Curvature perturbations from ekpyrotic collapse with multiple fields

TL;DR

The work shows that curvature perturbations can be generated in a contracting ekpyrotic universe with two fields bearing steep negative exponential potentials. A multi-field scaling solution is tachyonically unstable, driving a transition to a late-time single-field ekpyrotic attractor; this transition automatically converts scale-invariant isocurvature perturbations into comoving curvature perturbations, with the final amplitude governed by the Hubble scale at transition, $H_T$. By reformulating perturbations in rotated variables $\varphi$ and $\chi$, the authors derive decoupled dynamics and relate the final curvature perturbation to the initial entropy perturbations, yielding $|{\cal R}_c| = \frac{1}{\sqrt{c_1^2+c_2^2}} |\delta\chi|$ at transition and $|{\cal R}_c| = \frac{c^2}{2\sqrt{c_1^2+c_2^2}} \left| \frac{H}{2\pi} \right|_T$. The scheme predicts an almost scale-invariant spectrum for curvature perturbations and negligible tensor modes, offering a viable alternative to inflation under certain pre-transition conditions. The results highlight a natural, dynamical mechanism for generating primordial perturbations in a contracting phase without altering the potential shape during the transition.

Abstract

A scale-invariant spectrum of isocurvature perturbations is generated during collapse in the ekpyrotic scaling solution in models where multiple fields have steep negative exponential potentials. The scale invariance of the spectrum is realized by a tachyonic instability in the isocurvature field. This instability drives the scaling solution to the late time attractor that is the old ekpyrotic collapse dominated by a single field. We show that the transition from the scaling solution to the single field dominated ekpyrotic collapse automatically converts the initial isocurvature perturbations about the scaling solution to comoving curvature perturbations about the late-time attractor. The final amplitude of the comoving curvature perturbation is determined by the Hubble scale at the transition.

Figures (3)

• Figure 1: Left: Numerical solutions for $x_1(N)$. The horizontal axis is $N =\log a$ and we take $c_1 = 40$ and $c_2 = 30$. The initial time is $N=0.05$. Note that $N$ decreases towards the future in a collapsing universe. Right: The corresponding phase space trajectories in $(x_1,x_2)$ plane.
• Figure 2: Left: Solutions for $\delta r(N)$, using the same parameters as in Figure 1. The corresponding background solutions are shown in Figure 1. Right: The ratio between $\delta r$ and $\delta s$. The ratio approaches a constant given by Eqs. (\ref{['ratio1']}) and (\ref{['ratio2']}). In this case $1.3333$ for $B_1$ and $-0.75$ for $B_2$.
• Figure 3: Left: Solutions for $\delta \chi(N)$. We used the same parameters as Figures 1 and 2. Right: Solutions for $\log \vert H \vert$ with three different background solutions. We also show $\log \vert H_T \vert$ that is determined from the numerical solutions for $\delta \chi$.