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

178 cosmological intimations of infinityPerhaps even more troubling, because of the finite speed of light, a finite age meansa finite distance over which information can propagate. This means that such a modelcontains an infinite number of regions that can never have communicated with eachother or exerted any causal influence over one another. It then seems exceedinglystrange that they should, at any given time, 3 all have just the same properties, asdemanded by the cosmological principle and FLRW metric. This strangeness (whichafflicts the k =+1 model as well, but with only finitely many disconnected regions) isoften termed the “horizon problem” and was part of the impetus behind the inventionof inflation, which is discussed in Section 8.3.8.2.3 The Steady-State ModelThe Steady-State model of Hoyle, Bondi, Gold, Narlikar, and others (e.g. Bondi andGold (1948), Hoyle and Narlikar (1996), Hoyle (1948)), took a somewhat differentstandpoint. There, the cosmological principle was extended to time as well as space,so that the universe at all times looks essentially the same on large enough scales. Thisis a very constraining symmetry: it leads both to k = 0 and to an exponential formfor R(t).A fascinating aspect of the Steady-State model is that although the universe isexpanding it does not get any bigger! That is, the scale factor in the metric growsexponentially, so the space between galaxies increases (as we observe). But in themodel, new matter is created to fill in the gaps, so as to maintain a constant averagedensity of matter. So (by construction) the universe is exactly the same (statistically,and on very large scales) at each successive time, and nothing has really happened asthe universe has expanded.It would certainly seem that at a later time the universe must have more matter in itthan at an earlier time (since, after all, it is both expanding and acquiring more matter),but this is one of the very neat things about infinity: an infinite set can be a subsetof itself! For example, we can take the set of integers, double each element (and thushave precisely as many as before), then add an additional (odd) integer to go with eachelement, and end up precisely with what we started. So also with the Steady-State.Although beautiful in many ways, the Steady-State is simply not correct as a descriptionof the observed universe; however, as we will see, it has returned in a differentform through the idea of inflation.8.3 Infinity in Inflationary Cosmology8.3.1 The Idea of InflationPuzzles – like the horizon problem – concerning the “initial” conditions in the BigBang model led a number of scientists in the early 1980s to develop the theory that3 “At a given time” is an ambiguous phrase in GR, but the point here is that even if we assume strict thermalstatistical equilibrium so that the particle/field content is basically determined by bulk temperature and density,it is extremely special if there exists any coordinatization such that constant time surfaces are statisticallyhomogeneous and isotropic.