Adiabatic perturbations in pre big bang models: matching conditions and scale invariance

TL;DR

This paper investigates how adiabatic scalar perturbations behave in four-dimensional pre-big bang and ekpyrotic models, focusing on the transition (bounce) from a collapsing to an expanding Friedmann universe. By analyzing both the background evolution and linear perturbations in the Einstein frame, and by applying Israel junction conditions across a thin matching surface with a possible surface tension, the authors show that a scale-invariant spectrum ($n=1$) naturally emerges under generic matching when the dilaton potential is exponential (slow collapse, small $q$). They further demonstrate that the final spectral index in the radiation era depends sensitively on the chosen matching hypersurface and its perturbation, with common prescriptions (e.g., constant-energy surfaces) potentially transferring the growing mode into a blue spectrum, while more generic matching yields $n=1-2q$ for ekpyrotic/modified pre-big bang. The results highlight the crucial role of high-energy corrections in the matching layer and suggest these models as viable alternatives to inflation for generating scale-invariant perturbations, albeit with open questions about the precise transition dynamics and additional modes.

Abstract

At low energy, the four-dimensional effective action of the ekpyrotic model of the universe is equivalent to a slightly modified version of the pre big bang model. We discuss cosmological perturbations in these models. In particular we address the issue of matching the perturbations from a collapsing to an expanding phase in full generality. We show that, generically, one obtains $n=0$ for the spectrum of scalar perturbations in the original pre big model (with vanishing potential). When an exponential potential for the dilaton is included, a scale invariant spectrum ($n=1$) of adiabatic scalar perturbations is produced under very generic matching conditions, both in a modified pre big bang and ekpyrotic scenario. We also derive general results valid for power law scale factors matched to a radiation dominated era.

Figures (1)

• Figure 1: The spectral indices $n_\Psi$ (solid) and $n_\zeta$ (dashed) are shown as a function of $q ={\cal H} \eta$. As argued in the text, we expect a resulting spectral index $n=n_\Psi$ in the radiation era.