Hessian eigenvalue distribution in a random Gaussian landscape

• Published in: The journal of high energy physics Vol. 2018; no. 3; pp. 1 - 33
• Main Authors: Yamada, Masaki, Vilenkin, Alexander
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 06.03.2018; Springer Nature B.V; SpringerOpen
• Subjects: Approximation; Brownian motion; Classical and Quantum Gravitation; Cosmology; Cosmology of Theories beyond the SM; Dynamic stability; Eigenvalues; Elementary Particles; Hessian matrices; High energy physics; Mathematical analysis; Normal distribution; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Random Systems; Regular Article - Theoretical Physics; Relativity Theory; Saddle points; String Theory; Random Systems; Cosmology of Theories beyond the SM
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP03(2018)029

A bstract 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.