a)k v (k,χ) = − iG-(a)k2δ (k, a), (1.7.14 )
where δ(k,a) is the density contrast, G (a) is the growth function, and GΛ™(a) its derivative with respect to χ. With this we can express the angular power spectrum as
16πδ2 (α − 1)2GΛ™(a )2( 1 )2 ∫ Cvβ„“el(zS) = ----H-4-S----------S--- ------− 1 dkkT 2(k)j′β„“(k χS)2. (1.7.15 ) H 0G2 (a = 1) β„‹S χS
Here δ H is the density contrast at horizon and T (k) is the transfer function defined through, see e.g., [317]
9 G (a) Ψ(k, a) = 10Ψp (k)T (k)--a--. (1.7.16 )
We assume a flat power spectrum, ns = 1, for the primordial potential Ψp(k ). We want to compare this contribution with the standard contribution
2 2 2 2 2 ∫ [∫ χ ]2 st 36π-δH(αS-−--1)Ω-m-β„“(β„“-+-1)- dk- 2 S χS-−-χ-G-(a) Cβ„“ (zS) = G2 (a = 1) k T (k) 0 dχ χ χS a jβ„“(kχ) . (1.7.17 )
View Image

Figure 9: Left: The velocity contribution Cveβ„“l as a function of β„“ for various redshifts. Right: The standard contribution Cst β„“ as a function of β„“ for various redshifts.

We evaluate vel C β„“ and st Cβ„“ in a ΛCDM universe with Ωm = 0.25, Ω Λ = 0.75 and δH = 5.7 ⋅ 10 −5. We approximate the transfer function with the BBKS formula, see [85]. In Figure 9View Image, we plot Cvβ„“el and Cstβ„“ for various source redshifts. The amplitude of Cvβ„“el and Csβ„“t depends on (α − 1)2, which varies with the redshift of the source, the flux threshold adopted, and the sky coverage of the experiment. Since 2 (α − 1) influences vel C β„“ and st C β„“ in the same way we do not include it in our plot. Generally, at small redshifts, (α − 1 ) is smaller than 1 and consequently the amplitude of both Cvβ„“el and Cstβ„“ is slightly reduced, whereas at large redshifts (α − 1) tends to be larger than 1 and to amplify Cveβ„“l and Cstβ„“, see e.g., [978]. However, the general features of the curves and more importantly the ratio between Cvel β„“ and Cst β„“ are not affected by (α − 1).

Figure 9View Image shows that vel C β„“ peaks at rather small β„“, between 30 and 120 depending on the redshift. This corresponds to rather large angle πœƒ ∼ 90 –360 arcmin. This behavior differs from the standard term (Figure 9View Image) that peaks at large β„“. Therefore, it is important to have large sky surveys to detect the velocity contribution. The relative importance of Cvel β„“ and Cst β„“ depends strongly on the redshift of the source. At small redshift, zS = 0.2, the velocity contribution is about − 5 4 ⋅ 10 and is hence larger than the standard contribution which reaches −6 10. At redshift zS = 0.5, vel C β„“ is about 20% of st C β„“, whereas at redshift zS = 1, it is about 1% of Csβ„“t. Then at redshift zS = 1.5 and above, Cvβ„“el becomes very small with respect to Cst β„“: Cvel ≤ 10− 4Cst β„“ β„“. The enhancement of Cvel β„“ at small redshift together with its fast decrease at large redshift are due to the prefactor ( )2 β„‹S1χS-− 1 in Eq. (1.7.15View Equation). Thanks to this enhancement we see that if the magnification can be measured with an accuracy of 10%, then the velocity contribution is observable up to redshifts z ≤ 0.6. If the accuracy reaches 1% then the velocity contribution becomes interesting up to redshifts of order 1.

The shear and the standard contribution in the convergence are not independent. One can easily show that their angular power spectra satisfy the consistency relation, see [449]

β„“(β„“ + 1) C κβ„“st = --------------Cγβ„“ . (1.7.18 ) (β„“ + 2)(β„“ − 1 )
This relation is clearly modified by the velocity contribution. Using that the cross-correlation between the standard term and the velocity term is negligible, we can write a new consistency relation that relates the observed convergence C κtot β„“ to the shear
β„“(β„“ + 1) γ -------------C β„“ = C κβ„“tot − Cβ„“κvel. (1.7.19 ) (β„“ + 2)(β„“ − 1)
Consequently, if one measures both the shear C γβ„“ and the magnification C κβ„“tot as functions of the redshift, Eq. (1.7.19View Equation) allows to extract the peculiar velocity contribution C κvel β„“. This provides a new way to measure peculiar velocities of galaxies.

Note that in practice, in weak lensing tomography, the angular power spectrum is computed in redshift bins and therefore the square bracket in Eq. (1.7.17View Equation) has to be integrated over the bin

∫ ∞ ∫ χ χ − χ ′G(χ ′) dχni(χ ) dχ′----′-----′-jβ„“(kχ′), (1.7.20 ) 0 0 χ χ a(χ )
where ni is the galaxy density for the i-th bin, convolved with a Gaussian around the mean redshift of the bin. The integral over χ ′ is then simplified using Limber approximation, i.e.,
∫ ( ) χ ′ ′ ′ 1 β„“ dχ F (χ )J β„“(k χ ) ≃ kF k- πœƒ(kχ − β„“), (1.7.21 ) 0
where Jβ„“ is the Bessel function of order β„“. The accuracy of Limber approximation increases with β„“. Performing a change of coordinate such that k = β„“βˆ• χ, Eq. (1.7.17View Equation) can be recast in the usual form used in weak lensing tomography, see e.g., Eq. (1.8.4).

1.7.3 Observing modified gravity with redshift surveys

Wide-deep galaxy redshift surveys have the power to yield information on both H (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 distance DA (z) (and possibly distance ratios from weak lensing). Most importantly, Euclid survey enables us to measure the cosmic growth history using two independent methods: f (z) g 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.

View Image

Figure 10: Matter power spectrum form measured from SDSS [720]

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:

∫ z -dz′-- χ (z) ≡ E(z′). (1.7.22 ) 0
The dimensionless version of the comoving distance (defined in the previous section by the same symbol χ) is:
[∫ z ] 2 (0) 3 (0) 3(1 +-w(&tidle;z))- E (z) = Ωm (1 + z ) + (1 − Ωm )exp 0 1 + &tidle;z dz&tidle; . (1.7.23 )
The standard cosmological distances are related to χ(z) via
( ) D (z) = -------c-√------sin ∘ −-Ω-χ (z) (1.7.24 ) A H0 (1 + z ) − Ωk k
where the luminosity distance, DL (z), is given by the distance duality:
DL (z) = (1 + z)2DA (z). (1.7.25 )
The coupling between D (z ) A and D (z) L 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 both DA (z) and H (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) = cΔz βˆ•s||(z), while the tangential mode measures the angular diameter distance D (z) = s βˆ•Δ πœƒ(1 + z) A ⊥.

One can then use the power spectrum to derive predictions on the parameter constraining power of the survey (see e.g., [46, 418, 938, 945Jump To The Next Citation Point, 308Jump To The Next Citation Point]).

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 spectrum Pg(k) at a given wave-number bin is [359]

[ΔPg ]2 2 (2π )2 [ 1 ]2 ----- = --------2------ 1 + ----- , (1.7.26 ) Pg Vsurveyk Δk Δμ ngPg
where ng is the mean number density of galaxies, Vsurvey is the comoving survey volume of the galaxy survey, and μ is the cosine of the angle between k and the line-of-sight direction βƒ— μ = k ⋅rˆβˆ•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., [815Jump To The Next Citation Point])

DA (z)2refH (z) Pobs(kref⊥,krefβˆ₯,z) = ------2-------Pg (kref⊥,krefβˆ₯,z) + Pshot, (1.7.27 ) DA (z) H (z)ref
[ ] k2 2 Pg(kref⊥, krefβˆ₯,z) = b(z)2 1 + β(z) ----refβˆ₯----- × Pmatter(k,z ). (1.7.28 ) k2ref⊥ + k2refβˆ₯
In Eq. (1.7.27View Equation), H (z) and DA (z) are the Hubble parameter and the angular diameter distance, respectively, and the prefactor (DA (z )2refH (z ))βˆ•(DA (z )2H (z)ref) encapsulates the geometrical distortions due to the Alcock–Paczynski effect [815Jump To The Next Citation Point, 81]. Their values in the reference cosmology are distinguished by the subscript ‘ref’, while those in the true cosmology have no subscript. k⊥ and k βˆ₯ 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 by kref⊥ = k⊥DA (z )βˆ•DA (z)ref and k = k H (z) βˆ•H (z) refβˆ₯ βˆ₯ ref. P shot is the unknown white shot noise that remains even after the conventional shot noise of inverse number density has been subtracted [815Jump To The Next Citation Point]. In Eq. (1.7.28View Equation), b(z ) is the linear bias factor between galaxy and matter density distributions, fg(z) is the linear growth rate,4 and β(z) = fg(z)βˆ•b(z) is the linear redshift-space distortion parameter [485Jump To The Next Citation Point]. The linear matter power spectrum Pmatter(k,z ) in Eq. (1.7.27View Equation) takes the form
2 4 2 [ ]2 ( )ns Pmatter(k,z) = 8π--ck0-Δβ„›-(k0)T 2(k ) --G-(z)-- k-- e−k2μ2σ2r, (1.7.29 ) 25H40Ω2m G (z = 0) k0
where G(z) is the usual scale independent linear growth-factor in the absence of massive neutrino free-streaming (see Eq. (25) in [337Jump To The Next Citation Point]), whose fiducial value in each redshift bin is computed through numerical integration of the differential equations governing the growth of linear perturbations in presence of dark energy [588] or employing the approximation of Eq. (1.3.22View Equation). T(k) depends on matter and baryon densities5 (neglecting dark energy at early times), and is computed in each redshift bin using a Boltzmann code like camb6 [559Jump To The Next Citation Point] or cmbfast.

In Eq. (1.7.29View Equation) a damping factor e−k2μ2σ2r has been added, due to redshift uncertainties, where σr = (∂r βˆ•∂z)σz, r(z) being the comoving distance [940, 815Jump To The Next Citation Point], and assumed that the power spectrum of primordial curvature perturbations, Pβ„› (k), is

k3P (k) ( k )ns Δ2β„› (k ) ≡ ---β„›-2---= Δ2β„›(k0) --- , (1.7.30 ) 2π k0
where k = 0.002βˆ•Mpc 0, Δ2 (k )| = 2.45 × 10 −9 β„› 0 fid is the dimensionless amplitude of the primordial curvature perturbations evaluated at a pivot scale k0, and ns is the scalar spectral index [548].

In the limit where the survey volume is much larger than the scale of any features in Pobs(k), it has been shown that the redshift survey Fisher matrix for a given redshift bin can be approximated as [880Jump To The Next Citation Point]

∫ 1 ∫ kmax 2 F LSS = ∂-lnPobs(k,μ-)∂-ln-Pobs(k,μ)Veff(k, μ)2πk--dkdμ-, (1.7.31 ) ij − 1 kmin ∂pi ∂pj 2(2π)3
where the derivatives are evaluated at the parameter values pi of the fiducial model, and Veff is the effective volume of the survey:
[ ]2 --ngPg(k,μ-)-- Veff(k,μ) = n P (k,μ ) + 1 Vsurvey, (1.7.32 ) g g
where the comoving number density ng (z) is assumed to be spatially constant. Due to azimuthal symmetry around the line of sight, the three-dimensional galaxy redshift power spectrum Pobs(βƒ—k) depends only on k and μ, i.e., is reduced to two dimensions by symmetry [815Jump To The Next Citation Point]. 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 that kmax is given by requiring that the variance of matter fluctuations in a sphere of radius R is, for instance, σ2(R ) = 0.25 for R = πβˆ•(2kmax). Or one could model the nonlinear distortions as in [338Jump To The Next Citation Point]. On scales larger than (−1 ∼ 100h 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 [338Jump To The Next Citation Point, 816], to model nonlinear effect we multiply P (k) by the factor

{ [ 2 2 2 2]} exp − k2 (1 −-μ-)Σ⊥- + μ-Σ-βˆ₯ , (1.7.33 ) 2 2
where Σ⊥ and Σ βˆ₯ 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 Σ = Σ G (1 + f ) βˆ₯ 0 g. The value of Σ 0 is proportional to σ 8. For a reference cosmology where σ = 0.8 8 [526Jump To The Next Citation Point], we have −1 Σ0 = 11h Mpc.

Finally, we note that when actual data are available, the usual way to measure β = fgβˆ•b is by fitting the measured galaxy redshift-space correlation function ξ (σ, π) to a model [717]:

∫ ∞ &tidle; ξ(σ,π ) = dvf (v)ξ(σ,π − vβˆ•H0 ), (1.7.34 ) −∞
where f(v) describes the small-scale random motion (usually modeled by a Gaussian that depends on the galaxy pairwise peculiar velocity dispersion), and &tidle;ξ(σ,π ) is the model accounting for coherent infall velocities:7
&tidle; ξ(σ,π ) = ξ0(s)P0 (μ ) + ξ2(s)P2(μ ) + ξ4(s)P4(μ). (1.7.35 )
Pl(μ ) are Legendre polynomials; μ = cosπœƒ, where πœƒ denotes the angle between r and π; ξ0(s), ξ (s) 2, and ξ (s) 4 depend on β and the real-space correlation function ξ(r).

The bias between galaxy and matter distributions can be estimated from either galaxy clustering, or weak lensing. To determine bias, we can assume that the galaxy density perturbation δg is related to the matter density perturbation δm (x) as [371]:

δg = bδm(x) + b2δ2(x )βˆ•2. (1.7.36 ) m

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

{ [ ] ⟨δgk1δgk2δgk1⟩ = (2π)3 P(k1)P (k2) J (k1,k2)βˆ•b + b2βˆ•b2 D +cyc. }δ (k1 + k2 + k3), (1.7.37 )
where J is a function that depends on the shape of the triangle formed by (k1, k2, k3) in k space, but only depends very weakly on cosmology [648, 925Jump To The Next Citation Point].

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 factor G (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.

1.7.4 Cosmological bulk flows

As we have seen, the additional redshift induced by the galaxy peculiar velocity field generates the redshift distortion in the power spectrum. In this section we discuss a related effect on the luminosity of the galaxies and on its use to measure the peculiar velocity in large volumes, the so-called bulk flow.

In the gravitational instability framework, inhomogeneities in the matter distribution induce gravitational accelerations g, which result in galaxies having peculiar velocities v that add to the Hubble flow. In linear theory the peculiar velocity field is proportional to the peculiar acceleration

2fg H0fg ∫ (r′ − r) v(r) = -------g (r) = ----- δm(r′)--′----3d3r′, (1.7.38 ) 3H0 Ωm 4π |r − r|
and the bulk flow of a spherical region is solely determined by the gravitational pull of the dipole of the external mass distribution. For this reason, bulk flows are reliable indicators to deviations from homogeneity and isotropy on large scale, should they exist.

Constraints on the power spectrum and growth rate can be obtained by comparing the bulk flow estimated from the volume-averaged motion of the sphere of radius R:

∫ 3 BR ≡ --v∫(x)W-(x-βˆ•R)d-x-, (1.7.39 ) W (xβˆ•R )d3x
with expected variance:
H2 f2 ∫ σ2B,R = --0-g- P (k)𝒲 (kR )2(k)dk, (1.7.40 ) 6π2
where the window function W (xβˆ•R ) and its Fourier transform 𝒲 (kR ) describe the spatial distribution of the dataset.

Over the years the bulk flows has been estimated from the measured peculiar velocities of a large variety of objects ranging from galaxies [397, 398, 301, 256, 271, 788] clusters of galaxies [549, 165, 461] and SN Ia [766]. Conflicting results triggered by the use of error-prone distance indicators have fueled a long lasting controversy on the amplitude and convergence of the bulk flow that is still on. For example, the recent claim of a bulk flow of −1 407 ± 81 km s within R = 50 −1 h Mpc [947Jump To The Next Citation Point], inconsistent with expectation from the ΛCDM model, has been seriously challenged by the re-analysis of the same data by [694] who found a bulk flow amplitude consistent with ΛCDM expectations and from which they were able to set the strongest constraints on modified gravity models so far. On larger scales, [493] claimed the detection of a dipole anisotropy attributed to the kinetic SZ decrement in the WMAP temperature map at the position of X-ray galaxy clusters. When interpreted as a coherent motion, this signal would indicate a gigantic bulk flow of − 1 1028 ± 265 km s within R = 528 −1 h Mpc. This highly debated result has been seriously questioned by independent analyses of WMAP data [see, e.g., 699])

The large, homogeneous dataset expected from Euclid has the potential to settle these issues. The idea is to measure bulk flows in large redshift surveys, based on the apparent, dimming or brightening of galaxies due to their peculiar motion. The method, originally proposed by [875], has been recently extended by [693] who propose to estimate the bulk flow by minimizing systematic variations in galaxy luminosities with respect to a reference luminosity function measured from the whole survey. It turns out that, if applied to the photo-z catalog expected from Euclid, this method would be able to detect at 5σ significance a bulk flow like the one of [947Jump To The Next Citation Point] over ∼ 50 independent spherical volumes at z ≥ 0.2, provided that the systematic magnitude offset over the corresponding areas in the sky does not exceed the expected random magnitude errors of 0.02 – 0.04 mag. Additionally, photo-z or spectral-z could be used to validate or disproof with very large (> 7σ) significance the claimed bulk flow detection of [493] at z = 0.5.

Closely related to the bulk flow is the Local Group peculiar velocity inferred from the observed CMB dipole [483]

H f vCMB = vLG,R − --0-gxc.m.+ BR, (1.7.41 ) 3
where vLG,R is the Local Group velocity resulting from the gravitational pull of all objects in the sample within the radius R, x c.m. is the position of the center of mass of the sample and v CMB is the LG velocity inferred from the CMB dipole [121]. The convergence of vLG,R with the radius and its alignment with the CMB dipole direction indicates a crossover to homogeneity [793] and allows to constrain the growth rate by comparing vCMB with vLG,R. The latter can be estimated from the dipole in the distribution of objects either using a number-weighting scheme if redshifts are available for all objects of the sample, or using a flux-weighting scheme. In this second case the fact that both gravitational acceleration and flux are inversely proportional to the distance allows to compute the dipole from photometric catalogs with no need to measure redshifts. The drawback is that the information on the convergence scale is lost.

As for the bulk flow case, despite the many measurements of cosmological dipoles using galaxies [972, 283, 654, 868, 801, 513] there is still no general consensus on the scale of convergence and even on the convergence itself. Even the recent analyses of measuring the acceleration of the Local Group from the 2MASS redshift catalogs provided conflicting results. [344] found that the galaxy dipole seems to converge beyond R = 60h− 1 Mpc, whereas [552] find no convergence within R = 120h −1 Mpc.

Once again, Euclid will be in the position to solve this controversy by measuring the galaxy and cluster dipoles not only at the LG position and out to very large radii, but also in several independent ad truly all-sky spherical samples carved out from the the observed areas with |b| > 20∘. In particular, coupling photometry with photo-z one expects to be able to estimate the convergence scale of the flux-weighted dipole over about 100 independent spheres of radius 200h− 1 Mpc out to z = 0.5 and, beyond that, to compare number-weighted and flux-weighted dipoles over a larger number of similar volumes using spectroscopic redshifts.

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