On the asymptotic assumptions for Milne-like spacetimes

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