Perhaps the most important problem faced by any attempt to get a predictive multiverse framework is what is known as the “measure problem”: How do we count universes in the multiverse? In particular, for any particular observable, we would like to know what fraction of universes have a given range of values for this observable. If the number of universes were finite, this would just be the ratio of the NO/Ntotal, where
NO = number of universes with specified values of observable O
Ntotal= total number of universes
When the number of universes is infinite, solving the measure problem becomes much more difficult. In their latest paper, Measure Problem for Eternal and Non-Eternal Inflation, Linde and Noorbala make some progress on the problem by investigating the predictions of four different possible measures (proper time, scale factor cutoff, stationary and causal diamond).
The authors begin by reviewing various problems with measures in the eternal inflation scenario, deciding to instead address what appears to be a much simpler problem, one without the infinities of eternal inflation:
…it may make sense to temporarily suppress our ambitions and try to understand non-eternal inflation, which at the first glance could seem quite trivial, and then return again to the investigation of eternal inflation. We will see that the stationary probability measures lead to similar predictions for eternal and non-eternal inflation. However, all other measures discussed above give very different predictions for models with eternal and non-eternal inflation. While this discontinuity does not necessarily mean that such measures are problematic, we think that this fact requires certain attention.
These results do not necessarily disfavor the first three measures since they have been invented for eternal inflation rather than for the non-eternal inflation. Nevertheless, dramatic discontinuity of predictions during the transition from non-eternal to eternal inflation is quite intriguing. We believe that at least with respect to the first two measures in the Table I this discontinuity can be traced back to the use of the asymptotic stationary distributions at the stage when the stationarity is not reached for some of the processes. Once this problem is taken care of [26, 27], the transition from the noneternal to eternal inflation becomes continuous. It would be interesting to see whether a similar modification can restore the continuity of predictions of other probability measures.