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 (15) now
reads
Note that the first two terms cancel, that is,
, if the background metric is a solution
of the semiclassical Einstein equations. Here the superscripts
and
refer to functions in the
background metric
and functions, which are linear in the metric perturbation
, respectively. The
stress tensor operator
for the minimally-coupled inflaton field in the perturbed metric is
Using the decomposition of the scalar field into its homogeneous and inhomogeneous parts, see
Equation (165), and of the metric
into its homogeneous background
and its perturbation
, 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
and its perturbation
. 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
, 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
and may be obtained from the expectation value of the linearized Klein–Gordon equation,
The last term in Equation (169) 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
defined in Equation (12) can be
written as
where we have used the fact that
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
, whereas the second one is
quartic. Both contributions to the noise kernel are separately conserved, since both
and
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
as a small perturbation the second term
in (171) is of lower order than the first and may be consistently neglected. This corresponds to neglecting
the last term of Equation (169). The stress tensor fluctuations due to a term of that kind were considered
in [314
].
We can now write down the Einstein–Langevin equations (167) to linear order in the inflaton
fluctuations. It is easy to check [316
] that the space-space components coming from the stress-tensor
expectation-value terms and the stochastic tensor are diagonal, i.e.,
for
. This, in
turn, implies that the two functions characterizing the scalar metric perturbations are equal:
in agreement with [270
]. The equation for
can be obtained from the
-component of
the Einstein–Langevin equation, which, neglecting a nonlocal term, reads in Fourier space as,
where
is the comoving momentum component associated to the comoving coordinate
, and we have
used the definition
. Here primes denote derivatives with respect to the
conformal time
and
. A nonlocal term of dissipative character, which comes from the second
term in Equation (169), should also appear on the left-hand side of Equation (172), 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
to the usual linear cosmological perturbations approach does not assume that simplification [316
]. To
solve Equation (172), whose left-hand side comes from the linearized Einstein tensor for the
perturbed metric [270
], we need the retarded propagator for the gravitational potential
,
where
is a homogeneous solution of Equation (172) related to the initial conditions chosen and
. For instance, if we take
, the solution would correspond to
“turning on” the stochastic source at
. With the solution of the Einstein–Langevin equation (172) for
the scalar metric perturbations we are in position to compute the two-point correlation functions for these
perturbations.