Statistics of Inflating Regions in Eternal Inflation

We compute the distribution of sizes of inflating and non-inflating regions in an eternally inflating Universe. As a first illustrative problem, we study a simple scenario of an eternally inflating Universe in the presence of a massless scalar field $\varphi$ whose field values lie within some finite domain $\varphi\in(-\varphi_{cr},\varphi_{cr})$, with $\pm\varphi_{cr}$ marking the onset of thermalization/crunching. We compute many important quantities, including the fractal dimension, distribution of field values among inflating regions, and the number of inflating and non-inflating Hubble regions. With the aid of simulations in 1 spatial dimension, we show this eternally inflating Universe reaches a steady state in which average sizes of inflating regions grow only as a power law in the field's crunch value $\sim \varphi_{cr}^2$ (extension to higher dimensions is $\sim\varphi^{2/D}$), contrary to a naive expectation of an exponential dependence. Furthermore, the distribution in sizes exhibits an exponential fall off for large distances (with an initial power law for inflating regions). We leave other interesting cases of more realistic potentials and time varying Hubble parameter for future work, with a possible application to the SM Higgs in the early Universe.