Nice Article. Vilenkin seems to have one of the better imaginative minds in physics. If large-scale topology is flat as WMAP indicated, then it has always been flat, which means the singularity of the big bang was/is infinitely extended. So there must be an infinite set of O-regions. I felt this was inevitable in the 60's when I first learned the big bang theory. And Vilenkin is right to question the breadth of possibilities. It is easy to conclude there are infinite many O-regions, the big question is, how infinite are they? We can imagine any conceivable world originating from a big bang if the laws that govern time are arbitrarily created. Otherwise along with eternal inflation you have eternal governing forces which are measurably predetermined, they either rigidly govern what is possible unilaterally limiting the multiverse to what is possible in the many-worlds theory (worlds equally possible to our own, which we know to exist), or eternal governing forces measurably allow certain constants or forces to congeal within the process of time. In this case there are primary fixed constraints as well as secondary variable constraints. The really difficult problem is explaining how secondary constraints can congeal and govern an entire O-region, which essentially amounts to a reduction in what is possible for an O-region in later stages of its evolution, a reduction that for some reason doesn't apply to all O-regions unilaterally.
I believe the answer to the "how infinite" question lies in the future of our own universe. If our universe collapses in a big crunch then the breadth of possible universes can be very wide, as this indicates governing forces are determined only in the past. If all laws or forces are determined by initial conditions or early conditions, then the futures of O-regions can be dramatically different. The only hope of modeling such a multiverse would be something like string theory.
If the universe expands forever without ever reaching an equilibrium or maximum entropy then this would suggest the second law correctly defines the reason for the arrow of time, disorder is more probable than order. In this case governing forces are restricted by the attraction of an always larger body of disordered possibilities that ultimately governs all universes, which for the most part rules out the potential of big crunch universes. If the strength of gravity is predetermined by initial conditions it might temporarily violate the second law and cause a big crunch or oscillating universes, but disorder would win out in the end. It is more likely that a quantum gravity is the product of a contrasting large body of ordered states which are past-like (more dense) in comparison to our present, in which case, from an eternal or top-down perspective, time probabilistically traces forward to disorder and traces backward to order. Note that this scenario does require an explanation for why complexity emerges in time over simple disorder. One would expect that adjacent the arrow of time other large groups of possibilities, such as complexity, act as attractors, until disorder finally dominates.
The newest and most interesting possible future to consider is the big rip scenario, where time ends in a finite period at what Caldwell and colleagues refer to as the ultimate singularity. We all know there is an absolute zero temperature but the big rip future ends at zero temperature and energy, zero density and mass, zero curvature and gravity, and zero volume and (space-)time. This future indicates that the evolution of the universe has a specific goal. Time ends at zero. In this case the second law is wrong, the arrow of time does not point to disorder, it points to zero, which is certainly not disorder. Zero is of course balance, and if time ends at zero then the cosmological arrow of time traces forward to balance and thus naturally traces backward to imbalance. This offers a very restricted view of possible universes, since zero is in this case a single governing attractor for all universes, or all time directions leaving eternal inflation. This also defines the initial state of time as the most imbalanced state in the field of all possibilities. So thinking eternally, time ends eternally at a zero, just as it begins eternally from the density of the early big bang which must either be positive or negative relative to a balanced zero. This predicts two arrows of time, one pointing from positive to zero (matter), another pointing from negative to zero (stable anti-matter). Any observer in an imbalanced state, such as matter over anti-matter, will thus trace history forward to zero and backward to an ever increasing state of imbalance. Time will appear to be eternally departing one single point, the most imbalanced1 state, just as time ends at one point. The most probable time pathways in between these two extremes defines a many-worlds partition or the multiverse.
I have already done a lot of work developing the probabilities of this multiverse scenario. Can I possibly get a post grant for my work?