The Einstein–Langevin equation for scalar metric perturbations

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 (14

) reads now

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

Using the decomposition of the scalar field into its homogeneous and inhomogeneous part, see Equation (141 ), 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

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,

The last term in Equation (145

) 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 (11 ) can be written as

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 (147

) is of lower order than the first and may be consistently neglected; this corresponds to neglecting the last term of Equation (145

). The stress tensor fluctuations due to a term of that kind were considered in [252

We can now write down the Einstein–Langevin equations (143 ) to linear order in the inflaton fluctuations. It is easy to check [254 ] 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 [218 ]. The equation for $Φ$ can be obtained from the $0i$ -component of the Einstein–Langevin equation, which in Fourier space reads

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 (145

) should also appear on the left-hand side of Equation (148

), but we have neglected it to simplify the forthcoming expressions. Its inclusion does not change the large scale spectrum in an essential way [254

]. 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 [254

]. To solve Equation (148

), whose left-hand side comes from the linearized Einstein tensor for the perturbed metric [218

], we need the retarded propagator for the gravitational potential $Φ k$ ,

where $f$ is a homogeneous solution of Equation (148

) 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 (148

) for the scalar metric perturbations we are in the position to compute the two-point correlation functions for these perturbations.