Cosmological structure problem of the ekpyrotic scenario

We address the perturbation power spectrum generated in the recently proposed ekpyrotic scenario by Khoury et al. The issue has been raised recently by Lyth who used the conventional method based on a conserved variable in the large-scale limit, and derived different results from Khoury et al. The calculation is straightforward in the uniform-curvature gauge where the generated blue spectrum with a suppressed amplitude survives as the final spectrum. Whereas, although the metric fluctuations become unimportant and a scale-invariant spectrum is generated in the zero-shear gauge, the mode does not survive the bounce, but has the same final result. Therefore, an exponential potential leads to a power-law expansion or contraction $a\propto |t{|}^p,$ and the power p dictates the final power spectra of both the scalar and tensor structures. If $p\ll 1$ as one realization of the ekpyrotic scenario suggests, the results are $n_S-1\simeq 2\simeq n_T$ and the amplitude of the scalar perturbation is suppressed relative to that of the gravitational wave by a factor $\sqrt{p}/2.\mathrm{}$ Both results confirm Lyth’s. An observation is made on the constraint on the dynamics of the seed generating stage from the requirement of a scale-invariant spectrum.