Boltzmann brains and the scale-factor cutoff measure of the multiverse

To make predictions for an eternally inflating “multiverse,” one must adopt a procedure for regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has emerged: normal observers—who evolve in pocket universes cooling from hot big bang conditions—must not be vastly outnumbered by “Boltzmann brains”—freak observers that pop in and out of existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all but vacuum energy has redshifted away, rather than the rich structure that we observe. Using the scale-factor cutoff measure, we calculate the ratio of Boltzmann brains to normal observers. We find the ratio to be finite, and give an expression for it in terms of Boltzmann brain nucleation rates and vacuum decay rates. We discuss the conditions that these rates must obey for the ratio to be acceptable, and we discuss estimates of the rates under a variety of assumptions.

Figure 1

The graph shows the holographic bound, the $D$ bound, and the $m$ bound for the entropy of an object that fills de Sitter space out to the horizon. The holographic and $D$ bounds are each shown as broken lines in the region where they are superseded by the other. Although the $m$ bound looks very much like a straight line, it is not.

Figure 2

The graph shows the ratio of $\Delta S$ to $I_{\mathrm{BB}}$, where the nucleation rate for Boltzmann brains is proportional to $e^{-\Delta S}$. All curves are based on the $I$ bound, as discussed in the text, but they differ by their assumptions about the size $R$ of the Boltzmann brain.