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

infinity in inflationary cosmology 181t 2Non-inflationt ’’ 2t ’’ 1t 1t 2’t 1’InflationFigure 8.2. A depiction of inflation in the double-well potential. Black regions are at φ F , whiteare at φ T , and gray regions are at field values between the two. The top is a spacetime diagramdrawn so that light travels on diagonal lines, depicting bubbles expanding at the speed oflight. (Note that the required scaling implies that the upper boundary signifies t →∞,andthatas it is neared, a finite distance on the diagram represents an infinite physical distance.) Thespacetime can be decomposed into space and time in many different ways, for example, bysurfaces like t 1 , t 2 , or by surfaces like t1 ′ and t′ 2 ,orevenbyt′′ 1and t′′2. In the two former cases thespatial slices would be finite; in the latter they would be infinite. The bottom diagram showsthe distribution in two spatial dimensions at a fixed t.universes,” or “pocket universes,” because, to an observer inside, they would appearhomogeneous and isotropic over large regions – and thus described by the FLRWmetric above – even while this homogeneity would break down on very large scales.To see in exactly what sense, however, as well as discuss infinity in this model, wemust confront the somewhat thorny issue that in GR there is no unique way in which todecompose spacetime into space and time. Any given observer has a well-defined time,but this cannot be extended to define a globally defined time, because other observerswill not agree. Thus, questions such as “What is the spatial volume of this region?” andeven “Is the universe spatially infinite or finite?” become dependent on how surfacesof simultaneity are defined (Fig. 8.2).With this in mind, let us examine the picture that develops from the double-wellpotential of Figure 8.1. Suppose that at some time we imagine the universe to be(a) spatially finite and (b) dominated by homogeneous inflaton vacuum energy withφ = φ F . Then it is the case that Aguirre and Gratton (2003), (Komatsu et al. 2008),Vilenkin (1992):