Inflation with a Graceful Exit in a Random Landscape

Abstract

We develop a stochastic description of small-field inationary histories witha graceful exit in a random potential whose Hessian is a Gaussian random matrix as amodel of the unstructured part of the string landscape. The dynamical evolution in sucha random potential from a small-field ination region towards a viable late-time de Sitter(dS) minimum maps to the dynamics of Dyson Brownian motion describing the relaxationof non-equilibrium eigenvalue spectra in random matrix theory. We analytically computethe relaxation probability in a saddle point approximation of the partition function ofthe eigenvalue distribution of the Wigner ensemble describing the mass matrices of thecritical points. When applied to small-field ination in the landscape, this leads to anexponentially strong bias against small-field ranges and an upper bound $N\ll$ 10 on thenumber of light fields $N$ participating during ination from the non-observation of negativespatial curvature