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

Abstract

The stochastic-$δ\mathcal{N}$ 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 $\mathcal{P}_ζ(k)$ 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 $δN$ from the paths, and then estimate $\mathcal{P}_ζ(k)$. 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 $\langle δ\mathcal{N}_{\mathbf{X}}^2\rangle$, the variance of $δ\mathcal{N}$ 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 $\mathcal{P}_ζ(k)$ for a single value of $k$ but an approximating function of $\mathcal{P}_ζ(k)$ over a range of $k$ of interest. We also conduct numerical demonstrations for concrete inflation models, which show the usefulness of our method.

Figures (4)

• Figure 1: The approximating function of $F_{\left\langle\delta\mathcal{N}^2\right\rangle}$ (resp. $\mathcal{P}_\zeta$) in chaotic inflation output by MCLSFit is shown in the left panel (resp. the right panel) as a blue curve with the error shown as a blue band. The estimations by MCBinAve with $7$ bins are shown as red curves with the error bars. All the errors are of $1\sigma$ level. In the right panel, $\mathcal{P}_\zeta$ calculated via solving the MS equation, is also shown in green.
• Figure 2: The same figure as FIG. \ref{['fig:chaotic']} but for Starobinsky's linear-potential model. The number of bins in MCBinAve is now $10$.
• Figure 3: The same figure as FIG. \ref{['fig:chaotic']} but for the flat quantum well model. The number of bins in MCBinAve is now $10$. The analytical formulae \ref{['eq: AV variance']} and \ref{['eq: AV power']} are shown in green. In the right panel, the MCBinAve result is not shown because the statistical error bar is so large that plotting the result is not illustrative.
• Figure 4: The same figure as FIG. \ref{['fig:chaotic']} but for hybrid inflation. The number of bins in MCBinAve is $10$. In the right panel, the power spectrum calculated by the method in Ref. Tada:2023fvd is shown in green.