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Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory

Shi Pi, Misao Sasaki

TL;DR

The paper analyzes how turning trajectories in two-field inflation—with a light curvature mode and a heavy isocurvature mode—modify the curvature perturbation spectrum. Using the in-in formalism in a constant-turn setup, it derives analytic corrections from isocurvature mediation and compares them with an EFT description obtained by integrating out the heavy field. In the light-isocurvature regime, it provides analytic expressions for the correction coefficient C(\nu) that reproduce prior numerical results; in the heavy-isocurvature regime it shows C(\mu) ~ 1/(4 \mu^2), yielding a power-spectrum correction consistent with EFT via c_s^{-2} \approx 1 + 4 H^2/(\tilde{M}_{\mathrm{eff}}^2) (\dot\theta/H)^2. The results validate the EFT approach in the large-mass limit and clarify the parameter ranges where the two methods agree, with implications for features in the power spectrum and potential non-Gaussian signals.

Abstract

We revisit a two-component inflaton model with a turning trajectory in the field space, where the field slowly rolls down along the trajectory. We consider the case when the effective mass in the direction perpendicular to the trajectory, namely the isocurvature direction, is either of the same order as or much larger than the Hubble parameter. Assuming that the turning angular velocity is small, we compute analytically the corrections to the power spectrum of curvature perturbation caused by the mediation of the heavy isocurvature perturbation, and compare our analytic results with the numerical ones. Especially, when M_{eff}^2>>H^2, we find that it is proportional to M_{eff}^{-2}. This result is consistent with the one obtained previously by an effective field theory approach.

Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory

TL;DR

The paper analyzes how turning trajectories in two-field inflation—with a light curvature mode and a heavy isocurvature mode—modify the curvature perturbation spectrum. Using the in-in formalism in a constant-turn setup, it derives analytic corrections from isocurvature mediation and compares them with an EFT description obtained by integrating out the heavy field. In the light-isocurvature regime, it provides analytic expressions for the correction coefficient C(\nu) that reproduce prior numerical results; in the heavy-isocurvature regime it shows C(\mu) ~ 1/(4 \mu^2), yielding a power-spectrum correction consistent with EFT via c_s^{-2} \approx 1 + 4 H^2/(\tilde{M}_{\mathrm{eff}}^2) (\dot\theta/H)^2. The results validate the EFT approach in the large-mass limit and clarify the parameter ranges where the two methods agree, with implications for features in the power spectrum and potential non-Gaussian signals.

Abstract

We revisit a two-component inflaton model with a turning trajectory in the field space, where the field slowly rolls down along the trajectory. We consider the case when the effective mass in the direction perpendicular to the trajectory, namely the isocurvature direction, is either of the same order as or much larger than the Hubble parameter. Assuming that the turning angular velocity is small, we compute analytically the corrections to the power spectrum of curvature perturbation caused by the mediation of the heavy isocurvature perturbation, and compare our analytic results with the numerical ones. Especially, when M_{eff}^2>>H^2, we find that it is proportional to M_{eff}^{-2}. This result is consistent with the one obtained previously by an effective field theory approach.

Paper Structure

This paper contains 7 sections, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Light Isocurvature Modes: Quasi-Single Field Inflation
  3. Heavy Isocurvature Modes: Inflation under Effective Field Theory
  4. Conclusions and Discussions
  5. The Asymptotic Form of Hankel Functions of Real Order
  6. Gamma Function and Hypergeometric Function
  7. Hankel Function of Imaginary Order

Figures (6)

  • Figure 1: The second order interacting vertex $\mathcal{H}_2$ which is proportional to $\dot\theta/H$.
  • Figure 2: The leading order to the 2-point function of curvature perturbation, mediated by a isocurvature field.
  • Figure 3: The blue curve depicts an approximate analytic expression for $\mathcal{C}(\nu)$ given by Eqs. (\ref{['A final']}) and (\ref{['B final']}), discarding the contributions coming from the ultraviolet part $x>1$. The dots are those read from the numerical result obtained in Chen:2009zp. The summation over $k$ in these equations is truncated at $k\leq N=10^3$.
  • Figure 4: The red curve shows $\mathcal{C}(\mu)$ given by our analytical approximation (\ref{['CmuResult']}), which is also the one derived from the effective single-field approach (\ref{['cs in EFT']}). The black dots are the result of numerical integrations of (\ref{['def:C(mu)']}). There is a small deviation when $\mu\sim\mathcal{O}(10)$
  • Figure 5: This is the main contribution to large non-Gaussianity evaluated in Chen:2009zp.
  • ...and 1 more figures