The Distribution of Vacua in Random Landscape Potentials

The density of empty space is not exactly zero, but is roughly 10^-123 in “natural” units. One possible explanation is that there is a “landscape” of 10⁵⁰⁰ or more possible vacua and, if the heights of the vacua are uniformly distributed, a vast number of candidate vacua can match this apparently special value. We look at a toy model where the landscape is a Gaussian random function where we can calculate the probability that any specific minimum is positive, as a function of the dimensionality N and a single free parameter.  In many cases the odds of a minimum being greater than zero are far less than 1 in 10⁵⁰⁰. In this case the landscape has no positive minima and cannot explain the apparent tuning of the vacuum energy. 

Abstract

Landscape cosmology posits the existence of a convoluted, multidimensional, scalar potential — the “landscape” — with vast numbers of metastable minima. Random matrices and random functions in many dimensions provide toy models of the landscape, allowing the exploration of conceptual issues associated with these scenarios. We compute the relative number and slopes of minima as a function of the vacuum energy Λ in an N-dimensional Gaussian random potential, quantifying the associated probability density, p(Λ). After normalisations p(Λ) depends only on the dimensionality N and a single free parameter γ, which is related to the power spectrum of the random function. For a Gaussian landscape with a Gaussian power spectrum, the fraction of positive minima shrinks super-exponentially with N; at N=100, p(Λ>0)≈10⁻¹¹⁹⁷. Likewise, typical eigenvalues of the Hessian matrices reveal that the flattest approaches to typical minima grow flatter with N, while the ratio of the slopes of the two flattest directions grows with N. We discuss the implications of these results for both swampland and conventional anthropic constraints on landscape cosmologies. In particular, for parameter values when positive minima are extremely rare, the flattest approaches to minima where Λ≈0 are much flatter than for typical minima, increasingly the viability of quintessence solutions.

• Low, Hotchkiss and Easther
• The Distribution of Vacua in Random Landscape Potentials
• JCAP 01 (2021) 029 or ArXiV:2004.04429

The log (base 10) of p(Λ) is plotted as a function of the number of fields and the γ parameter. For large values of γ p(Λ) decreases rapidly with N.