Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators

TL;DR

This paper develops exact solutions for the stochastic dynamics of scalar fields in de Sitter spacetime using quantum deficient oscillators, enabling analytical exploration of inflationary models with multiple potential wells.

Contribution

It introduces quantum deficient oscillators via Krein--Adler transformations to derive new exact solutions in stochastic inflation models, including multi-well potentials.

Findings

Exact solutions for single-well and double-well models

Extension to multi-well potentials for cosmological studies

Analytical tools for non-perturbative stochastic inflation

Abstract

The stochastic dynamics of a scalar field in de Sitter spacetime can be regarded as a non-perturbative diffusion process, to which exact distribution and correlation functions are constructed by utilising the correspondence between diffusion and Schr\"{o}dinger equations. The Krein--Adler transformation of the quantum harmonic oscillator deletes several pairs of the energy levels to define anharmonic oscillators that we dub quantum deficient oscillators, based on which this article constructs a new class of exact solutions in stochastic inflation. In addition to the simplest single-well model, an exactly solvable double-well model is also presented. The results are further extended to exactly solvable models with multiple wells, allowing analytical studies on various cosmological phenomenologies.