

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
, as the background metric
satisfies the semiclassical Einstein equations. Here the
superscripts
and
refer to functions
in the background metric
and 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 part, see Equation (141), and 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 is 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 (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
defined in Equation (11) 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
preferred in the de Sitter background, and this state is Gaussian.
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
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.,
for
. This, in turn, implies that
the two functions characterizing the scalar metric perturbations
are equal,
, in agreement with [218
]. The equation for
can be obtained from the
-component of the
Einstein-Langevin equation, which in Fourier space reads
where
is the comoving momentum component associated to the
comoving coordinate
, and where 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 (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
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
,
where
is a homogeneous solution of Equation (148) 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 (148) for the scalar metric
perturbations we are in the position to compute the two-point
correlation functions for these perturbations.

