## 1.3 Perturbations

This section is devoted to a discussion of linear perturbation theory in dark-energy models. Since we will discuss a number of non-standard models in later sections, we present here the main equations in a general form that can be adapted to various contexts. This section will identify which perturbation functions the Euclid survey [551] will try to measure and how they can help us to characterize the nature of dark energy and the properties of gravity.

### 1.3.1 Cosmological perturbation theory

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
where is the scalar potential; a vector shift; is the scalar perturbation to the spatial curvature; is the trace-free distortion to the spatial metric; is the conformal time.

We will assume that the universe is filled with perfect fluids only, so that the energy momentum tensor takes the simple form

where and are the density and the pressure of the fluid respectively, is the four-velocity and is the anisotropic-stress perturbation tensor that represents the traceless component of the .

The components of the perturbed energy momentum tensor can be written as:

Here and are the energy density and pressure of the homogeneous and isotropic background universe, is the density perturbation, is the pressure perturbation, is the velocity vector. Here we want to investigate only the scalar modes of the perturbation equations. So far the treatment of the matter and metric is fully general and applies to any form of matter and metric. We now choose the Newtonian gauge (also known as the longitudinal gauge), characterized by zero non-diagonal metric terms (the shift vector and ) and by two scalar potentials and ; the metric Eq. (1.3.2) then becomes
The advantage of using the Newtonian gauge is that the metric tensor is diagonal and this simplifies the calculations. This choice not only simplifies the calculations but is also the most intuitive one as the observers are attached to the points in the unperturbed frame; as a consequence, they will detect a velocity field of particles falling into the clumps of matter and will measure their gravitational potential, represented directly by ; corresponds to the perturbation to the spatial curvature. Moreover, as we will see later, the Newtonian gauge is the best choice for observational tests (i.e., for perturbations smaller than the horizon).

In the conformal Newtonian gauge, and in Fourier space, the first-order perturbed Einstein equations give [see 599, for more details]:

where a dot denotes , , the index indicates a sum over all matter components in the universe and is related to through:
The energy-momentum tensor components in the Newtonian gauge become:
where we have defined the variable that represents the divergence of the velocity field.

Perturbation equations for a single fluid are obtained taking the covariant derivative of the perturbed energy momentum tensor, i.e., . We have

The equations above are valid for any fluid. The evolution of the perturbations depends on the characteristics of the fluids considered, i.e., we need to specify the equation of state parameter , the pressure perturbation and the anisotropic stress . For instance, if we want to study how matter perturbations evolve, we simply substitute (matter is pressureless) in the above equations. However, Eqs. (1.3.17) – (1.3.18) depend on the gravitational potentials and , which in turn depend on the evolution of the perturbations of the other fluids. For instance, if we assume that the universe is filled by dark matter and dark energy then we need to specify and for the dark energy.

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.

### 1.3.2 Modified growth parameters

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:

• Model parameters capture the degrees of freedom of DE/MG and modify the evolution equations of the energy-momentum content of the fiducial model. They can be associated with physical meanings and have uniquely-predicted behavior in specific theories of DE and MG.
• Trigger relations are derived directly from observations and only hold in the fiducial model. They are constructed to break down if the fiducial model does not describe the growth of structure correctly.

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

These can be used to reduce the energy-momentum conservation of matter simply to the second-order growth equation
Primes denote derivatives with respect to and we define the time-dependent fractional matter density as . Notice that the evolution of is driven by and is scale-independent throughout (valid on sub- and super-Hubble scales after radiation-matter equality). We define the growth factor as . This is very well approximated by the expression
and
defines the growth rate and the growth index that is found to be for the CDM solution [see 937, 585, 466, 363].

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.

#### 1.3.2.1 Two new degrees of freedom

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

Non-trivial behavior of the two functions and can be due to a clustering dark-energy component or some modification to GR. In MG models the function represents a mass screening effect due to local modifications of gravity and effectively modifies Newton’s constant. In dynamical DE models represents the additional clustering due to the perturbations in the DE. On the other hand, the function parameterizes the effective anisotropic stress introduced by MG or DE, which is absent in CDM.

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])

where we define . Remember as defined above. Notice that it is that modifies the source term of the equation and therefore also the growth of . Together with the modified Einstein constraints (1.3.23) these evolution equations form a closed system for which can be solved for given .

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

and . The growth equation is only modified by the factor on the RHS with respect to CDM (1.3.20). On “super-Hubble” scales, , we have
and now create an additional drag term in the equation, except if when the drag term could flip sign. [737] also showed that the metric potentials evolve independently and scale-invariantly on super-Hubble scales as long as for . This is needed for the comoving curvature perturbation, , to be constant on super-Hubble scales.

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

A summary of different other variables used was given by [278]. For instance, the gravitational slip parameter introduced by [194] and widely used is related through . Recently [277] used , while [115] defined . All these variables reflect the same two degrees of freedom additional to the linear growth of structure in CDM.

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].

#### 1.3.2.2 Parameterizations and non-parametric approaches

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:

with constant , , , , and . Generally, the problem with any kind of parameterization is that it is difficult – if not impossible – for it to be flexible enough to describe all possible modifications.

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.

#### 1.3.2.3 Trigger relations

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

is accurate to the level compared with the actual solution of the growth equation (1.3.20). Generally, for a given solution of the growth equation the growth index can simply be computed using
The other way round, the modified gravity function can be computed for a given [737]

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:

• As only one additional parameter is introduced, a second parameter, such as , is needed to close the system and be general enough to capture all possible modifications.
• The growth factor is a solution of the growth equation on sub-Hubble scales and, therefore, is not general enough to be consistent on all scales.
• The framework is designed to describe the evolution of the matter density contrast and is not easily extended to describe all other energy-momentum components and integrated into a CMB-Boltzmann code.