LAUSR

We compute the distribution of sizes of inflating regions (surrounded by non inflating ones) 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 thermalized 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 grows 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 is initally a power law fall off, followed by an exponential fall off. 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.