On the Regularizability of the Big Bang Singularity

TL;DR

This paper investigates the conditions under which the big bang singularity can be mathematically regularized, revealing that for certain equations of state it is only regularizable if specific number-theoretic conditions are met.

Contribution

It introduces a novel application of McGehee transformation to analyze the regularizability of the big bang singularity based on the equation of state parameter w.

Findings

For w > 1, regularizability depends on number-theoretic conditions.

For w ≤ 1, the singularity can always be regularized.

The method connects cosmological singularities with dynamical systems techniques.

Abstract

The singularity for the big bang state can be represented using the generalized anisotropic Friedmann equation, resulting in a system of differential equations in a central force field. We study the regularizability of this singularity as a function of a parameter, the equation of state, $w$. We prove that for $w >1$ it is regularizable only for $w$ satisfying relative prime number conditions, and for $w \leq 1$ it can always be regularized. This is done by using a McGehee transformation, usually applied in the three and four-body problems. This transformation blows up the singularity into an invariant manifold. The relationship of this result to other cosmological models is briefly discussed.