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We study past-directed extendibility ofFriedmann-Lema\^{i}tre-Robertson-Walker (FLRW) and Bianchi type I spacetimeswith a scale factor vanishing in the past. We give criteria for determiningwhether a boundary for past-directed incomplete geodesics is a parallellypropagated curvature singularity, which cannot necessarily be read off fromscalar curvature invariants. It is clarified that, for incomplete FLRWspacetime to avoid the singularity, the spacetime necessarily reduces to theMilne universe or flat de Sitter universe toward the boundary. For incompleteBianchi type I spacetime to be free of singularity, it is necessary that thespacetime asymptotically fits into the product of the extendible isotropicgeometry (Milne or flat de Sitter) and flat space, or, anisotropic spacetimewith specific power law scale factors. Furthermore, we investigate in detailthe time-dependence of the scale factor compatible with the extendibility inboth spacetimes beyond the leading order.

Past extendibility and initial singularity in Friedmann-Lema\^{i}tre-Robertson-Walker and Bianchi I spacetimes