Numerical Investigation of Two-Dimensional Fokker-Planck Equation in Inflationary Models: Importance of Slow-Roll Parameters

TL;DR

This paper extends the Fokker-Planck equation to two dimensions for multi-field inflation models, using spectral and Crank-Nicolson methods, revealing the influence of potential and slow-roll parameters on probability distributions.

Contribution

It introduces a numerical approach to solve the two-dimensional Fokker-Planck equation in inflationary models, highlighting the role of potential and slow-roll parameters.

Findings

Distribution is determined by the two-dimensional potential and slow-roll parameters.

Volume weighting influences the distribution in the Fokker-Planck volume equation.

Spectral and Crank-Nicolson methods effectively solve the extended equation.

Abstract

In this study, we generalize the Fokker-Planck equation to two-dimensional cases, including potential functions with periodic boundary conditions and piecewise-defined structures, to analyze the probability distribution in multi-field inflationary models. We employ the spectral method for spatial derivatives and the Crank-Nicolson method for the time evolution to solve the equation numerically for the slow-roll inflation. We find that the distribution in the Fokker-Planck equation was determined by the two-dimensional potential combined slow-roll parameters. And the volume weighting effect makes the distribution in the Fokker-Planck Volume equation is determined by the potential.