Perturbations in a regular bouncing universe

We consider a simple toy model of a regular bouncing universe. The bounce is caused by an extra timelike dimension, which leads to a sign flip of the ${\rho }^2$ term in the effective four dimensional Randall Sundrum-like description. We find a wide class of possible bounces: big bang avoiding ones for regular matter content, and big rip avoiding ones for phantom matter. Focusing on radiation as the matter content, we discuss the evolution of scalar, vector and tensor perturbations. We compute a spectral index of $n_s=-1$ for scalar perturbations and a deep blue index for tensor perturbations after invoking vacuum initial conditions, ruling out such a model as a realistic one. We also find that the spectrum (evaluated at Hubble crossing) is sensitive to the bounce. We conclude that it is challenging, but not impossible, for cyclic/ekpyrotic models to succeed, if one can find a regularized version.

Figure 1

The numerical integration of (30) yields the spectrum (31) evaluated at $x=1/(4\tilde{k})$ (dots); the solid line corresponds to a spectral index of $n_s=-1$, in agreement with the analytic result (55). Higher-order effects in $\tilde{k}$ are accounted for by letting $\tilde{k}$ cover a broad range from 1 down to ${10}^{-10}$.

Figure 2

$\mathcal{V}(x)$ sets a scale for the behavior of each mode, it rises up to $\frac{1}{2}$ around the bounce but falls off very quickly as $x\to \pm \infty$.