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It is speculated that the correct theory of fundamental physics includes a large landscape of states, which can be described as a potential which is a function of N scalar fields and some number of discrete variables. The properties of such a landscape are crucial in determining key cosmological parameters including the dark energy density, the stability of the vacuum, the naturalness of inflation and the properties of the resulting perturbations, and the likelihood of bubble nucleation events. We codify an approach to landscape cosmology based on specifications of the overall form of the landscape potential and illustrate this approach with a detailed analysis of the properties of N-dimensional Gaussian random landscapes. We clarify the correlations between the different matrix elements of the Hessian at the stationary points of the potential. We show that these potentials generically contain a large number of minima. More generally, these results elucidate how random function theory is of central importance to this approach to landscape cosmology, yielding results that differ substantially from those obtained by treating the matrix elements of the Hessian as independent random variables.
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We develop analytic and numerical techniques for studying the statistics of slow-roll inflation in random Gaussian landscapes. As an illustration of these techniques, we analyze small-field inflation in a one-dimensional landscape. We calculate the probability distributions for the maximal number of e-folds and for the spectral index of density fluctuations $n_s$ and its running $alpha_s$. These distributions have a universal form, insensitive to the correlation function of the Gaussian ensemble. We outline possible extensions of our methods to a large number of fields and to models of large-field inflation. These methods do not suffer from potential inconsistencies inherent in the Brownian motion technique, which has been used in most of the earlier treatments.
We investigate slow-roll inflation in a multi-field random Gaussian landscape. The landscape is assumed to be small-field, with a correlation length much smaller than the Planck scale. Inflation then typically occurs in small patches of the landscape, localized near inflection or saddle points. We find that the inflationary track is typically close to a straight line in the field space, and the statistical properties of inflation are similar to those in a one-dimensional landscape. This picture of multi-field inflation is rather different from that suggested by the Dyson Brownian motion model; we discuss the reasons for this difference. We also discuss tunneling from inflating false vacua to the neighborhood of inflection and saddle points and show that the tunneling endpoints tend to concentrate along the flat direction in the landscape.
No-scale supergravity is the appropriate general framework for low-energy effective field theories derived from string theory. The simplest no-scale Kahler potential with a single chiral field corresponds to a compactification to flat Minkowski space with a single volume modulus, but generalizations to single-field no-scale models with de Sitter vacua are also known. In this paper we generalize these de Sitter constructions to two- and multi-field models of the types occurring in string compactifications with more than one relevant modulus. We discuss the conditions for stability of the de Sitter solutions and holomorphy of the superpotential, and give examples whose superpotential contains only integer powers of the chiral fields.
In this note we revisit some of the recent 10d and 4d arguments suggesting that uplifting of supersymmetric AdS vacua leads to flattening of the potential, preventing formation of dS vacua. We explain why the corresponding 10d approach is inconclusive and requires considerable modifications. We also show that while the flattening effects may occur for some extreme values of the parameters, they do not prevent the formation of dS vacua within the range of validity of the 4d KKLT models. The KL version of the KKLT scenario based on a racetrack superpotential requires parametrically small uplifting, which is not affected by flattening. We show that this scenario is compatible with the weak gravity conjecture for a broad choice of parameters of the KL model. Thus, the results of our analysis do not support the recent swampland conjecture.
We investigate the recent suggestion that a Minkowski vacuum is either absolutely stable, or it has a divergent decay rate and thus fails to have a locally Minkowski description. The divergence comes from boost integration over momenta of the vacuum bubbles. We point out that a prototypical example of false-vacuum decay is pair production in a uniform electric field, so if the argument leading to the divergence is correct, it should apply to this case as well. We provide evidence that no catastrophic vacuum instability occurs in a constant electric field, indicating that the argument cannot be right. Instead, we argue that the boost integration that leads to the divergence is unnecessary: when all possible fluctuations of the vacuum bubble are included, the quantum state of the bubble is invariant under Lorentz boosts.
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