Cosmological perturbation theory with trinity of scalar fields

TL;DR

This work extends cosmological perturbation theory to three-field inflation with flat field space by formulating a semikinematic basis that yields explicit, coupled equations for the curvature and two isocurvature modes. It provides a complete background and perturbative treatment, introduces a practical rotation matrix to define adiabatic and isocurvature directions, and computes power spectra and cross-correlations for three representative scenarios, including interacting and noninteracting cases. The study reveals that rapid turns can leave distinctive imprints in the curvature and isocurvature spectra, and that two isocurvature modes combined with three-field dynamics enrich the phenomenology beyond two-field models. It also connects inflationary perturbations to radiation-era observables via transfer matrices, showing that three-field models offer a broader parameter space to model reheating and post-inflationary evolution, with potential implications for isocurvature constraints and future observations.

Abstract

We present an explicit formulation of cosmological perturbation theory for three-field models with a flat field space. By performing rotations to align one field with the direction of curvature perturbations and applying the same rotations to the other two field directions, we introduce the semikinematic basis, which is applicable to models with more than two fields. We derive the governing equations in this basis. We also stress a characteristic property of more-than-two-field models: the freedom in choosing the isocurvature perturbations. This framework enables the computation of the curvature and two isocurvature power spectra for any given potential. We numerically solve the background and perturbation equations for three distinct scenarios. First, to validate the consistency of our three-field formalism, we examine an effective two-field model inspired by the two-block case of the multigiant vacua matrix inflation scenario. Next, we analyze a purely three-field system without direct interfield interactions. Finally, we study a three-field case that incorporates direct interactions. For all scenarios, we numerically compute the curvature perturbation power spectra and highlight the effects of rapid turns on the spectra. Finally, we investigate the relationship between these quantities and the observables in the early radiation-dominated era. Through both general arguments and a simple example, we show that three-field inflation can yield a much richer phenomenology. This is particularly true when we assume the initial perturbations in the radiation era include two isocurvature modes.

Figures (20)

• Figure 1: This figure shows the adiabatic direction (green vectors), defined at each point as the tangent direction to the field trajectory (shown in red). At each point on the field trajectory we have a field basis (shown in black), and at each point we have a field perturbation vector (shown in blue vectors). We should transform this vector to a new basis at each point of which one of its components is along the adiabatic direction. The other two components should be on the surface orthogonal to $\hat{l}$.
• Figure 2: Field basis (black) and semikinematic basis (green) are shown. The adiabatic direction is determined by two angles, $\alpha$ and $\beta$. The field perturbation vector is also shown in blue. This vector's direction is also determined by two angles, $\alpha'$ and $\beta'$.
• Figure 3: Two-Field Case: the evolution of $\phi$ and $\chi$ with respect to the number of e-folds, and the trajectory in the field space is shown. $N_e=0$ is the end of inflation.
• Figure 4: Two-Field Case: the evolution of the Hubble parameter, H, and the slow-roll parameter, $\epsilon_H$, with respect to the number of e-folds is shown.
• Figure 5: Two-Field Case: the evolution of the power spectra (left plot) of the curvature and isocurvature modes, and their correlations (right plot), defined in \ref{['Cij']}, for the specific mode that exits the horizon 60 e-folds before the end of inflation, $k= 0.002 \space {\rm Mpc}^{-1}$, with respect to $N_e$. The correlation including the second isocurvature mode vanishes ($\tilde{\mathcal{C}}_{\mathcal{R} \mathcal{S}_2}$ is invisible since it coincides with $\tilde{\mathcal{C}}_{\mathcal{S}_1 \mathcal{S}_2}$), and the correlation between the curvature and the first isocurvature mode increases during the inflation. The increase is related to the turn rate increase, shown in the left plot of Fig. \ref{['PS-TR-ns-2']}