Cosmological perturbations in the 5D big bang

Bucher has recently proposed an interesting brane-world cosmological scenario where the “big bang” hypersurface is the locus of collision of two vacuum bubbles which nucleate in a five-dimensional flat space. This gives rise to an open universe, where the curvature can be very small provided that ${d/R}_0$ is sufficiently large. Here, $d$ is the distance between bubbles and $R_0$ is their size at the time of nucleation. Quantum fluctuations develop on the bubbles as they expand towards each other, and these in turn imprint cosmological perturbations on the initial hypersurface. We present a simple formalism for calculating the spectrum of such perturbations and their subsequent evolution. We conclude that, unfortunately, the spectrum is very tilted, with a spectral index $n_s=3.\mathrm{}$ The amplitude of fluctuations at the horizon crossing is given by $〈(\delta \rho /\rho {)}^2〉\sim {(R}_0{/d)}^2{S}_E^{-1}{k}^2,$ where $S_E\gg 1$ is the Euclidean action of the instanton describing the nucleation of a bubble and $k$ is the wave number in units of the curvature scale. The spectrum peaks on the smallest possible relevant scale, whose wave number is given by $k\sim {d/R}_0.$ We comment on the possible extension of our formalism to more general situations where a big bang is ignited through the collision of 4D extended objects.