Here we provide the perturbation equations in a dark-energy dominated universe for a general fluid, focusing on scalar perturbations.
For simplicity, we consider a flat universe containing only (cold dark) matter and dark energy, so that the Hubble parameter is given by
We will consider linear perturbations on a spatially-flat background model, defined by the line of element whereWe will assume that the universe is filled with perfect fluids only, so that the energy momentum tensor takes the simple form
whereThe components of the perturbed energy momentum tensor can be written as:
Here In the conformal Newtonian gauge, and in Fourier space, the first-order perturbed Einstein equations
give [see 599, for more details]:
Perturbation equations for a single fluid are obtained taking the covariant derivative of the perturbed
energy momentum tensor, i.e., . We have
The problem here is not only to parameterize the pressure perturbation and the anisotropic stress for
the dark energy (there is not a unique way to do it, see below, especially Section 1.4.5 for what to do when
crosses
) but rather that we need to run the perturbation equations for each model we assume,
making predictions and compare the results with observations. Clearly, this approach takes too much
time. In the following Section 1.3.2 we show a general approach to understanding the observed
late-time accelerated expansion of the universe through the evolution of the matter density
contrast.
In the following, whenever there is no risk of confusion, we remove the overbars from the background quantities.
Even if the expansion history, , of the FLRW background has been measured (at least up to redshifts
by supernova data), it is not yet possible yet to identify the physics causing the recent
acceleration of the expansion of the universe. Information on the growth of structure at different
scales and different redshifts is needed to discriminate between models of dark energy (DE) and
modified gravity (MG). A definition of what we mean by DE and MG will be postponed to
Section 1.4.
An alternative to testing predictions of specific theories is to parameterize the possible departures from a fiducial model. Two conceptually-different approaches are widely discussed in the literature:
As the current observations favor concordance cosmology, the fiducial model is typically taken to be spatially
flat FLRW in GR with cold dark matter and a cosmological constant, hereafter referred to as
CDM.
For a large-scale structure and weak lensing survey the crucial quantities are the matter-density contrast
and the gravitational potentials and we therefore focus on scalar perturbations in the Newtonian gauge with
the metric (1.3.8).
We describe the matter perturbations using the gauge-invariant comoving density contrast
where
and
are the matter density contrast and the divergence of the
fluid velocity for matter, respectively. The discussion can be generalized to include multiple
fluids.
In CDM, after radiation-matter equality there is no anisotropic stress present and the Einstein
constraint equations at “sub-Hubble scales”
become
Clearly, if the actual theory of structure growth is not the CDM scenario, the constraints (1.3.19
)
will be modified, the growth equation (1.3.20
) will be different, and finally the growth factor (1.3.21
) is
changed, i.e., the growth index is different from
and may become time and scale dependent.
Therefore, the inconsistency of these three points of view can be used to test the
CDM
paradigm.
Any generic modification of the dynamics of scalar perturbations with respect to the simple scenario of a
smooth dark-energy component that only alters the background evolution of CDM can be represented
by introducing two new degrees of freedom in the Einstein constraint equations. We do this by replacing
(1.3.19
) with
Given an MG or DE theory, the scale- and time-dependence of the functions and
can be derived
and predictions projected into the
plane. This is also true for interacting dark sector models,
although in this case the identification of the total matter density contrast (DM plus baryonic matter) and
the galaxy bias become somewhat contrived [see, e.g., 848
, for an overview of predictions for different
MG/DE models].
Using the above-defined modified constraint equations (1.3.23), the conservation equations of matter
perturbations can be expressed in the following form (see [737
])
The influence of the Hubble scale is modified by , such that now the size of
determines the
behavior of
; on “sub-Hubble” scales,
, we find
Many different names and combinations of the above defined functions have been used in the
literature, some of which are more closely related to actual observables and are less correlated than others in
certain situations [see, e.g., 41
, 667, 848
, 737
, 278
, 277
, 363
].
For instance, as observed above, the combination modifies the source term in the growth
equation. Moreover, peculiar velocities are following gradients of the Newtonian potential,
, and
therefore the comparison of peculiar velocities with the density field is also sensitive to
. So we define
Weak lensing and the integrated Sachs–Wolfe (ISW) effect, on the other hand, are measuring
, which is related to the density field via
Any combination of two variables out of is a valid alternative to
. It turns out
that the pair
is particularly well suited when CMB, WL and LSS data are combined as it is less
correlated than others [see 980
, 277
, 68].
So far we have defined two free functions that can encode any departure of the growth of linear
perturbations from CDM. However, these free functions are not measurable, but have to be inferred via
their impact on the observables. Therefore, one needs to specify a parameterization of, e.g.,
such
that departures from
CDM can be quantified. Alternatively, one can use non-parametric
approaches to infer the time and scale-dependence of the modified growth functions from the
observations.
Ideally, such a parameterization should be able to capture all relevant physics with the least number of
parameters. Useful parameterizations can be motivated by predictions for specific theories of MG/DE [see
848] and/or by pure simplicity and measurability [see 41
]. For instance, [980
] and [278
] use
scale-independent parameterizations that model one or two smooth transitions of the modified growth
parameters as a function of redshift. [115
] also adds a scale dependence to the parameterization, while
keeping the time-dependence a simple power law:
Daniel et al. [278, 277] investigate the modified growth parameters binned in
and
. The
functions are taken constant in each bin. This approach is simple and only mildly dependent on the size
and number of the bins. However, the bins can be correlated and therefore the data might
not be used in the most efficient way with fixed bins. Slightly more sophisticated than simple
binning is a principal component analysis (PCA) of the binned (or pixelized) modified growth
functions. In PCA uncorrelated linear combinations of the original pixels are constructed. In the
limit of a large number of pixels the model dependence disappears. At the moment however,
computational cost limits the number of pixels to only a few. Zhao et al. [982, 980
] employ a
PCA in the
plane and find that the observables are more strongly sensitive to the
scale-variation of the modified growth parameters rather than the time-dependence and their
average values. This suggests that simple, monotonically or mildly-varying parameterizations as
well as only time-dependent parameterizations are poorly suited to detect departures from
CDM.
A useful and widely popular trigger relation is the value of the growth index in
CDM. It turns out
that the value of
can also be fitted also for simple DE models and sub-Hubble evolution in some MG
models [see, e.g., 585
, 466, 587
, 586, 692
, 363]. For example, for a non-clustering perfect fluid DE model
with equation of state
the growth factor
given in (1.3.21
) with the fitting formula
The fact that the value of is quite stable in most DE models but strongly differs in MG scenarios
means that a large deviation from
signifies the breakdown of GR, a substantial DE clustering or a
breakdown of another fundamental hypothesis like near-homogeneity. Furthermore, using the growth factor
to describe the evolution of linear structure is a very simple and computationally cheap way to carry out
forecasts and compare theory with data. However, several drawbacks of this approach can be identified:
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