Volume-distance-ratio asymptote and spacetime inextendibility for spatially flat and hyperbolic FLRW spacetimes

Abstract

We study the volume-distance-ratio (VDR) asymptote at the past timelike boundary point for spatially flat FLRW spacetime with scale factor $a(t) = t^{\alpha}$, and spatially hyperbolic FLRW spacetime with scale factor $a(t) = a_0 t^{\alpha}$. We employ the spacetime inextendibility criteria via the volume-distance-ratio asymptote to deduce their inextendibility. We show that for spatial dimensions greater than $2$, there exists at least one critical exponent $\alpha \in (0,1)$ for which the VDR asymptote along the $\Sigma_t$-orthogonal geodesic equals the Minkowski value.