On the Transfer of Metric Fluctuations when Extra Dimensions Bounce or Stabilize

TL;DR

The work analyzes a $5$-dimensional cosmology with one compact extra dimension that either bounces nonsingularly or stabilizes, using a gas of massless string modes to realize the low-energy background. It shows that, for long-wavelength perturbations, the spectrum of the Bardeen potential $\Phi$ is preserved through the transient dynamics of the extra dimension, with radion fluctuations (encoded in $\\xi$) possessing a stable constant mode while the metric perturbation $\\Gamma$ decays. The results hold both in analytic late-time approximations and in numerical solutions, indicating that the pre-bounce spectrum can survive into the post-bounce/late-time era without spectral distortion. This has important implications for ekpyrotic/cyclic models and string-gas cosmology, enabling proposed mechanisms to generate a scale-invariant spectrum and motivating two concrete pathways to integrate inflation or cyclic dynamics with internal-dimension stabilization.

Abstract

In this report, we study within the context of general relativity with one extra dimension compactified either on a circle or an orbifold, how radion fluctuations interact with metric fluctuations in the three non-compact directions. The background is non-singular and can either describe an extra dimension on its way to stabilization, or immediately before and after a series of non-singular bounces. We find that the metric fluctuations transfer undisturbed through the bounces or through the transients of the pre-stabilization epoch. Our background is obtained by considering the effects of a gas of massless string modes in the context of a consistent 'massless background' (or low energy effective theory) limit of string theory. We discuss applications to various approaches to early universe cosmology, including the ekpyrotic/cyclic universe scenario and string gas cosmology.

Figures (6)

• Figure 1: The analytic approximation (\ref{['analyticb']}) (solid line) is compared with the numeric solution of (\ref{['eom']}) (circles).
• Figure 2: $\log(|\Phi_k^2)|$ is plotted for different values of $k^*$, with the initial conditions given in section \ref{['num']}. Black: analytic solution of (\ref{['analyticphi']}); Grey (bending curve): numerical solution of (\ref{['eomgamma']})-(\ref{['eomrhoeta']}).
• Figure 3: The spectrum of (a) $\xi_k$, (b) $\Gamma_k$ and (c) $\Phi_k=(\Gamma_k-\xi_k)/2$ is evaluated at Hubble radius crossing $\eta_r=2/k^*$, with the initial conditions given in section \ref{['num']}. If the long wavelength regime $k^*\ll1$ is approached, all oscillations are damped away such that only the constant mode of $\xi_k$ survives, as expected.
• Figure 4: (a) and (b): $\tilde{b}$ over $k^{*}\eta$ is plotted for $k^*=0.005$, with the initial conditions given in section \ref{['num']}. (c): $\xi_k$ is plotted over the same time range as $b$; Note how the oscillations in the background scale factor source transient oscillations in $\xi_k$.
• Figure 5: The perturbation variables $\xi_k$, $\Gamma_k$ and $N_k$ plotted for $k^*=0.005$, with the initial conditions given in section \ref{['num']}.