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

TL;DR

The work addresses exact, nonperturbative stochastic dynamics of a light scalar field in de Sitter spacetime by exploiting the diffusion–Schrödinger correspondence and the Krein–Adler transformation of the harmonic oscillator to generate quantum deficient oscillators. It constructs time-dependent PDFs and statistical moments for both single- and multi-well potentials, using two explicit solvable models (a single-well deficient oscillator I and a double-well deficient oscillator II) and a general Krein–Adler framework to build further multi-well cases. The approach maps exactly solvable quantum-mechanical systems to stochastic-inflation dynamics, enabling analytic study of cosmological phenomenologies such as vacuum decay and curvaton dynamics, with potential extensions to other exceptional orthogonal-polynomial families. These results provide a powerful analytic toolkit for probing nonperturbative effects in stochastic inflation and facilitate applications to primordial-structure formation and related high-energy cosmology.

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 Schrödinger 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.

Figures (8)

• Figure 1: The solid curve shows the potential (Eq. (\ref{['eq:c1_pot']}) in the right panel), while the dashed curves do the wavefunctions (Eq. (\ref{['eq:df1_wf_nom']}) in the right panel) squared. Each wavefunction squared is rescaled by factor two for illustrative purpose, and is displayed with the offset of its energy. It is $\lambda_{n} / \sqrt{A} = 0$ for $n = 0$ and $\lambda_{n} / \sqrt{A} = 2 (n + 2)$ for $n \geq 1$ in the right panel.
• Figure 2: In the right panel, the solid curve shows the potential (\ref{['eq:df2_pot']}), while the dashed curves do the wavefunction (\ref{['eq:df2_wf_nom']}) squared. Each wavefunction is displayed with the offset of its energy, $\lambda_{n} / \sqrt{A} = 0$ for $n = 0$ and $\lambda_{n} / \sqrt{A} = 2 (n + 2)$ for $n \geq 1$.
• Figure 3: Typical energy spectra of the quantum harmonic oscillator, the Crum-transformed, and the Krein--Adler-transformed oscillators. The dotted horizontal line indicates that the eigenstate is absent in each transformed system.
• Figure 4: The scalar-field potential (\ref{['eq:df1_pot_cos']}) (solid curve) and the distribution function of the scalar field (dashed curves) for $\sqrt{A} \, (N - N_{0}) = 0.05, \, 0.1, \, 0.2$, and $N \to \infty$ (thick dashed blue curve). The potential is rescaled by ten times for illustrative purpose.
• Figure 5: The scalar-field potential (\ref{['eq:df2_pot_cos']}) (solid curve) and the distribution function of the scalar field (dashed curves) for $\sqrt{A} \, (N - N_{0}) = 0.08, \, 0.1, \, 0.2$, and $N \to \infty$ (thick dashed blue curve). The potential is rescaled by ten times for illustrative purpose.