6.1 Perturbation equations

We start with a general perturbed metric about the flat FLRW background [57Jump To The Next Citation Point352231232437Jump To The Next Citation Point]
2 2 i 2 i j ds = − (1 + 2α )dt − 2a (t)(∂iβ − Si)dtdx + a (t)(δij + 2ψδij + 2 ∂i∂jγ + 2 ∂jFi + hij)dx dx ,(6.1 )
where α, β, ψ, γ are scalar perturbations, Si, Fi are vector perturbations, and hij is the tensor perturbations, respectively. In this review we focus on scalar and tensor perturbations, because vector perturbations are generally unimportant in cosmology [71Jump To The Next Citation Point].

For generality we consider the following action

[ ] ∫ 4 √--- 1 1 μν S = d x − g ---2f(R, ϕ) − -ω (ϕ)g ∂μϕ∂ νϕ − V (ϕ ) + SM (gμν,ΨM ), (6.2 ) 2κ 2
where f(R, ϕ) is a function of the Ricci scalar R and the scalar field ϕ, ω(ϕ ) and V (ϕ) are functions of ϕ, and SM is a matter action. We do not take into account an explicit coupling between the field ϕ and matter. The action (6.2View Equation) covers not only f (R) gravity but also other modified gravity theories such as Brans–Dicke theory, scalar-tensor theories, and dilaton gravity. We define the quantity F (R, ϕ) ≡ ∂f ∕∂R. Varying the action (6.2View Equation) with respect to gμν and ϕ, we obtain the following field equations
1- FR μν − 2f gμν − ∇ μ∇ νF + gμν□F [ ( 1 ) ] = κ2 ω ∇ μϕ ∇νϕ − -gμν∇ λϕ∇ λϕ − V gμν + T(μMν ) , (6.3 ) ( 2 ) 1 λ f,ϕ □ϕ + --- ω,ϕ∇ ϕ ∇ λϕ − 2V,ϕ + -2- = 0, (6.4 ) 2ω κ
where T(μMν ) is the energy-momentum tensor of matter.

We decompose ϕ and F into homogeneous and perturbed parts, ¯ ϕ = ϕ + δϕ and ¯ F = F + δF, respectively. In the following we omit the bar for simplicity. The energy-momentum tensor of an ideal fluid with perturbations is

0 0 i i T0 = − (ρM + δρM ), Ti = − (ρM + PM )∂iv, Tj = (PM + δPM )δj, (6.5 )
where v characterizes the velocity potential of the fluid. The conservation of the energy-momentum tensor (μ ∇ Tμν = 0) holds for the theories with the action (6.2View Equation[357Jump To The Next Citation Point].

For the action (6.2View Equation) the background equations (without metric perturbations) are given by

[ ] 2 1 2 1 2 3F H = 2(RF − f ) − 3H F˙ + κ 2ω ˙ϕ + V (ϕ ) + ρM , (6.6 ) − 2F H˙ = F¨ − H F˙+ κ2 ωϕ˙2 + κ2(ρM + PM ), (6.7 ) 1 ( f ) ¨ϕ + 3H ˙ϕ + --- ω,ϕϕ˙2 + 2V,ϕ − -,ϕ2- = 0, (6.8 ) 2ω κ ˙ρM + 3H (ρM + PM ) = 0, (6.9 )
where R is given in Eq. (2.13View Equation).

For later convenience, we define the following perturbed quantities

Δ χ ≡ a (β + a˙γ), A ≡ 3(H α − ψ˙) − -2χ. (6.10 ) a
Perturbing Einstein equations at linear order, we obtain the following equations [316317Jump To The Next Citation Point] (see also [436566355438312313314Jump To The Next Citation Point49213833441328])
[( ) Δ 1 2 Δ 1( 2 2 2 ) a2-ψ + HA = − 2F- 3H + 3 ˙H + a2- δF − 3H δ˙F + 2- κ ω,ϕ ˙ϕ + 2κ V,ϕ − f,ϕ δϕ ] 2 ˙ ˙ ˙ 2 ˙2 ˙ 2 + κ ω ϕδϕ + (3H F − κ ωϕ )α + FA + κ δρM , (6.11 ) 1 [ ] H α − ψ˙ = --- κ2ωϕ˙δϕ + δ˙F − H δF − F˙α + κ2(ρM + PM )v , (6.12 ) 2F χ˙+ H χ − α − ψ = 1(δF − F˙χ), (6.13 ) F ( Δ ) 1 [ ( Δ ) A˙+ 2HA + 3H + -2- α = --- 3δ¨F + 3H δ˙F − 6H2 + -2- δF + 4 κ2ωϕ˙˙δϕ a 2F a + (2κ2ω,ϕϕ˙2 − 2κ2V,ϕ + f,ϕ )δϕ − 3 ˙Fα˙− ˙FA ] − (4κ2ω ˙ϕ2 + 3H F˙+ 6F¨)α + κ2(δρM + δPM ) , (6.14 )
( ) ¨ ˙ Δ-- R- 2- 2 ˙ ˙ 1- 2 ˙2 2 δF + 3H δF − a2 + 3 δF + 3 κ ϕδϕ + 3 (κ ω,ϕϕ − 4κ V,ϕ + 2f,ϕ)δϕ ( ) = 1-κ2(δρ − 3δP ) + ˙F(A + ˙α) + 2 ¨F + 3H F˙ + 2κ2 ωϕ˙2 α − 1F δR, (6.15 ) 3 M M 3 3 ( ) [ ( ) 2 ( ) ] δ¨ϕ + 3H + ω,ϕ ˙ϕ δ ˙ϕ + − Δ--+ ω,ϕ ϕ˙-+ 2V,ϕ-−-f,ϕ- δϕ ω a2 ω ,ϕ 2 2ω ,ϕ ( ) = ϕ˙α˙+ 2¨ϕ + 3H ϕ˙+ ω,ϕ˙ϕ2 α + ϕ˙A + -1-F δR, (6.16 ) ω ( 2ω ,ϕ ) Δ δρ˙M + 3H (δρM + δPM ) = (ρM + PM ) A − 3H α + --2v , (6.17 ) a ------1-------d- 3 ---δPM--- a3(ρM + PM )dt [a (ρM + PM )v] = α + ρM + PM , (6.18 )
where δR is given by
[ ( ) ] ˙ Δ-- ˙ Δ-- δR = − 2 A + 4HA + a2 + 3H α + 2a2ψ . (6.19 )

We shall solve the above equations in two different contexts: (i) inflation (Section 7), and (ii) the matter dominated epoch followed by the late-time cosmic acceleration (Section 8).


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