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7.4 Summary and outlook

Stochastic gravity provides an alternative framework to study the generation of primordial inhomogeneities in inflationary models. Besides the interest of the problem in its own right, there are also other reasons that make this problem worth discussing from the point of view of stochastic gravity. The Einstein–Langevin equation is not restricted by the use of linearized perturbations of the inflaton field. In practice this may not be very important for inflationary models, which are driven by an inflaton field, which takes a non-zero expectation value, because the linear perturbations will give the leading contribution. However, the importance of considering corrections due to one-loop contributions from the inflaton-field perturbations, beyond the tree level of the linear cosmological perturbation theory, has recently been emphasized by Weinberg [368, 369] as a means to understand the consequences of the theory even where in practice its predictions may not be observed. Note also [225, 226] for the effect of the quantum gravitational loop corrections on the dynamics of the inflaton field.

In the stochastic gravity approach some insights on the exact treatment of the inflaton scalar field perturbations have been discussed in [316Jump To The Next Citation Point, 353, 354]. The main features that would characterize an exact treatment of the inflaton perturbations are the following. First, the three types of metric perturbations (scalar, vectorial and tensorial perturbations) couple to the perturbations of the inflaton field. Second, the corresponding Einstein–Langevin equation for the linear metric perturbations will explicitly couple to the scalar and tensorial metric perturbations. Furthermore, although the Fourier modes (with respect to the spatial coordinates) for the metric perturbations will still decouple in the Einstein–Langevin equation, any given mode of the noise and dissipation kernels will get contributions from an infinite number of Fourier modes of the inflaton field perturbations. This fact will imply, in addition, the need to properly renormalize the ultraviolet divergences arising in the dissipation kernel, which actually correspond to the divergences associated with the expectation value of the stress-tensor operator of the quantum matter field evolving on the perturbed geometry.

We should remark that although the gravitational fluctuations are here assumed to be classical, the correlation functions obtained correspond to the expectation values of the symmetrized quantum metric perturbations [66Jump To The Next Citation Point, 316Jump To The Next Citation Point]. This means that even in the absence of decoherence, the fluctuations predicted by the Einstein–Langevin equation still give the correct symmetrized quantum two-point correlation functions. In [66] it was explained how a stochastic description based on a Langevin-type equation could be introduced to gain information on fully quantum properties of simple linear open systems. In a forthcoming paper [317] it will be shown that, by carefully dealing with the gauge freedom and the consequent dynamical constraints, this result can be extended to the case of N free quantum matter fields interacting with the metric perturbations around a given background. In particular, the correlation functions for the metric perturbations obtained using the Einstein–Langevin equation are equivalent to the correlation functions that would follow from a purely quantum field theory calculation up to the leading order contribution in the large N limit. This will generalize the results already obtained on a Minkowski background [203, 204].

These results have important implications on the use of the Einstein–Langevin equation to address situations in which the background configuration for the scalar field vanishes. This includes not only the case of a Minkowski background spacetime, but also the remarkably interesting case of the trace anomaly-induced inflation. That is, inflationary models driven by the vacuum polarization of a large number of conformal fields [162Jump To The Next Citation Point, 339Jump To The Next Citation Point, 355Jump To The Next Citation Point], in which the usual approaches based on the linearization of both the metric perturbations and the scalar field perturbations and their subsequent quantization can no longer be applied. More specifically, the semiclassical Einstein equations (8View Equation) for massless quantum fields conformally coupled to the gravitational field admit an inflationary solution that begins in an almost de Sitter-like regime and ends up in a matter-dominated-like regime [339, 355]. In these models the standard approach based on the quantization of the gravitational and the matter fields to linear order cannot be used because the calculation of the metric perturbations correspond to having only the last term in the noise kernel in Equation (171View Equation), since there is no homogeneous field φ(η) as the expectation value ˆ ⟨φ⟩ = 0 and linearization becomes trivial.

In the trace anomaly induced inflation model Hawking et al. [162] were able to compute the two-point quantum correlation function for scalar and tensorial metric perturbations in a spatially-closed de Sitter universe, making use of the anti-de Sitter/conformal field theory correspondence. They find that short-scale metric perturbations are strongly suppressed by the conformal matter fields. This is similar to what we obtained in Section 6 for the induced metric fluctuations in Minkowski spacetime. In the stochastic gravity context, the noise kernel in a spatially-closed de Sitter background was derived in [314], and in a spatially-flat arbitrary Friedmann–Robertson–Walker model the Einstein–Langevin equations describing the metric perturbations were first obtained in [73Jump To The Next Citation Point]. The computation of the corresponding two-point correlation functions for the metric perturbations is now in progress.

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