6.3 The Einstein–Langevin equation

With the previous results for the kernels we can now write the -dimensional Einstein–Langevin equation (94), previous to the renormalization. Also taking into account Equations (89) and (90), we may finally write:
Notice that the terms containing the bare cosmological constant have cancelled. These equations can now be renormalized; that is, we can now write the bare coupling constants as renormalized coupling constants plus some suitably-chosen counterterms, and take the limit as . In order to carry out such a procedure, it is convenient to distinguish between massive and massless scalar fields. The details of the calculation can be found in [259].

It is convenient to introduce the two new kernels

where is given by the restriction to of expression (109). The renormalized coupling constants , and are easily computed as in Equation (92). Substituting their expressions into Equation (111), we can take the limit as . Using the fact that, for , , we obtain the corresponding semiclassical Einstein–Langevin equation.

For the massless case one needs the limit as of Equation (111). In this case it is convenient to separate in Equation (93) as , where

and use that, from Equation (109), we have
The coupling constants are then easily renormalized. We note that in the massless limit, the Newtonian gravitational constant is not renormalized and, in the conformal coupling case, , we have that . Note also that, by making in Equation (101), the noise and dissipation kernels can be written as
where
It is also convenient to introduce the new kernel
This kernel is real and can be written as the sum of an even part and an odd part in the variables , where the odd part is the dissipation kernel . The Fourier transforms (116) and (117) can actually be computed and, thus, in this case, we have explicit expressions for the kernels in position space; see, for instance, [71, 169, 220].

Finally, the Einstein–Langevin equation for the physical stochastic perturbations can be written in both cases, for and for , as

where, in terms of the renormalized constants and , the new constants are and . The kernels and are given by Equations (112) when , and by and when . In the massless case, we can use the arbitrariness of the mass scale to eliminate one of the parameters or . The components of the Gaussian stochastic source have zero mean value, and their two-point correlation functions are given by , where the noise kernel is given in Equation (102), which in the massless case reduces to Equation (115).

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 to 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 perturbations 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 (102) and (115) 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 [73], where the coupling constant was fixed to be zero. See also [208] for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [162, 339, 355].

Note that the expectation value of the renormalized stress-energy tensor for a scalar field can be obtained by comparing Equation (118) with the Einstein–Langevin equation (15); its explicit expression is given in [259]. The results agree with the general form found by Horowitz [169, 170] using an axiomatic approach and coincide with that given in [110]. The particular cases of conformal coupling, , and minimal coupling, , are also in agreement with the results for these cases given in [72, 169, 170, 223, 340], 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 [347].