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.
