Cosmological Measures With Volume Averaging

Abstract

It has been common for cosmologists to advocate volume weighting for the cosmological measure problem, weighting spatial hypersurfaces by their volume. However, this often leads to the Boltzmann brain problem, that almost all observations would be by momentary Boltzmann brains that arise very briefly as quantum fluctuations in the late universe when it has expanded to a huge size, so that our observations (too ordered for Boltzmann brains) would be highly atypical and unlikely. Here it is suggested that volume weighting may be a mistake. Volume averaging is advocated as an alternative. One consequence may be a loss of the argument that eternal inflation gives a nonzero probability that our universe now has infinite volume.

Possibilities for probabilities

“…Therefore, to avoid Boltzmann brain domination in the sum or integral of localized projection operators over an asymptotically empty spacetime, it seems necessary to choose weight factors that give convergent integrals that are not dominated by the asymptotically empty regions where almost all observations would be by Boltzmann brains. For spacetimes with preferred spatial hypersurfaces (e.g., each at some proper time from a big bang or bounce minimal hypersurface) that each have finite 3-volume (though perhaps tending to infinity asymptotically with time), one simple procedure for greatly ameliorating the divergent integrals over spacetime is to divide the contribution over each of these preferred spatial hypersurfaces by the 3-volume V , thus taking the contribution at each time to be the volume average of the expectation value of the localized projection operator [17,18]. This will make the contribution of each of the preferred spatial hypersurfaces finite, but since this contribution seems likely to go to a constant at late times if the universe becomes asymptotically empty with an asymptotically constant spacetime density of contributions to Boltzmann brain observations, the sum or integral over times would diverge if the universe lasts forever.…”

Cosmological Ontology and Epistemology

“…In 2008 I proposed the alternative of Volume Averaging [20,21], which gives a contribution to the measure for an observation from a hypersurface that is proportional to the spatial density of the occurrences of the observation on the hypersurface.…”

Spacetime Average Density (SAD) cosmological measures

“…As an example of what I mean, in [77] (see also [2][3][4]78] for further discussion and motivation) I proposed volume averaging instead of volume weighting to avoid divergences in the measure of Boltzmann brain observations on spatial hypersurfaces as they expand to become infinitely large. However, summing up over all hypersurfaces still gave a divergence if that were done by a uniform integral over proper time t and if indeed the proper time goes to infinity.…”