Relativistic Cosmological Perturbation Theory and the Evolution of Small-Scale Inhomogeneities

TL;DR

This paper develops a gauge-invariant relativistic perturbation theory for cosmology, revealing new insights into density perturbations, their evolution, and implications for early universe structure formation, challenging traditional Newtonian approaches.

Contribution

It introduces a unique gauge-invariant quantity for density perturbations, applicable across all scales, and demonstrates its implications for structure formation without Cold Dark Matter.

Findings

Identifies a single gauge-invariant density perturbation quantity.

Shows the inadequacy of Newtonian perturbation theory for early universe studies.

Suggests heat exchange in perturbations enhances early structure formation.

Abstract

It is shown that a first-order relativistic perturbation theory for the open, flat or closed Friedmann-Lemaitre-Robertson-Walker universe admits one, and only one, gauge-invariant quantity which describes the perturbation to the energy density and which becomes equal to the usual Newtonian energy density in the non-relativistic limit. The same holds true for the perturbation to the particle number density. These facts exclude all definitions of gauge-invariant quantities used to describe density perturbations in former theories. Using these two new quantities, a manifestly covariant and gauge-invariant cosmological perturbation theory, adapted to non-barotropic equations of state for the pressure, has been developed. The new theory is valid for all scales since metric gradients do not occur in the final evolution equations. The new theory has an exact non-relativistic limit with a time-independent Newtonian potential. The usual Newtonian perturbation theory is inadequate to study the evolution of density perturbations. In the radiation-dominated era, perturbations in the particle number density are gravitationally coupled to perturbations in the total energy density, irrespective of the nature of the particles. This implies that structure formation can commence only after decoupling of matter and radiation. After decoupling of matter and radiation density perturbations evolve diabatically, i.e., they exchange heat with their environment. This heat loss of a perturbation may enhance the growth rate of its mass sufficiently to explain stellar formation in the early universe, a phenomenon not understood, as yet, without the additional assumption of the existence of Cold Dark Matter. This theoretical observation is the main result of this article.