Hessian eigenvalue distribution in a random Gaussian landscape

Abstract

The energy landscape of multiverse cosmology is often modeled by a multidimensional 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.

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

“…While these results predict large scale properties of the low energy configurations, little is known about the detailed statistical structure of the complex energy landscape of pinned manifolds. This relates to the broad effort of understanding the statistical structure of stationary points (minima, maxima and saddles) of random landscapes which is of steady interest in theoretical physics [30][31][32][33][34][35][36][37][38][39], with recent applications to statistical physics [10,[34][35][36][38][39][40][41], neural networks and complex dynamics [42][43][44][45][46], string theory [47,48] and cosmology [49,50]. It is also of active current interest in pure and applied mathematics [51][52][53][54][55][56][57][58][59][60], For the model (1)- (2) in the simplest case d = 0 (x is a single point), the mean number of stationary points and of minima of the energy function was investigated in the limit of large N 1 in [35,38,39], see also [37,…”

“…Let us mention here some works on related models, although they are more similar to the case d = 0, and not the manifold. In [41,50] the Hessian statistics is sampled over all saddle-points or minima at a given value of the potential H(u) = E = const, a priori quite different from imposing the absolute minimum. The spectrum of the soft modes was also calculated in a mean-field model of the jamming transition, the 'soft spherical perceptron'.…”

“…with χ aa = 0 by definition. The saddle point equation (50) leads then to the explicit formula for σ, which determines completely the solution…”

Anthropic bound on dark radiation and its implications for reheating

“…In fact, the flux compactification of the higher-dimensional space in the string theory predicts a large number of axions and gauged dark sectors in the low-energy effective field theory. Inflation may occur in the axion field space, the so-called axion landscape [43][44][45][46][47][48]. For instance, the reheating could occur via the decay into gauge fields.…”

Inflation in multi-field modified DBM potentials

“…Following the argument of Bjorkmo and Marsh, this is expected: our potentials do not include a welldefined third derivative term, and it is precisely the structure of this term which gives rise to the eigenvalue gap in their construction. Vilenkin and Yamada [44] predicted the eigenvalue gap in a VY distribution to go like 1/ N f , by taking the width of the distribution to go like N f and dividing by N f for the N f eigenvalues. A power law fit of N 0.73 f is a decent approximation over this range of N f .…”