Calculating the power spectrum in stochastic inflation by Monte Carlo simulation and least squares curve fitting

TL;DR

This paper introduces a novel Monte Carlo method with least squares fitting to efficiently compute the primordial curvature perturbation spectrum in stochastic inflation models, reducing computational costs.

Contribution

The paper presents a new Monte Carlo approach that avoids nested simulations by using a variance estimator and least squares fitting to approximate the spectrum over a range of scales.

Findings

The method successfully estimates the power spectrum with reduced computational effort.

Numerical demonstrations confirm the accuracy and efficiency of the proposed approach.

The approach is applicable to multifield inflation models with quantum diffusion effects.

Abstract

The stochastic-$\delta \mathcal{N}$ formalism is widely used to study inflation models in which the quantum diffusion of inflatons dominates the background dynamics, leading to interesting phenomena such as the production of primordial black holes. Among numerical approaches to calculate the curvature perturbation spectrum $\mathcal{P}_\zeta(k)$ in this formalism, the Monte Carlo simulation-based approach has been proposed as a promising choice, especially in multifield cases. In this approach, we generate many paths of inflatons from the initial points to the end of inflation, obtain statistics of $\delta N$ from the paths, and then estimate $\mathcal{P}_\zeta(k)$. However, this method involves a nested Monte Carlo simulation, which requires generating many branch paths from each trunk path at the point corresponding to the scale $k$ of interest, resulting in a high computational cost. In this paper, we propose a new Monte Carlo-based approach that utilizes least squares fitting, introducing two novel features for reducing computational cost. First, we devise a simple estimator of a key statistic $\langle \delta\mathcal{N}_{\mathbf{X}}^2\rangle$, the variance of $\delta \mathcal{N}$ conditioned on the branching point, to avoid nesting path generation. Second, via least squares fitting of a parametric function to the sampled values of the estimator, we obtain not just an estimate of $\mathcal{P}_\zeta(k)$ for a single value of $k$ but an approximating function of $\mathcal{P}_\zeta(k)$ over a range of $k$ of interest. We also conduct numerical demonstrations for concrete inflation models, which show the usefulness of our method.