Topological implications of inhomogeneity

The approximate homogeneity of spatial sections of the Universe is well supported observationally, but the inhomogeneity of the spatial sections is even better supported. Here, we consider the implications of inhomogeneity in dust models for the connectedness of spatial sections at early times. We consider a non-global Lemaitre-Tolman-Bondi (LTB) model designed to match observations, a more general, heuristic model motivated by the former, and two specific, global LTB models. We propose that the generic class of solutions of the Einstein equations projected back in time from the spatial section at the present epoch includes subclasses in which the spatial section evolves (with increasing time) smoothly (i) from being disconnected to being connected, or (ii) from being simply connected to being multiply connected, where the coordinate system is comoving and synchronous. We show that (i) and (ii) each contain at least one exact solution. These subclasses exist because the Einstein equations allow non-simultaneous big bang times. The two types of topology evolution occur over time slices that include significantly post-quantum epochs if the bang time varies by much more than a Planck time. In this sense, it is possible for cosmic topology evolution to be "mostly" classical.