6.2 Gauge-invariant quantities

Before discussing the detail for the evolution of cosmological perturbations, we construct a number of gauge-invariant quantities. This is required to avoid the appearance of unphysical modes. Let us consider the gauge transformation
ˆt = t + δt, ˆxi = xi + δij∂jδx, (6.20 )
where δt and δx characterize the time slicing and the spatial threading, respectively. Then the scalar metric perturbations α, β, ψ and E transform as [5771Jump To The Next Citation Point412Jump To The Next Citation Point]
ˆα = α − δ˙t, (6.21 ) − 1 ˆβ = β − a δt + aδ˙x, (6.22 ) ˆψ = ψ − H δt, (6.23 ) ˆγ = γ − δx. (6.24 )

Matter perturbations such as δϕ and δρ obey the following transformation rule

δˆϕ = δϕ − ϕ˙δt, (6.25 ) ˆ δρ = δρ − ρ˙δt. (6.26 )
Note that the quantity δF is also subject to the same transformation: δˆF = δF − F˙δt. We express the scalar part of the 3-momentum energy-momentum tensor δT 0 i as
0 δT i = ∂iδq. (6.27 )
For the scalar field and the perfect fluid one has δq = − ˙ϕδϕ and δq = − (ρM + PM )v, respectively. This quantity transforms as
ˆ δq = δq + (ρ + P )δt. (6.28 )

One can construct a number of gauge-invariant quantities unchanged under the transformation (6.20View Equation):

d [ 2 ] 2 Φ = α − --- a (γ + β∕a ) , Ψ = − ψ + a H (˙γ + β∕a ), (6.29 ) dt ℛ = ψ + -H----δq, ℛ δϕ = ψ − H-δϕ, ℛ δF = ψ − H--δF, (6.30 ) ρ + P ˙ϕ F˙ δρq = δρ − 3H δq. (6.31 )
Since δq = − ϕ˙δϕ for single-field inflation with a potential V (ϕ), ℛ is identical to ℛδϕ [where we used ρ = ϕ˙2 ∕2 + V (ϕ ) and P = ˙ϕ2∕2 − V (ϕ)]. In f (R) gravity one can introduce a scalar field ϕ as in Eq. (2.31View Equation), so that ℛ δF = ℛ δϕ. From the gauge-invariant quantity (6.31View Equation) it is also possible to construct another gauge-invariant quantity for the matter perturbation of perfect fluids:
δρM-- δM = ρ + 3H (1 + wM )v, (6.32 ) M
where wM = PM ∕ ρM.

We note that the tensor perturbation hij is invariant under the gauge transformation [412].

We can choose specific gauge conditions to fix the gauge degree of freedom. After fixing a gauge, two scalar variables δt and δx are determined accordingly. The Longitudinal gauge corresponds to the gauge choice ˆβ = 0 and ˆγ = 0, under which δt = a(β + a ˙γ) and δx = γ. In this gauge one has ˆΦ = αˆ and ˆΨ = − ψˆ, so that the line element (without vector and tensor perturbations) is given by

2 2 2 i j ds = − (1 + 2Φ )dt + a (t)(1 − 2Ψ )δijdx dx , (6.33 )
where we omitted the hat for perturbed quantities.

The uniform-field gauge corresponds to ˆδϕ = 0 which fixes δt = δϕ ∕ ˙ϕ. The spatial threading δx is fixed by choosing either ˆ β = 0 or ˆγ = 0 (up to an integration constant in the former case). For this gauge choice one has ℛˆδϕ = ˆψ. Since the spatial curvature (3) ℛ on the constant-time hypersurface is related to ψ via the relation (3)ℛ = − 4∇2 ψ∕a2, the quantity ℛ is often called the curvature perturbation on the uniform-field hypersurface. We can also choose the gauge condition ˆδq = 0 or δˆF = 0.


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