The Distribution of Vacua in Random Landscape Potentials

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 $\Lambda$ in an $N$-dimensional Gaussian random potential, quantifying the associated probability density, $p(\Lambda )$. After normalisations $p(\Lambda )$ depends only on the dimensionality $N$ and a single free parameter $\gamma$, 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 $N=100$, $p(\Lambda >0)\approx 10^{-1197}$. 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 $\Lambda \approx 0$ are much flatter than for typical minima, increasingly the viability of quintessence solutions.

22 pages, 11 figures. $P(\Lambda >0$ at $N=100$ updated from $10^{-780}$ to $10^{-1197}$; the original error was due to an incorrect value of $\gamma$

References in corpus (8)

• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Cosmology With Many Light Scalar Fields: Stochastic Inflation and Loop Corrections
• Cosmology From Random Multifield Potentials
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Exploring a string-like landscape
• Multi-field Inflation with a Random Potential
• On the Unlikeliness of Multi-Field Inflation: Bounded Random Potentials and our Vacuum
• Slepian models for Gaussian Random Landscapes

Cited by in corpus (4)

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• Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity
• Inflation in a Gaussian Random Landscape