Self-consistent invariant dynamics of scalar perturbations in the inflationary cosmology

TL;DR

This paper develops a gauge-independent approach to scalar perturbations in inflationary cosmology, deriving invariant equations and solutions that clarify physical effects and provide analytical spectra, including the Harrison-Zeldovich form.

Contribution

It introduces a self-consistent, gauge-invariant framework for analyzing scalar perturbations during inflation, correcting overestimations from traditional gauges and providing analytical solutions for various potentials.

Findings

Invariant equations for scalar perturbations are derived.

Analytical solutions for all wavelengths during inflation are obtained.

The perturbation spectrum is nearly flat, matching the Harrison-Zeldovich form.

Abstract

The gauge-independent invariant approach to investigation of the linear scalar perturbations of inflaton and gravitational fields is developed in self-consistent way. This approach allows to compare various gauges used by other researchers and to find unambiguous selection criteria of physical and coordinate effects. We have shown that the so-called longitudinal gauge commonly used for studying the gravitational instability leads to overestimation of physical effects due to the presence of nonphysical proper time perturbations. Equation of invariant dynamics (EID) is derived. The general long-wave solution of EID for an arbitrary potential U(phi) has been obtained. We have also found analytical solutions for all wave lengths at all stages of the universe evolution in the framework of simplest potential U(phi)=m^2*phi^2/2. We have constructed the analytical expressions for the energy density perturbations spectrum Delta(k,t) at all possible k and t. Amplitude of the long-wave spectrum in the case of the transition from short waves to long ones occurs at the inflationary stage is almost flat, i.e. has the Harrison-Zeldovich form, for arbitrary potential U(phi).