Geometric distances between closed universes

Abstract

The large-scale structure of the Universe is well approximated by the Friedmann equations, parametrized by several energy densities which can be observationally inferred. A natural question to ask is: How different would the Universe be if these densities took on other values? While there are many ways this can be approached depending on interpretation and mathematical rigor, we attempt an answer by building a "history space" of different cosmologies. A metric is introduced after overcoming technical hurdles related to Lorentzian signature and infinite volume, at least for topologically closed cases. Geodesics connecting two points on the superspace are computed to express how distant--in a purely geometric sense--two universes are. Age can be treated as a free parameter in such an approach, leading to a more general mathematical construct relative to geometrodynamical configuration space. Connections to bubble inflation and phase transitions are explored.