Inflation in a Gaussian Random Landscape

Random, multifield functions can set generic expectations for landscape-style cosmologies. We consider the inflationary implications of a landscape defined by a Gaussian random function, which is perhaps the simplest such scenario. Many key properties of this landscape, including the distribution of saddles as a function of height in the potential, depend only on its dimensionality, $N$, and a single parameter, ${\gamma}$, which is set by the power spectrum of the random function. We show that for saddles with a single downhill direction the negative mass term grows smaller, relative to the average mass, as $N$ increases, a result with potential implications for the ${\eta}$-problem in landscape scenarios. For some power spectra Planck-scale saddles have ${\eta} \sim 1$ and eternal, topological inflation would be common in these scenarios. Lower-lying saddles typically have large ${\eta}$, but the fraction of these saddles which would support inflation is computable, allowing us to identify which scenarios can deliver a universe that resembles ours. Finally, by drawing inferences about the relative viability of different multiverse proposals we also illustrate ways in which quantitative analyses of multiverse scenarios are feasible.