Heating up the cold bounce

Self-dual string cosmological models provide an effective example of bouncing solutions where a phase of accelerated contraction smoothly evolves into an epoch of decelerated Friedmann–Robertson–Walker expansion dominated by the dilaton. While the transition to the expanding regime occurs at sub-Planckian curvature scales, the Universe emerging after the bounce is cold, with sharply growing gauge coupling. However, since massless gauge bosons (as well as other massless fields) are super-adiabatically amplified, the energy density of the maximally amplified modes re-entering the horizon after the bounce can efficiently heat the Universe. As a consequence the gauge coupling reaches a constant value, which can still be perturbative. 1 Formulation of the problem One of the inspiring symmetries of the pre-big bang scenario is the scale factor duality (SFD), allowing the connection of different cosmological solutions of the low-energy string effective action [1]. For instance, in the string frame description, accelerated solutions with growing curvature are connected via SFD to decelerated Friedmann–Robertson–Walker (FRW) models with decreasing curvature and decreasing dilaton energy density. The same solutions can also be described in the so-called Einstein frame, where the dilaton is not directly coupled to the Einstein–Hilbert term [2]. In this case a solution exhibiting accelerated contraction evolves into a decelerated FRW expansion. The connection among different solutions provided by the combined action of SFD and time-reversal does not imply an analytical connection of the duality-related solutions, as clearly noted in the early days of the pre-big bang cosmology [3]. In spite of this, it is known that solutions connecting smoothly two duality-related regimes exist in the presence of dilaton potentials which are explicitly invariant under the SFD symmetry [3, 4, 5]. Recently its was shown [6, 7] that these models can be derived from a generally covariant action supplemented by a non-local dilaton potential. In the absence of fluid sources and in the Einstein frame metric, these solutions smoothly interpolate between a phase of accelerated contraction valid for negative values of the conformal-time coordinate and a phase of decelerated expansion valid for positive values of the conformal time coordinate. The intermediate high-curvature phase is regular and can be interpreted as cold bounce (CB) since the Universe is never dominated by radiation, be it before or after the bounce. Furthermore, the nature of CB solutions also implies that the dilaton field never becomes constant for η → +∞ but is always growing logarithmically. A possible way out is to consider the effects of the high-frequency (small-scale) modes amplified during the epoch preceding the bounce. In the low-energy effective action also Abelian gauge bosons may be present and since their coupling to the dilaton field breaks conformal invariance [8] the various modes of the “photon” field can be super-adiabatically amplified. While inside the horizon the quantum mechanical fluctuations related to these modes are adiabatically damped. On the contrary, as the modes get outside the horizon, their energy density is super-adiabatically amplified, thanks to the coupling of the kinetic term of the gauge bosons to the dilaton field. When the fluctuations re-enter the horizon, after the bounce, their energy density red-shifts as radiation while the energy density of the background, still dominated by the dilaton field, red-shifts as stiff matter. Since the energy density of the background decreases faster than radiation, there will be a moment where the energy density of the radiation background will become dominant, providing a mechanism for the gravitational heating of the cold bounce [9]. Well after the bounce the