Quasi-Single Field Inflation with Large Mass

TL;DR

This work analytically quantifies how a heavy isocurvaton with mass $M$ affects density perturbations in quasi-single-field inflation, providing exact leading-order corrections to the power spectrum for all masses. By decomposing the correction into two contributions ${\cal C}_1$ and ${\cal C}_2$ within the in-in formalism, the authors derive closed-form expressions (in terms of polygamma functions) and establish the large-$M$ behavior ${\cal C}\to H^2/(4M^2)$, with ${\cal C}_1$ Boltzmann-suppressed and subdominant to ${\cal C}_2$. They validate the analytic results against previous numerics for $0<M<3H/2$ via analytic continuation to imaginary $\mu$, and present an infrared expansion and a history-truncation analysis showing that the dominant heavy-field contributions arise near horizon crossing over a window of roughly $\ln M/H$ e-folds. The findings demonstrate decoupling of heavy fields in the power spectrum and clarify the mass-dependent, time-localized nature of heavy-field effects on curvature perturbations.

Abstract

We study the effect of massive isocurvaton on density perturbations in quasi-single field inflation models, when the mass of the isocurvaton M becomes larger than the order of the Hubble parameter H. We analytically compute the correction to the power spectrum, leading order in coupling but exact for all values of mass. This verifies the previous numerical results for the range 0<M<3H/2 and shows that, in the large mass limit, the correction is of order H^2/M^2.

Figures (6)

• Figure 1: The correction to the leading power spectrum.
• Figure 2: The final result ${\cal C}$ (solid lines), ${\cal C}_1$ and ${\cal C}_2$ (dashed lines) as a function of $M/H$ for both $M\sim H$ and $M\gg H$. The dots are numerical results.
• Figure 3: Comparison of analytical and numerical results. The left, middle, right panels are analytical and numerical plots for $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}\equiv \mathcal{C}_1 + \mathcal{C}_2$ respectively. It is clear from the plot that $\mathcal{C}_1$ is decaying exponentially as a function of $\mu$. However the decay of $\mathcal{C}_2$ is not as fast as exponential.
• Figure 4: The analytical continuation of ${\cal C}_1(\mu)$ and ${\cal C}_2(\mu)$ to imaginary $\mu$ (corresponds to real $0<\nu<3/2)$. The solid line is ${\cal C}={\cal C}_1+{\cal C}_2$. The dots are numerical results from Chen:2009zp.
• Figure 5: The coefficients $\mathcal{C}$, $\mathcal{C}_1$ and $\mathcal{C}_2$ as a function of IR cutoff $x_c$. The left, middle, right panels are with $\mu=1$, $\mu=5$ and $\mu=10$ respectively.