Volume-weighted measure for eternal inflation

I propose a new volume-weighted probability measure for cosmological “multiverse” scenarios involving eternal inflation. The “reheating-volume (RV) cutoff” calculates the distribution of observable quantities on a portion of the reheating hypersurface that is conditioned to be finite. The RV measure is gauge-invariant, does not suffer from the “youngness paradox,” and is independent of initial conditions at the beginning of inflation. In slow-roll inflationary models with a scalar inflaton, the RV-regulated probability distributions can be obtained by solving nonlinear diffusion equations. I discuss possible applications of the new measure to “landscape” scenarios with bubble nucleation. As an illustration, I compute the predictions of the RV measure in a simple toy landscape.

Figure 1

A $1+1$-dimensional slice of the spacetime in an eternally inflating universe (numerical simulation in Ref. 40). Shades of different color represent different regions where reheating took place. The reheating surface is the line separating the white (inflating) domain and the shaded domains.

Figure 2

A schematic representation of the ensemble $E_{\mathcal{V}}$ in comoving coordinates $(t,x)$. Lightly shaded vertical strips represent the comoving future of various initial $H$ regions from $E_{\mathcal{V}}$; dark shades represent reheated domains; the boundary of the dark shaded domains is the reheating surface. Vertical dashed lines are the eternally inflating comoving worldlines that never cross the reheating surface. The 3-volumes of the reheating surfaces in the comoving future of the pictured $H$ regions are large but finite because these $H$ regions are located near eternal worldlines but do not contain any such worldlines.