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 assume that the universe is filled with perfect fluids only, so that the energy momentum tensor takes the simple form
The components of the perturbed energy momentum tensor can be written as:
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 )
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,  and  use scale-independent parameterizations that model one or two smooth transitions of the modified growth parameters as a function of redshift.  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:
Living Rev. Relativity 16, (2013), 6
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