Curvature and isocurvature perturbations in two-field inflation

TL;DR

This paper systematically analyzes curvature and isocurvature perturbations in two-field inflation with non-standard kinetic terms. It derives analytic expressions for the spectra and cross-correlations at Hubble crossing and performs detailed numerical evolutions from sub- to super-Hubble scales, showing that isocurvature perturbations can significantly transfer power to curvature during inflation. Through canonical, non-canonical, and roulette inflation examples, it demonstrates that final curvature perturbations can be dominated by curvature–isocurvature coupling rather than fluctuations along the inflationary trajectory, impacting the predicted spectral index. The results stress the importance of including isocurvature dynamics when comparing multi-field inflation models to observations and highlight potential constraints from post-inflation reheating on any surviving isocurvature modes.

Abstract

We study cosmological perturbations in two-field inflation, allowing for non-standard kinetic terms. We calculate analytically the spectra of curvature and isocurvature modes at Hubble crossing, up to first order in the slow-roll parameters. We also compute numerically the evolution of the curvature and isocurvature modes from well within the Hubble radius until the end of inflation. We show explicitly for a few examples, including the recently proposed model of `roulette' inflation, how isocurvature perturbations affect significantly the curvature perturbation between Hubble crossing and the end of inflation.

Figures (6)

• Figure 1: Examples of classical inflationary trajectories for double inflation with canonical kinetic terms (left), double inflation with non-canonical kinetic terms (center) and roulette inflation (right). The details of the models are described in Section \ref{['exex']}. Subsequent tens of efolds are indicated along the curves.
• Figure 2: Predictions for the spectra and correlations of the perturbations in double inflation with canonical kinetic terms. Thick lines show the numerical results for ${\cal P}_{\cal R}$, ${\cal P}_{\cal S}$ and ${\cal C}_{\cal{RS}}$ or $\tilde{\cal C}$ normalized to the single-field result (\ref{['sifi']}), respectively. Circles, stars and squares indicate the predictions of eqs. (\ref{['pra']}), (\ref{['pca']}) and (\ref{['psa']}), respectively. Thin dashed lines indicate the predictions of eqs. (\ref{['shr']}), (\ref{['shc']}) and (\ref{['shs']}), respectively. The coupling $B$ between the curvature and isocurvature perturbations is also shown.
• Figure 3: Predictions for the spectra and correlations of the perturbations in double inflation with non-canonical kinetic terms. Thick lines show the numerical results for ${\cal P}_{\cal R}$, ${\cal P}_{\cal S}$ and ${\cal C}_{\cal{RS}}$ or $\tilde{\cal C}$ normalized to the single-field result (\ref{['sifi']}), respectively. Circles, stars and squares indicate the predictions of eqs. (\ref{['pra']}), (\ref{['pca']}) and (\ref{['psa']}), respectively. Thin dashed lines indicate the predictions of eqs. (\ref{['shr']}), (\ref{['shc']}) and (\ref{['shs']}), respectively. The coupling $B$ between the curvature and isocurvature perturbations is also shown.
• Figure 4: Predictions for the spectra and correlations of the perturbations in roulette inflation. Thick lines show the numerical results for ${\cal P}_{\cal R}$, ${\cal P}_{\cal S}$ and ${\cal C}_{\cal{RS}}$ or $\tilde{\cal C}$ normalized to the single-field result (\ref{['sifi']}), respectively. Circles, stars and squares indicate the predictions of eqs. (\ref{['pra']}), (\ref{['pca']}) and (\ref{['psa']}), respectively. Thin dashed lines indicate the predictions of eqs. (\ref{['shr']}), (\ref{['shc']}) and (\ref{['shs']}), respectively. The coupling $B$ between the curvature and isocurvature perturbations is also shown.
• Figure 5: Evolution of the coefficients $C^{\rm (c)}_{\sigma s}$ and $C^{\rm (nc)}_{\sigma s}$ defined in (\ref{['coedefnew']}), parametrizing the coupling between the curvature and isocurvature modes: (a) for double inflation with non-canonical kinetic terms and (b) for roulette inflation.