Cosmology From Random Multifield Potentials

TL;DR

This work analyzes the statistical structure of random multifield potentials, motivated by the string landscape, to understand how many vacua exist, how inflation can proceed, and how far apart extrema are in field space. By employing random-matrix theory and the central limit theorem, it shows that if cross-couplings are not hierarchically suppressed, almost all extrema become saddles and minima vanish, with a scale separation akin to protecting Newton's constant needed to retain any minima. It demonstrates that even with an enormous number of extrema, typical separations in field space are large, which has strong implications for inflationary model-building in landscapes, including the feasibility of chained or folded inflation. The analysis distinguishes universal high-dimensional statistical effects from model-specific string-theoretic details and suggests that realistic realizations of inflation in a landscape likely require either suppressed cross-couplings or correlated multi-field dynamics.

Abstract

We consider the statistical properties of vacua and inflationary trajectories associated with a random multifield potential. Our underlying motivation is the string landscape, but our calculations apply to general potentials. Using random matrix theory, we analyze the Hessian matrices associated with the extrema of this potential. These potentials generically have a vast number of extrema. If the cross-couplings (off-diagonal terms) are of the same order as the self-couplings (diagonal terms) we show that essentially all extrema are saddles, and the number of minima is effectively zero. Avoiding this requires the same separation of scales needed to ensure that Newton's constant is stable against radiative corrections in a string landscape. Using the central limit theorem we find that even if the number of extrema is enormous, the typical distance between extrema is still substantial -- with challenging implications for inflationary models that depend on the existence of a complicated path inside the landscape.

Figures (3)

• Figure 1: The eigenvalue distribution of a $1000 \times 1000$ symmetric matrix with entries drawn from a standard normal distribution is shown, with the theoretical distribution plotted for reference.
• Figure 2: We show the proportion, $f$ of randomly generated symmetric $N \times N$ matrices for which all eigenvalues have the same sign. The curve is well fit by $\sim \exp(- b N^2)$, with the key observation being that $f$ is decreasing faster than exponentially with $N$.
• Figure 3: We plot the proportion of matrices $H= X+U$ whose lowest eigenvalue is negative, where $X$ is a $\hbox{diag}(1,\cdots,N)/N$ and $U$ is a symmetric perturbation whose terms have standard deviation $\sigma$. These results are drawn from a sample of 10000 matrices at each value of $\sigma$, with $N=100$.