Time evolution of the perturbations for a complex scalar field in Friedmann-Lemaitre universe

TL;DR

This paper analyzes the evolution of small perturbations in a complex scalar field within a Friedmann-Lemaître universe, extending previous real scalar field studies and deriving the Jeans wavenumber without the Jeans swindle.

Contribution

It generalizes scalar field perturbation analysis to complex fields and derives the Jeans wavenumber from general relativity, avoiding common approximations.

Findings

Perturbation behavior during inflation is similar for real and complex scalar fields.

Analytic solutions for perturbations are obtained in the oscillatory phase for large bosonic charge.

The length of inflation depends on the bosonic charge of the scalar field.

Abstract

We study the time evolution of small classical perturbations in a gauge invariant way for a complex scalar field in the early zero curvature Friedmann-Lema\^{\i}tre universe. We, thus, generalize the analysis which has been done so far for a real scalar field. We give also a derivation of the Jeans wavenumber in the Newtonian regime starting from the general relativistic equations, avoiding the so-called Jeans swindle. During the inflationary phase, whose length depends on the value of the bosonic charge, the behavior of the perturbations turns out to be the same as for a real scalar field. In the oscillatory phase the time evolution of the perturbations can be determined analytically as long as the bosonic charge of the corresponding background solution is sufficiently large. This is not possible for the real scalar field, since the corresponding bosonic charge vanishes.