On Past Singularities in $k=0$ FLRW Cosmologies

The fundamental singularity theorem of FLRW cosmologies assumes that the matter content in the cosmological model obeys the strong energy condition along with a nonpositive cosmological constant which gives rise to an irrotational geodesic singularity. In this paper, we show that the important case of a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker universe with barotropic matter obeying only the \emph{weak} energy condition with a nonnegative cosmological constant also contains a past singularity. We accomplish this using topological methods from dynamical systems theory. The methods employed in this paper are sufficiently general that they could be extended to other models to study the existence of past singularities.