On the asymptotic assumptions for Milne-like spacetimes

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 $a(\tau )=\tau +o({\tau }^{1+\epsilon })$ as $\tau \to 0$ for some $\epsilon >0$. 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 $\epsilon$ in $a(\tau )=\tau +o({\tau }^{1+\epsilon })$ is not necessary to achieve an extension through $\tau =0$.