Volume-Distance-Ratio Asymptote and Spacetime Inextendibility for FLRW Spacetimes

This paper examines the volume-distance-ratio (VDR) asymptote at the past timelike boundary for Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes. We consider spatially flat FLRW spacetimes with scale factor $a(t)\sim t^{\alpha }$, as well as spatially hyperbolic and spherical FLRW spacetimes with scale factor $a(t)\sim a_0{t}^{\alpha }$. Using criteria for spacetime inextendibility based on the VDR asymptote, we investigate the conditions under which these FLRW spacetimes are past inextendible.