Exactly solvable stochastic spectator

TL;DR

This paper explores the stochastic inflation formalism, establishing a correspondence with quantum mechanics to derive exact analytical solutions for scalar-field dynamics using shape-invariant Hamiltonians.

Contribution

It introduces a method to construct exact solutions in stochastic inflation via isospectral Hamiltonians with shape invariance, linking quantum mechanics and inflationary cosmology.

Findings

Exact solutions for stochastic inflation models derived

Distribution and correlation functions characterized analytically

Connection established between stochastic inflation and quantum mechanics

Abstract

The stochastic formalism of inflation allows us to describe the scalar-field dynamics in a non-perturbative way. The correspondence between the diffusion and Schr\"{o}dinger equations makes it possible to exhaustively construct analytical solutions in stochastic inflation. Those exact statistical quantities such as distribution and correlation functions have one-to-one correspondence to the exactly solvable solutions in non-relativistic quantum mechanics in terms of classical orthogonal polynomials. A class of such solutions is presented by means of isospectral Hamiltonians with an underlying symmetry called shape invariance.