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 [551Jump To The Next Citation Point] 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

( )2 [ ( ∫ a ′ ) ] 2 1da- 2 −3 1 +-w(a-) H = adt = H0 Ωm0a + (1 − Ωm0 )exp − 3 1 a ′ da . (1.3.1 )
We will consider linear perturbations on a spatially-flat background model, defined by the line of element
2 2 [ 2 i j] ds = a − (1 + 2A )d η + 2Bidηdx + ((1 + 2HL )δij + 2HT ij)dxidx , (1.3.2 )
where A is the scalar potential; Bi a vector shift; HL is the scalar perturbation to the spatial curvature; ij H T is the trace-free distortion to the spatial metric; dη = dt∕a 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

μν μ ν μν μν T = (ρ + p)u u + p g + Π , (1.3.3 )
where ρ and p are the density and the pressure of the fluid respectively, u μ is the four-velocity and Π μν is the anisotropic-stress perturbation tensor that represents the traceless component of the T i j.

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

T 0= − (¯ρ + δ ρ) (1.3.4 ) 00 Tj = (ρ¯+ ¯p) (vj − Bj ) (1.3.5 ) T i= (ρ¯+ ¯p) vi (1.3.6 ) 0i i i T j = (p¯+ δp) δj + ¯p Π j. (1.3.7 )
Here ¯ρ and ¯p are the energy density and pressure of the homogeneous and isotropic background universe, δρ is the density perturbation, δp is the pressure perturbation, vi 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 Bi = 0 and HijT = 0) and by two scalar potentials Ψ and Φ; the metric Eq. (1.3.2View Equation) then becomes
ds2 = a2 [− (1 + 2Ψ )dη2 + (1 − 2Φ) dx dxi]. (1.3.8 ) i
The advantage of using the Newtonian gauge is that the metric tensor gμν 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 Ψ; Φ 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 599Jump To The Next Citation Point, for more details]:

( ) 2 a˙ a˙ 2 ∑ k Φ + 3a- ˙Φ + a-Ψ = − 4πGa ρ¯αδα, (1.3.9 ) ( ) α 2 a˙ 2∑ k ˙Φ + --Ψ = 4πGa (¯ρα + ¯pα)𝜃α, (1.3.10 ) ( ) a α a˙ a¨ ˙a2 k2 2∑ ¨Φ + --( ˙Ψ + 2Φ˙) + 2--− -2- Ψ + --(Φ − Ψ) = 4πGa δpα, (1.3.11 ) a a a 3 α 2 2∑ k (Φ − Ψ) = 12πGa (¯ρα + ¯pα)πα, (1.3.12 ) α
where a dot denotes d∕dη, δα = δρα∕ρ¯α, the index α indicates a sum over all matter components in the universe and π is related to Πi j through:
( ) ˆ ˆ 1- i (ρ¯+ ¯p) π = − kikj − 3δij Π j. (1.3.13 )
The energy-momentum tensor components in the Newtonian gauge become:
T00= − (¯ρ + δ ρ) (1.3.14 ) ik T i= − ik T0 = (¯ρ + ¯p) 𝜃 (1.3.15 ) i 0i i i i i T j = (p¯+ δp) δj + ¯pΠ j (1.3.16 )
where we have defined the variable j 𝜃 = ikjv 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., μ Tν;μ = 0. We have

( ) ( ) δ˙= − (1 + w ) 𝜃 − 3˙Φ − 3 ˙a- δp-− wδ for ν = 0 (1.3.17 ) a ¯ρ a˙ w˙ δp∕¯ρ 𝜃˙= − --(1 − 3w )𝜃 − ------𝜃 + k2------+ k2Ψ − k2 π for ν = i. (1.3.18 ) a 1 + w 1 + w
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 w, the pressure perturbation δp and the anisotropic stress π. For instance, if we want to study how matter perturbations evolve, we simply substitute w = δp = π = 0 (matter is pressureless) in the above equations. However, Eqs. (1.3.17View Equation) – (1.3.18View Equation) depend on the gravitational potentials Ψ and Φ, 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 δp and π 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 w crosses − 1) 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, H (z), of the FLRW background has been measured (at least up to redshifts ∼ 1 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 ΛCDM.

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 (1.3.8View Equation).

We describe the matter perturbations using the gauge-invariant comoving density contrast ΔM ≡ δM + 3aH 𝜃M ∕k2 where δM and 𝜃M 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 ΛCDM, after radiation-matter equality there is no anisotropic stress present and the Einstein constraint equations at “sub-Hubble scales” k ≫ aH become

2 2 − k Φ = 4πGa ρM ΔM , Φ = Ψ. (1.3.19 )
These can be used to reduce the energy-momentum conservation of matter simply to the second-order growth equation
3 Δ ′M′ + [2 + (lnH )′]Δ ′M = -ΩM (a )ΔM . (1.3.20 ) 2
Primes denote derivatives with respect to ln a and we define the time-dependent fractional matter density as ΩM (a) ≡ 8 πG ρM (a )∕(3H2 ). Notice that the evolution of ΔM is driven by ΩM (a ) and is scale-independent throughout (valid on sub- and super-Hubble scales after radiation-matter equality). We define the growth factor G (a) as Δ = Δ0G (a). This is very well approximated by the expression
{∫ a ′ } G (a) ≈ exp da- [Ω (a′)γ] (1.3.21 ) 1 a′ M
d log G fg ≡ ------- ≈ ΩM (a)γ (1.3.22 ) d loga
defines the growth rate and the growth index γ that is found to be γΛ ≃ 0.545 for the ΛCDM solution [see 937Jump To The Next Citation Point, 585Jump To The Next Citation Point, 466Jump To The Next Citation Point, 363Jump To The Next Citation Point].

Clearly, if the actual theory of structure growth is not the ΛCDM scenario, the constraints (1.3.19View Equation) will be modified, the growth equation (1.3.20View Equation) will be different, and finally the growth factor (1.3.21View Equation) is changed, i.e., the growth index is different from γΛ and may become time and scale dependent. Therefore, the inconsistency of these three points of view can be used to test the ΛCDM paradigm. 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 ΛCDM can be represented by introducing two new degrees of freedom in the Einstein constraint equations. We do this by replacing (1.3.19View Equation) with

− k2Φ = 4πGQ (a,k)a2ρM ΔM , Φ = η(a,k)Ψ. (1.3.23 )
Non-trivial behavior of the two functions Q and η can be due to a clustering dark-energy component or some modification to GR. In MG models the function Q(a,k ) represents a mass screening effect due to local modifications of gravity and effectively modifies Newton’s constant. In dynamical DE models Q represents the additional clustering due to the perturbations in the DE. On the other hand, the function η(a,k ) parameterizes the effective anisotropic stress introduced by MG or DE, which is absent in ΛCDM.

Given an MG or DE theory, the scale- and time-dependence of the functions Q and η can be derived and predictions projected into the (Q, η) 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., 848Jump To The Next Citation Point, for an overview of predictions for different MG/DE models].

Using the above-defined modified constraint equations (1.3.23View Equation), the conservation equations of matter perturbations can be expressed in the following form (see [737Jump To The Next Citation Point])

′ 1∕η-−-1-+-(ln-Q-)′9- x2Q-−--3(ln-H-)′∕Q- 𝜃M-- Δ M = − x2 + 9Ω 2ΩM ΔM − x2 + 9Ω aH Q 2 M Q 2 M ′ 3- Q- 𝜃M = − 𝜃M − 2aH ΩM η ΔM , (1.3.24 )
where we define √-- xQ ≡ k∕(aH Q ). Remember ΩM = ΩM (a) as defined above. Notice that it is Q ∕η that modifies the source term of the 𝜃M equation and therefore also the growth of ΔM. Together with the modified Einstein constraints (1.3.23View Equation) these evolution equations form a closed system for (ΔM ,𝜃M ,Φ,Ψ ) which can be solved for given (Q, η).

The influence of the Hubble scale is modified by Q, such that now the size of xQ determines the behavior of ΔM; on “sub-Hubble” scales, xQ ≫ 1, we find

Δ′′ + [2 + (ln H )′]Δ ′ = 3ΩM (a)Q-ΔM (1.3.25 ) M M 2 η
and 𝜃 = − aH Δ ′ M M. The growth equation is only modified by the factor Q ∕η on the RHS with respect to ΛCDM (1.3.20View Equation). On “super-Hubble” scales, xQ ≪ 1, we have
′ ′ 2 (ln H )′ 1 Δ M = − [1 ∕η − 1 + (ln Q) ]ΔM + 3-Ω----aH---Q-𝜃M , M 𝜃′ = − 𝜃 − 3-Ω aH Q-Δ . (1.3.26 ) M M 2 M η M
Q and η now create an additional drag term in the ΔM equation, except if η > 1 when the drag term could flip sign. [737Jump To The Next Citation Point] also showed that the metric potentials evolve independently and scale-invariantly on super-Hubble scales as long as xQ → 0 for k → 0. This is needed for the comoving curvature perturbation, ζ, to be constant on super-Hubble scales.

Many different names and combinations of the above defined functions (Q, η) 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., 41Jump To The Next Citation Point, 667, 848Jump To The Next Citation Point, 737Jump To The Next Citation Point, 278Jump To The Next Citation Point, 277Jump To The Next Citation Point, 363Jump To The Next Citation Point].

For instance, as observed above, the combination Q ∕η modifies the source term in the growth equation. Moreover, peculiar velocities are following gradients of the Newtonian potential, Ψ, and therefore the comparison of peculiar velocities with the density field is also sensitive to Q ∕η. So we define

μ ≡ Q∕η ⇒ − k2Ψ = 4πGa2 μ(a, k)ρM ΔM . (1.3.27 )

Weak lensing and the integrated Sachs–Wolfe (ISW) effect, on the other hand, are measuring (Φ + Ψ )∕2, which is related to the density field via

1- 1- 2 2 Σ ≡ 2 Q(1 + 1∕η) = 2μ(η + 1) ⇒ − k (Φ + Ψ ) = 8 πGa Σ(a,k )ρM ΔM . (1.3.28 )
A summary of different other variables used was given by [278Jump To The Next Citation Point]. For instance, the gravitational slip parameter introduced by [194] and widely used is related through ϖ ≡ 1 ∕η − 1. Recently [277Jump To The Next Citation Point] used {𝒢 ≡ Σ, μ ≡ Q, 𝒱 ≡ μ}, while [115Jump To The Next Citation Point] defined R ≡ 1∕η. All these variables reflect the same two degrees of freedom additional to the linear growth of structure in ΛCDM.

Any combination of two variables out of {Q, η, μ,Σ, ...} is a valid alternative to (Q, η). It turns out that the pair (μ, Σ) is particularly well suited when CMB, WL and LSS data are combined as it is less correlated than others [see 980Jump To The Next Citation Point, 277Jump To The Next Citation Point, 68]. 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 ΛCDM. 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., (Q, η) such that departures from ΛCDM 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 848Jump To The Next Citation Point] and/or by pure simplicity and measurability [see 41Jump To The Next Citation Point]. For instance, [980Jump To The Next Citation Point] and [278Jump To The Next Citation Point] use scale-independent parameterizations that model one or two smooth transitions of the modified growth parameters as a function of redshift. [115Jump To The Next Citation Point] also adds a scale dependence to the parameterization, while keeping the time-dependence a simple power law:

[ − k∕kc −k∕kc ] s Q (a,k) ≡ 1 + [Q0e + Q ∞(1 − e ) − 1] a , η (a, k)−1 ≡ 1 + R0e− k∕kc + R∞ (1 − e−k∕kc) − 1 as, (1.3.29 )
with constant Q0, Q ∞, R0, R ∞, s and kc. 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, 277Jump To The Next Citation Point] investigate the modified growth parameters binned in z and k. 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, 980Jump To The Next Citation Point] employ a PCA in the (μ, η) 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 ΛCDM. Trigger relations

A useful and widely popular trigger relation is the value of the growth index γ in ΛCDM. It turns out that the value of γ can also be fitted also for simple DE models and sub-Hubble evolution in some MG models [see, e.g., 585Jump To The Next Citation Point, 466, 587Jump To The Next Citation Point, 586, 692Jump To The Next Citation Point, 363]. For example, for a non-clustering perfect fluid DE model with equation of state w(z) the growth factor G (a) given in (1.3.21View Equation) with the fitting formula

γ = 0.55 + 0.05 [1 + w (z = 1)] (1.3.30 )
is accurate to the 10−3 level compared with the actual solution of the growth equation (1.3.20View Equation). Generally, for a given solution of the growth equation the growth index can simply be computed using
ln(Δ ′ ) − ln ΔM γ(a,k) = -----M-----------. (1.3.31 ) lnΩM (a)
The other way round, the modified gravity function μ can be computed for a given γ [737Jump To The Next Citation Point]
2 γ−1 γ ′ ′ μ = -Ω M (a)[ΩM (a) + 2 + (ln H ) − 3γ + γ ln γ]. (1.3.32 ) 3

The fact that the value of γ is quite stable in most DE models but strongly differs in MG scenarios means that a large deviation from γΛ 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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