Hessian eigenvalue distribution in a random Gaussian landscape

Abstract

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of $1/N$ expansion, where $N$ is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

33 pages, 10 figures

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• Inflation in multi-field random Gaussian landscapes
• Anthropic Bound on Dark Radiation and its Implications for Reheating
• Inflation in Random Landscapes with two energy scales
• Complexity of Gaussian random fields with isotropic increments: critical points with given indices
• Hessian Eigenspectra of More Realistic Nonlinear Models
• Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum
• The Distribution of Vacua in Random Landscape Potentials
• Hessian characterization of a vortex in a maze
• Inflation in a Gaussian Random Landscape