It is convenient to introduce the two new kernels
whereFor the massless case one needs the limit of Equation (104
). In this case it is
convenient to separate
in Equation (86
) as
, where
Finally, the Einstein-Langevin equation for the
physical stochastic perturbations can be written in
both cases, for
and for
, as
It is interesting to consider the massless
conformally coupled scalar field, i.e., the case , which is of particular interest because of its
similarities with the electromagnetic field, and also because of
its interest in cosmology: Massive fields become conformally
invariant when their masses are negligible compared to the
spacetime curvature. We have already mentioned that for a
conformally coupled field, the stochastic source tensor must be
traceless (up to first order in perturbation theory around
semiclassical gravity), in the sense that the stochastic variable
behaves deterministically as a
vanishing scalar field. This can be directly checked by noticing,
from Equations (95
) and (108
), that, when
, one has
, since
and
. The Einstein-Langevin equations for
this particular case (and generalized to a spatially flat
Robertson-Walker background) were first obtained in [58
], where the coupling
constant
was fixed to be zero. See also [169
] for a discussion of
this result and its connection to the problem of structure
formation in the trace anomaly driven inflation [269
, 280
, 132
].
Note that the expectation value of the
renormalized stress-energy tensor for a scalar field can be
obtained by comparing Equation (111) with the
Einstein-Langevin equation (14
), its explicit
expression is given in [209
]. The results agree
with the general form found by Horowitz [137
, 138
] using an axiomatic
approach, and coincides with that given in [91
]. The particular
cases of conformal coupling,
, and minimal
coupling,
, are also in agreement with the results
for these cases given in [137
, 138
, 270, 57, 182
], modulo local terms
proportional to
and
due to
different choices of the renormalization scheme. For the case of a
massive minimally coupled scalar field,
, our result
is equivalent to that of [276].