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7.2 Einstein–Langevin equation for scalar metric perturbations

The Einstein–Langevin equation as described in Section 3 is gauge invariant, and thus we can work in a desired gauge and then extract the gauge invariant quantities. The Einstein–Langevin equation (15View Equation) now reads
G (0)− 8πG ⟨ˆT(0)⟩ + G (1)(h) − 8πG ⟨ˆT (1)(h)⟩ = 8πG ξ . (167 ) ab ab ab ab ab
Note that the first two terms cancel, that is, G (a0)b − 8πG ⟨ˆTa(0b)⟩ = 0, if the background metric is a solution of the semiclassical Einstein equations. Here the superscripts (0) and (1) refer to functions in the background metric g ab and functions, which are linear in the metric perturbation h ab, respectively. The stress tensor operator ˆ Tab for the minimally-coupled inflaton field in the perturbed metric is
1 Tˆab = ∇&tidle;a ˆφ∇&tidle;b ˆφ + --&tidle;gab(∇&tidle;c ˆφ∇&tidle;c ˆφ + m2 ˆφ2). (168 ) 2
Using the decomposition of the scalar field into its homogeneous and inhomogeneous parts, see Equation (165View Equation), and of the metric &tidle;gab into its homogeneous background gab and its perturbation hab, the renormalized expectation value for the stress-energy tensor operator can be written as
ˆR ˆ ˆ ˆR ⟨Tab[&tidle;g]⟩ = ⟨Tab[g&tidle;]⟩φφ + ⟨Tab[&tidle;g ]⟩φϕ + ⟨Tab[&tidle;g ]⟩ϕϕ, (169 )
where the subindices indicate the degree of dependence on the homogeneous field φ and its perturbation ϕ. The first term in this equation depends only on the homogeneous field and it is given by the classical expression. The second term is proportional to ⟨ϕˆ[g&tidle;]⟩, which is not zero because the field dynamics are considered on the perturbed spacetime, i.e., this term includes the coupling of the field with hab and may be obtained from the expectation value of the linearized Klein–Gordon equation,
( 2) □g+h − m ϕˆ = 0. (170 )
The last term in Equation (169View Equation) corresponds to the expectation value of the stress tensor for a free scalar field on the spacetime of the perturbed metric.

After using the previous decomposition, the noise kernel Nabcd[g;x,y) defined in Equation (12View Equation) can be written as

⟨{ˆtab[g;x),ˆtcd[g;y )}⟩ = ⟨{ ˆtab[g;x),ˆtcd[g;y)}⟩(φϕ)2 + ⟨{ˆtab[g;x ),ˆtcd[g;y)}⟩(ϕϕ)2, (171 )
where we have used the fact that ⟨ˆϕ ⟩ = 0 = ⟨ϕˆˆϕϕˆ⟩ for Gaussian states on the background geometry. We consider the vacuum state to be the Euclidean vacuum, which is Gaussian and is the preferred state in the de Sitter background. In the above equation the first term is quadratic in ˆϕ, whereas the second one is quartic. Both contributions to the noise kernel are separately conserved, since both φ(η) and ˆϕ satisfy the Klein–Gordon field equations on the background spacetime. Consequently, the two terms can be considered separately. On the other hand, if one treats ϕˆ as a small perturbation the second term in (171View Equation) is of lower order than the first and may be consistently neglected. This corresponds to neglecting the last term of Equation (169View Equation). The stress tensor fluctuations due to a term of that kind were considered in [314Jump To The Next Citation Point].

We can now write down the Einstein–Langevin equations (167View Equation) to linear order in the inflaton fluctuations. It is easy to check [316Jump To The Next Citation Point] that the space-space components coming from the stress-tensor expectation-value terms and the stochastic tensor are diagonal, i.e., ˆ ⟨Tij⟩ = 0 = ξij for i ⁄= j. This, in turn, implies that the two functions characterizing the scalar metric perturbations are equal: Φ = Ψ in agreement with [270Jump To The Next Citation Point]. The equation for Φ can be obtained from the 0i-component of the Einstein–Langevin equation, which, neglecting a nonlocal term, reads in Fourier space as,

′ 2iki(ℋ Φk + Φk) = 8πG (ξ0i)k, (172 )
where ki is the comoving momentum component associated to the comoving coordinate xi, and we have used the definition ∫ 3 ⃗ Φk (η) = d xexp (− ik ⋅⃗x)Φ (η,⃗x). Here primes denote derivatives with respect to the conformal time η and ′ ℋ = a ∕a. A nonlocal term of dissipative character, which comes from the second term in Equation (169View Equation), should also appear on the left-hand side of Equation (172View Equation), but we have neglected it to simplify the forthcoming expressions (the large scale spectrum does not change in a substantial way). We must emphasize, however, that the proof of the equivalence of the stochastic approach to linear order in ϕˆ to the usual linear cosmological perturbations approach does not assume that simplification [316Jump To The Next Citation Point]. To solve Equation (172View Equation), whose left-hand side comes from the linearized Einstein tensor for the perturbed metric [270Jump To The Next Citation Point], we need the retarded propagator for the gravitational potential Φk,
( ′ ) Gk (η,η′) = − i-4-π- θ(η − η′)a-(η-) + f(η,η ′) , (173 ) kim2P a(η)
where f is a homogeneous solution of Equation (172View Equation) related to the initial conditions chosen and 2 m P = 1∕G. For instance, if we take ′ ′ ′ f(η,η ) = − θ(η0 − η )a(η)∕a (η ), the solution would correspond to “turning on” the stochastic source at η0. With the solution of the Einstein–Langevin equation (172View Equation) for the scalar metric perturbations we are in position to compute the two-point correlation functions for these perturbations.
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