Multiple Fields in Stochastic Inflation

TL;DR

This paper explores the complex stochastic effects in multi-field inflation, revealing how multiple fields influence the probability of large-field exploration and the distribution of inflationary e-folds, with implications for quantum effects in early universe cosmology.

Contribution

It derives a hierarchy of diffusion equations for multi-field inflation, analyzing how additional fields affect inflationary dynamics and the distribution of e-folds, including cases with infinite mean e-folds.

Findings

Number of fields critically affects inflationary outcomes.

Multiple fields enable exploration of large-field regions inaccessible to single-field models.

Multi-field stochastic effects can be significant even at sub-Planckian energies.

Abstract

Stochastic effects in multi-field inflationary scenarios are investigated. A hierarchy of diffusion equations is derived, the solutions of which yield moments of the numbers of inflationary $e$-folds. Solving the resulting partial differential equations in multi-dimensional field space is more challenging than the single-field case. A few tractable examples are discussed, which show that the number of fields is, in general, a critical parameter. When more than two fields are present for instance, the probability to explore arbitrarily large-field regions of the potential, otherwise inaccessible to single-field dynamics, becomes non-zero. In some configurations, this gives rise to an infinite mean number of $e$-folds, regardless of the initial conditions. Another difference with respect to single-field scenarios is that multi-field stochastic effects can be large even at sub-Planckian energy. This opens interesting new possibilities for probing quantum effects in inflationary dynamics, since the moments of the numbers of $e$-folds can be used to calculate the distribution of primordial density perturbations in the stochastic-$\delta N$ formalism.