On the asymptotic assumptions for Milne-like spacetimes

Abstract

Milne-like spacetimes are a class of hyperboloidal FLRW spacetimes which admit continuous spacetime extensions through the big bang, $\tau = 0$. In a previous paper [27], it was advocated that the existence of this big bang extension could have applications to fundamental problems in cosmology, which illustrates the physical importance of such extensions. By definition, the scale factor for a Milne-like spacetime satisfies the past asymptotic assumption $a(\tau ) = \tau + o(\tau ^{1+\varepsilon })$ as $\tau \rightarrow 0$ for some $\varepsilon > 0$. The existence of the big bang extension follows from writing the metric in conformal Minkowskian coordinates and using the past asymptotic assumption of the scale factor. This asymptotic assumption implies $a(\tau ) = \tau + o(\tau )$ as $\tau \rightarrow 0$. In this paper, we show that $a(\tau ) = \tau + o(\tau )$ is not sufficient to achieve a big bang extension, but it is necessary (provided its derivative converges as $\tau \rightarrow 0$). We also show that the $\varepsilon$ in $a(\tau ) = \tau + o(\tau ^{1+\varepsilon })$ is not necessary to achieve a big bang extension.