Bucher ( ( 1 ) ) 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 spectral index n _ { s } = 3 .
The amplitude of fluctuations at 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 wavenumber 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 .