In the next subsections we explore how dark energy or modified gravity effects can be detected through weak lensing and redshift surveys.
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, (1.4.74). 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., 11
, 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 CDM 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.
The magnification matrix is a 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,
, integrated along the photon trajectory. At large scales
however, this consistency relation does not hold anymore. Various relativistic effects contribute to
the convergence, see [150
]. 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. [175]. The
standard method extracts the magnification from correlations of distant quasars with foreground clusters,
see [804
, 657
]. Recently, [977
, 978
] 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.
We are interested in computing the magnification matrix 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.
can be computed by solving Sachs
equation, see [775], that governs propagation of light in a generic geometry. The convergence
and the
shear
are then defined respectively as the trace and the symmetric traceless part of
We consider a spatially flat () Friedmann universe with scalar perturbations. We start from the
usual longitudinal (or Newtonian) gauge where the metric is given by
Eq. (1.7.3) and the first term in Eq. (1.7.4
) are the standard contributions of the shear and the
convergence, but expressed here with the full-sky transverse operators
The other terms in Eq. (1.7.4) 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
rather that at a fixed conformal time
. 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
and
as
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
The magnification modifies the size of a source: The two components of the convergence and
(and consequently the galaxy number
overdensity) are functions of redshift
and direction of observation
. We can therefore determine the
angular power spectrum
So far the derivation has been completely generic. Eqs. (1.7.3) and (1.7.4
) are valid in any theory of
gravity whose metric can be written as in Eq. (1.7.2
). 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
. We use
the Fourier transform convention
We evaluate and
in a
CDM universe with
,
and
. We approximate the transfer function with the BBKS formula, see [85]. In Figure 9
, we
plot
and
for various source redshifts. The amplitude of
and
depends on
, which varies with the redshift of the source, the flux threshold adopted, and the sky coverage of
the experiment. Since
influences
and
in the same way we do not include it in our
plot. Generally, at small redshifts,
is smaller than 1 and consequently the amplitude of
both
and
is slightly reduced, whereas at large redshifts
tends to be
larger than 1 and to amplify
and
, see e.g., [978]. However, the general features
of the curves and more importantly the ratio between
and
are not affected by
.
Figure 9 shows that
peaks at rather small
, between 30 and 120 depending on the redshift.
This corresponds to rather large angle
. This behavior differs from the standard term
(Figure 9
) that peaks at large
. Therefore, it is important to have large sky surveys to detect the
velocity contribution. The relative importance of
and
depends strongly on the redshift of the
source. At small redshift,
, the velocity contribution is about
and is hence larger than
the standard contribution which reaches
. At redshift
,
is about 20% of
,
whereas at redshift
, it is about
of
. Then at redshift
and above,
becomes very small with respect to
:
. The enhancement of
at small
redshift together with its fast decrease at large redshift are due to the prefactor
in
Eq. (1.7.15
). 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
. 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]
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 Note that in practice, in weak lensing tomography, the angular power spectrum is computed in
redshift bins and therefore the square bracket in Eq. (1.7.17) has to be integrated over the bin
Wide-deep galaxy redshift surveys have the power to yield information on both and
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
and
to test
gravity. As a wide-deep survey, Euclid allows us to measure
directly from BAO, but
also indirectly through the angular diameter distance
(and possibly distance ratios
from weak lensing). Most importantly, Euclid survey enables us to measure the cosmic growth
history using two independent methods:
from galaxy clustering, and
from weak
lensing. In the following we discuss the estimation of
and
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:
The dimensionless version of the comoving distance (defined in the previous section by the same symbol 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
() measures
through
, while the tangential mode measures the angular
diameter distance
.
One can then use the power spectrum to derive predictions on the parameter constraining power of the
survey (see e.g., [46, 418, 938, 945, 308
]).
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 at a given wave-number bin is [359]
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., [815])
In Eq. (1.7.29) a damping factor
has been added, due to redshift uncertainties, where
,
being the comoving distance [940, 815
], and assumed that the power spectrum of
primordial curvature perturbations,
, is
In the limit where the survey volume is much larger than the scale of any features in , it has
been shown that the redshift survey Fisher matrix for a given redshift bin can be approximated as [880
]
To minimize nonlinear effects, one should restrict wave-numbers to the quasi-linear regime, e.g.,
imposing that is given by requiring that the variance of matter fluctuations in a sphere of radius
is, for instance,
for
. Or one could model the nonlinear distortions as in
[338
]. On scales larger than (
) 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 [338
, 816], to model nonlinear effect we multiply
by the factor
Finally, we note that when actual data are available, the usual way to measure is
by fitting the measured galaxy redshift-space correlation function
to a model [717]:
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 is related to the
matter density perturbation
as [371]:
Bias can be derived from galaxy clustering by measuring the galaxy bispectrum:
where 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
can be derived from weak lensing by comparing the cross-correlations of multiple redshift
slices.
Of course, if bias is assumed to be linear () 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.
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 , which result in galaxies having peculiar velocities
that add to the
Hubble flow. In linear theory the peculiar velocity field is proportional to the peculiar acceleration
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 :
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 within
[947
], inconsistent with expectation from
the
CDM model, has been seriously challenged by the re-analysis of the same data by [694] who found a
bulk flow amplitude consistent with
CDM 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
within
. 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- catalog expected from Euclid,
this method would be able to detect at
significance a bulk flow like the one of [947
] over
independent spherical volumes at
, 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-
or spectral-
could be used to validate
or disproof with very large (
) significance the claimed bulk flow detection of [493] at
.
Closely related to the bulk flow is the Local Group peculiar velocity inferred from the observed CMB dipole [483]
where 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 , whereas [552] find no convergence within
.
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 . In particular, coupling
photometry with photo-
one expects to be able to estimate the convergence scale of the flux-weighted
dipole over about 100 independent spheres of radius
out to
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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Living Rev. Relativity 16, (2013), 6
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