Cosmological perturbations in the (1+3+6)-dimensional space-times

TL;DR

This paper analyzes cosmological perturbations in a (1+3+6)-dimensional universe, deriving solutions for both small and large wave-numbers, and examines how these perturbations evolve and influence the spectrum over time.

Contribution

It extends Abbott et al.'s formalism by deriving large wave-number solutions and analyzing their dependence on two wave-numbers, revealing their dominance in different regimes.

Findings

Large wave-number solutions depend on two wave-numbers, k_r and k_R.

Perturbation dominance varies with the ratio of wave-numbers and scale-factors.

The spectrum of outer space perturbations evolves over time based on these solutions.

Abstract

Cosmological perturbations in the (1+3+6)-dimensional space-times including photon gas without viscous processes are studied on the basis of Abbott et al.'s formalism. Space-times consist of the outer space (the 3-dimensional expanding section) and the inner space (the 6-dimensional section). The inner space expands initially and contracts later. Abbott et al. derived only power-type solutions in the small wave-number limit which appear at the final stage of the space-times. In this paper, we derive not only small wave-number solutions, but also large wave-number solutions. It is found that the latter solutions depend on the two wave-numbers k_r and k_R (which are defined in the outer and inner spaces, respectively), and that the k_r-dependent and k_R-dependent parts dominate the total perturbations when (k_r/r(t))/(k_R/R(t)) >> 1 or << 1, respectively, where r(t) and R(t) are the scale-factors in the outer and inner spaces. By comparing the behaviors of these perturbations, moreover, changes in the spectrum of perturbations in the outer space with time are discussed.