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

182 cosmological intimations of infinitySlices of constantand constant t’’True vacuumtBubble wall( = M )xFalse vacuum(inflation)Nucleation eventFigure 8.3. The geometry of a nucleated bubble. From the nucleation event, the bubble wallexpands at (approximately) the speed of light. Nestled into this light cone are hyperboloidsof constant φ, each of which has the geometry of an infinite negatively curved homogenousspace. The sequence of nested hyperboloids corresponds to the time sequence during whichthe inflaton rolls down the hill toward φ T in Figure 8.1.1. There will be a spacetime decomposition such that our region’s physical volume Uexpands exponentially with time t.2. At each time interval dt, a number of bubbles of φ T will nucleate, with the number givenby λUf inf dt, where f inf is the fraction of our region that has not thus far converted fromφ F to φ T , and λ is a rate determined by the form of the potential in Figure 8.2.3. Each such nucleated bubble will expand at the speed of light.4. The fraction f inf will approach zero as t →∞.5. Nonetheless, the inflating volume Uf inf will increase exponentially as t →∞, so thatthe number of bubbles nucleated in each time interval dt also increases exponentiallywith time.6. There will exist other “slicings” of the spacetime in which the volume of our regionincreases more slowly, or even decreases with time Winitzki (2005).7. Nonetheless, it will always be true that an infinite number of bubbles eventually form.Note that it is also possible (as discussed in the next section) to choose the initialvolume to be infinite, in which case infinitely many bubbles would form at each of theinfinitely many times t Aguirre and Gratton (2002, 2003), Winitzki (2005).8.3.4 Pocket Universes Are (Probably) InfiniteSo far we have been discussing the inflating region outside of the bubbles. Whathappens inside? For this we must return to the issue of time slicing; but here thereis a natural answer – not a unique way to decompose spacetime, but a particularlynatural and physically meaningful way. Figure 8.3 shows the spacetime created by abubble nucleation. The bubble wall accelerates away from the nucleation site, quickly