The mode solution of the wave equation in Kasner spacetimes and redshift

TL;DR

This paper analyzes scalar wave solutions in Kasner spacetimes, providing explicit mode solutions, asymptotic behaviors, and linking mode frequency to cosmological redshift, enhancing understanding of wave propagation in anisotropic cosmological models.

Contribution

It offers explicit mode solutions in axisymmetric Kasner spacetimes and characterizes their asymptotic behaviors, connecting wave modes to cosmological redshift.

Findings

Modes grow logarithmically at small times in non-flat Kasner spacetimes

Modes oscillate with decreasing amplitude at large times

Redshift derived from mode analysis matches standard cosmological redshift

Abstract

We study the mode solution to the Cauchy problem of the scalar wave equation $\Box \varphi = 0$ in Kasner spacetimes. As a first result, we give the explicit mode solution in axisymmetric Kasner spacetimes, of which flat Kasner spacetimes are special cases. Furthermore, we give the small and large time asymptotics of the modes in general Kasner spacetimes. Generically, the modes in non-flat Kasner spacetimes grow logarithmically for small times, while the modes in flat Kasner spacetimes stay bounded for small times. For large times, however, the modes in general Kasner spacetimes oscillate with a polynomially decreasing amplitude. This gives a notion of large time frequency of the modes, which we use to model the wavelength of light rays in Kasner spacetimes. We show that the redshift one obtains in this way actually coincides with the usual cosmological redshift.