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

infinity in classic cosmological models 177time describes an expanding universe such as we see. At a given time (dt = 0), thismetric describes a homogenous and isotropic space, and there are only three types: flat“Euclidean” space with k = 0, the positively curved k =+1 space of a three-sphere,and the space of constant negative curvature with k =−1. For k = 0ork =−1 theuniverse is, at any time, spatially infinite. Moreover, Einstein’s equations yield a relationdRdtbetween H ≡ 1 , the energy density ρ, and k, so that by observationally measuringRρ and H, k can be determined.This leads to the intoxicating possibility of determining observationally whether theuniverse is infinite or finite. But this is an illusion: it rests on the very strong assumptionthat the FLRW metric holds not just inside the observed region of the universe, butalso everywhere outside it. Without this assumption, it is perfectly possible to be in alocally overdense region and determine k =+1 even while the universe is infinite, orto locally determine k =−1 in a very large but finite universe.Despite this caveat, the FLRW metric has served as a workhorse in cosmology andunderlies the two main classic cosmological models: the Big Bang, and the Steady-State. Both have interesting connections with the idea of a physically realized infinity.8.2.2 The Big Bang ModelIn the Big Bang model, it is assumed that the FLRW metric with k =±1 or 0 appliesback to very early times. Under certain assumptions regarding the energy density andpressure of the cosmic “fluid,” this implies that there exists a time t = 0, a finiteduration to the past, at which R(t) → 0. Cosmological attributes such as the value of kare simply prescribed as “initial” conditions shortly after t = 0. The cosmic evolutioncan then be evolved forward until today, when, for example, R 0 is an observationalquantity related to the time-varying scale over which the universe is curved (for k = 0,this curvature scale is always infinite).It is, however, important to distinguish the Big Bang theory (a very well-tested andobservationally successful theory of the universe’s evolution from an early, hot, denseepoch) from the idea that the FLRW metric applies at arbitrarily early times. Amongother reasons, if we strictly maintain the FLRW metric, the finite age of the universeleads to a rather strange situation for the spatially infinite k = 0 and k =−1 versions:although the physical distance between any two objects of fixed coordinate separationgoes to zero as t → 0, nonetheless, there will always be objects, at any time, that arephysically arbitrarily far apart; thus, there is no sense in which the universe is at anytime “small” or “at a point.” Rather, the whole infinite system springs into being all atonce at t = 0. It is unclear to what degree this troubled various cosmologists over theyears. Einstein had a strong preference for k =+1, but for reasons more connectedwith “Mach’s principle” 1 than with the Big Bang. 21 Roughly speaking, Mach’s principle asserts that nonaccelerated (inertial) frames should be determined by thematter distribution of the universe, as those that are moving at constant velocity with respect to the rest frameof that matter distribution.2 Einstein also initially preferred the idea of a static, eternal universe and introduced the cosmological constantto make this possible. Aside from being disproven observationally, I have often wondered how Einstein did notrealize that this model is (a) unstable to small perturbations or (b) in violation of Olber’s paradox.