On the asymptotic assumptions for Milne-like spacetimes

TL;DR

This paper investigates the precise asymptotic conditions on the scale factor in Milne-like spacetimes that determine whether these spacetimes can be extended through the big bang, refining previous assumptions and identifying necessary and sufficient conditions.

Contribution

The paper clarifies the minimal asymptotic conditions on the scale factor needed for spacetime extension through the big bang in Milne-like spacetimes.

Findings

Showing that $a( au) = au + o( au)$ is necessary but not sufficient for extension.

Demonstrating that the parameter $ ext{ extepsilon}$ in the asymptotic expansion is not necessary for extension.

Providing conditions under which the spacetime admits a continuous extension through $ au=0$.

Abstract

Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, $\tau = 0$. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies $a(\tau) = \tau + o(\tau^{1+\varepsilon})$ as $\tau \to 0$ for some $\varepsilon > 0$. This asymptotic assumption implies $a(\tau) = \tau + o(\tau)$. In this paper, we show that $a(\tau) = \tau + o(\tau)$ is not sufficient to achieve an extension through $\tau = 0$, but it is necessary provided its derivative converges as $\tau \to 0$. We also show that the $\varepsilon$ in $a(\tau) = \tau + o(\tau^{1+\varepsilon})$ is not necessary to achieve an extension through $\tau = 0$.