Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

CHAPTER 8Cosmological Intimationsof InfinityAnthony Aguirre8.1 IntroductionThe question of whether the universe is infinite or finite is an old and contentious onein philosophy. Einstein’s general relativity (GR) has, however, brought the questionlargely into the domain of science by giving us mathematical models of just what aninfinite (or finite) universe would look like. It has also provided genuinely new waysof conceptualizing old questions. For example, a seemingly insuperable objection to afinite universe would be to ask what is beyond the edge of the finite universe; however,GR teaches us that the universe may be finite but without boundary, in just the way thatthe surface of a sphere is. And, in contrast to Newtonian gravity, GR has no troubledealing with an infinite spacetime.In this chapter I explore the question of infinity in cosmology, in terms of infinitespace, infinite time, and an infinite number of states.8.2 Infinity in Classic Cosmological Models8.2.1 Homogeneous and Isotropic SpacetimesCosmological models describing our observed universe have been based almost entirelyon two ingredients: Einstein’s theory of general relativity (GR) and the “cosmologicalprinciple.” The latter takes a number of specific forms, but the general idea is that onvery large spatial scales, the universe is homogeneous and isotropic. This leads, in GR(or any other metric theory of gravity), to a spacetime described by the “Friedmann-Lemaître-Robertson-Walker” (FLRW) metric:[ drds 2 =−dt 2 + R 2 2](t)1 − kr + 2 r2 (dθ 2 + sin 2 θdφ 2 ) ,where k =±1 or 0, and R(t) is a “scale factor” that essentially converts coordinatedistances (such as radial distances in r) into physical distances; R(t) increasing in176