It is convenient to introduce the two new kernels
For the massless case one needs the limit of Equation (104). In this case it is convenient to separate in Equation (86) as , where[179, 56, 137].
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 , where the coupling constant was fixed to be zero. See also  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 . The results agree with the general form found by Horowitz [137, 138] using an axiomatic approach, and coincides with that given in . 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 .
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