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7.2 The 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 (14View Equation) reads now
⟨ ⟩ ⟨ ⟩ G (0) − 8πG ˆT (0) + G (1)(h) − 8πG ˆT(1)(h ) = 8πG ξab, (143 ) ab ab ab ab
where the two first terms cancel, that is ⟨ ⟩ G (0)− 8πG ˆT(0) = 0 ab ab, as the background metric satisfies the semiclassical Einstein equations. Here the superscripts (0) and (1) refer to functions in the background metric gab and linear in the metric perturbation hab, respectively. The stress tensor operator Tˆab for the minimally coupled inflaton field in the perturbed metric is
ˆ &tidle; ˆ&tidle; ˆ 1- ( &tidle; ˆ &tidle;cˆ 2ˆ2) Tab = ∇a φ ∇bφ + 2&tidle;gab ∇c φ∇ φ + m φ . (144 )

Using the decomposition of the scalar field into its homogeneous and inhomogeneous part, see Equation (141View Equation), and 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[&tidle;g] φφ+ Tab[&tidle;g] φϕ+ Tab[&tidle;g] ϕ ϕ, (145 )
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[&tidle;g]⟩ which is not zero because the field dynamics is 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,
( ) □g+h − m2 ϕˆ = 0. (146 )
The last term in Equation (145View Equation) corresponds to the expectation value to 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 (11View Equation) can be written as

⟨{ˆt [g;x),ˆt [g; y)}⟩ = ⟨{ˆt [g;x),ˆt [g;y)}⟩ + ⟨{ ˆt [g;x),ˆt [g;y )} ⟩ , (147 ) ab cd ab cd (φϕ)2 ab cd (ϕ ϕ)2
where we have used the fact that ⟨ϕˆ⟩ = 0 = ⟨ˆϕˆϕ ˆϕ⟩ for Gaussian states on the background geometry. We consider the vacuum state to be the Euclidean vacuum which is preferred in the de Sitter background, and this state is Gaussian. In the above equation the first term is quadratic in ϕˆ, 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 ˆϕ as a small perturbation, the second term in Equation (147View Equation) is of lower order than the first and may be consistently neglected; this corresponds to neglecting the last term of Equation (145View Equation). The stress tensor fluctuations due to a term of that kind were considered in [252Jump To The Next Citation Point].

We can now write down the Einstein–Langevin equations (143View Equation) to linear order in the inflaton fluctuations. It is easy to check [254Jump 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., ⟨ˆTij⟩ = 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 [218Jump To The Next Citation Point]. The equation for Φ can be obtained from the 0i-component of the Einstein–Langevin equation, which in Fourier space reads

2iki(ℋ Φk + Φ′k) = 8πG (ξ0i)k, (148 )
where ki is the comoving momentum component associated to the comoving coordinate i x, and where we have used the definition ∫ Φk(η) = d3x exp(− i⃗k ⋅⃗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 (145View Equation) should also appear on the left-hand side of Equation (148View Equation), but we have neglected it to simplify the forthcoming expressions. Its inclusion does not change the large scale spectrum in an essential way [254Jump To The Next Citation Point]. Note, however, that the equivalence of the stochastic approach to linear order in ˆϕ and the usual linear cosmological perturbations approach is independent of that approximation [254Jump To The Next Citation Point]. To solve Equation (148View Equation), whose left-hand side comes from the linearized Einstein tensor for the perturbed metric [218Jump To The Next Citation Point], we need the retarded propagator for the gravitational potential Φ k,
( ′ ) Gk (η,η′) = − i-4-π θ(η − η ′)a(η-) + f(η,η′) , (149 ) kim2P a(η)
where f is a homogeneous solution of Equation (148View 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 (148View Equation) for the scalar metric perturbations we are in the position to compute the two-point correlation functions for these perturbations.
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