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

180 cosmological intimations of infinity8.3.2 An Infinite Time of InflationConsider a region of the universe, with volume U, that is inflating. Now you supposedivide it into eight regions of equal volume U/8 and wait for a time t necessary forthe scale factor to double. During this time, the classical tendency will be for φ to roll tolower values in all eight subvolumes, but the quantum fluctuations mean that in some,occasionally, the field will go up instead. Now, as it turns out, the quantum fluctuationsbecome stronger at larger values of V (φ); thus, if the region starts with a high enoughvalue, at least one of the eight volumes will, on average, fluctuate up during the timet. During the same time, that subvolume will grow into just the original volume U.Thus, while seven-eighths of the universe has gone down in field value, the total volumethat is inflating has stayed constant. Now if you consider a region starting at a fieldvalue φ i slightly above this, it can be shown (e.g., Linde (1986)) that the volume of theuniverse at φ>φ i will, in fact, increase exponentially in time. This phenomenon isoften called “everlasting inflation” (or often “eternal inflation,” but I will reserve thatterm for a different use) and means that while inflation may end locally to create a“Big Bang-like” region such as we observe, globally it will go on forever, continuallyspawning such regions. The universe is thus infinite in time, and at each time, theuniverse contains exponentially more volume of both inflating space and noninflatingspace than at the time before.The back-reaction of quantum effects on the large-scale classical geometry makesit rather tricky to rigorously work out the structure of these “stochastic inflation”spacetimes. It is therefore useful to look at a related example in which these issues areless troublesome.The model is a “double-well” potential as shown in Figure 8.1. The upper well atφ = φ F corresponds to an inflationary vacuum energy, and the lower well at φ = φ Tto no vacuum energy (or perhaps the very small one that we appear to observe in ouruniverse now). A region of the universe starting at φ F would classically inflate forever.Because of quantum mechanics, however, the field can tunnel through the barrier tocreate a region of φ T .This leads to a first-order phase transition in which the universe would seem toconvert from the φ F phase into the φ T phase. Because the space in the φ F phase isinflating, however, it can be shown that even though the nucleated bubbles of the φ Tphase grow at the speed of light, they nonetheless fail to take over all of the inflatingvolume Aguirre and Gratton (2003), Guth and Weinberg (1983), Vilenkin (1992).Instead, the volume with φ = φ F increases exponentially with time, just as in thefluctuation-driven version described earlier.The usefulness of this example is that the process of decay from φ F to φ T is welldescribed mathematically (it was first treated by Coleman and Deluccia Coleman andLuccia (1980)), so (neglecting collisions between bubbles; see next section) we canassemble a fairly accurate global picture of the inflationary universe.8.3.3 An Infinite Number of Pocket Universes Form at Each TimeThe regions of the universe in which inflation ends (when the inflaton either rollsor tunnels to near the bottom of the well) might be (and often are) called “bubble