Density Perturbations in the Ekpyrotic Scenario

TL;DR

Addressing density perturbations in a slowly contracting ekpyrotic universe, the paper shows that a nearly scale-invariant spectrum can be generated during contraction and survive a nonsingular bounce when matched using gauge-invariant variables. Gravitational backreaction is found to be negligible in the contracting phase, while the curvature perturbation ζ does not reflect the growing mode; perturbations instead appear in the Newtonian potential Φ and are transferred through the bounce via a non-singular matching condition that relies on a jump in the equation of state. The resulting density perturbation amplitude is naturally small due to multiple suppression factors, offering an alternative to inflation for producing structure. The work highlights the need for microscopic validation of the matching procedure in a string/M-theory context and points to further issues such as tensor modes and the detailed inter-brane potential.

Abstract

We study the generation of density perturbations in the ekpyrotic scenario for the early universe, including gravitational backreaction. We expose interesting subtleties that apply to both inflationary and ekpyrotic models. Our analysis includes a detailed proposal of how the perturbations generated in a contracting phase may be matched across a `bounce' to those in an expanding hot big bang phase. For the physical conditions relevant to the ekpyrotic scenario, we re-obtain our earlier result of a nearly scale-invariant spectrum of energy density perturbations. We find that the perturbation amplitude is typically small, as desired to match observation.

Figures (1)

• Figure 1: Sketch of a nearly light-like collision between two boundary branes. The potential employed was $a^4V(\phi) = -a_1^4 (a_1/a_0)^4 e^{-f}$ where $f= {1\over 10}\left((a_0/a_1)-1\right)^{-1}$, chosen so that when expressed in terms of $\phi$, the potential $V$ vanishes as $\phi \rightarrow -\infty$, in a manner mimicking the vanishing of a non-perturbative potential $V \propto e^{-{1\over g^2}}$ or $V \propto e^{-{1\over g}}$. At collision, $a_0'$ and $a_1'$ are equal and opposite. The matching rule we propose is given in equation (\ref{['eq:nullb']}); in the Figure it is assumed that the efficiency $\xi$ and $Q$-violation parameter $\Delta$ are small. The scale factor $a_1$ is decreasing after collision: we assume it couples to a massless modulus $\chi$ which causes $a_1$ to be repelled from $a_1=0$. In the final state both $a_0$ and $a_1$ are expanding, and the universe becomes radiation-dominated with the outer-brane separation tending to a finite constant.