Squeezed thermal vacuum and the maximum scale for inflation.

We consider the stimulated emission of gravitons from an initial state of thermal equilibrium, under the action of the cosmic gravitational background field. We find that the low-energy graviton spectrum is enhanced if compared with spontaneous creation from the vacuum; as a consequence, the scale of inflation must be lowered, in order not to exceed the observed CMB quadrupole anisotropy. This effect is particularly important for models based on a symmetry-breaking transition which require, as initial condition, a state of thermal equilibrium at temperatures of the order of the inflation scale. To appear in Phys. Rev. D (Rapid Communications) * Permanent address: Dipartimento di Fisica Teorica, Via P.Giuria 1, 10125 Turin, Italy It has recently been argued that the Universe cannot be in a state of “eternal” inflation: not only a primordial de Sitter exponential expansion, without a beginning in time, is impossible [1], but also there are quantum cosmological arguments [2] suggesting for the inflationary phase just the minimum duration required to bring in causal contact scales that are not much larger than our present Hubble radius H 0 . If this is the case, the initial state, whose perturbations are amplified by the subsequent inflationary evolution, should be fixed by the dynamics of the preinflationary era and could differ considerably from the usually assumed vacuum. This difference influences the shape of the perturbation spectrum, as first pointed out in [3] for the case of tensor perturbations (graviton production), and recently discussed in [4] for the scalar perturbation case. It is well known, in particular, that any inflationary model based on a temperature dependent phase transition necessarily requires a homogeneous thermal state as initial condition. The existence of initial thermal equilibrium constrains the parameters of such models, and a recent discussion [5] suggests that, in their context, a sufficient duration of inflation can be arranged only for low enough energy scales. The aim of this paper is to point out that also the conventional upper bounds on the scale arising from the gravity-wave contribution to the CMB quadrupole anisotropy are to be lowered, if the relic graviton background is produced from an initial thermal bath rather than from the zero-temperature vacuum. Indeed, the inflationary amplification of the vacuum fluctuations can be properly represented, in second quantized formalism, as a process of pair production under the action of the external gravitational field [6], with the produced particles necessarily appearing in a final “squeezed vacuum” quantum state [6,7]. If the initial vacuum is changed into a state of thermal equilibrium, while the inflationary dynamics is left unchanged, the pair production process leads eventually to a ther-