Self-resonance during preheating: The case of $\alpha$-attractor models

TL;DR

This paper introduces a new class of solutions for Hill-type equations in the preheating phase of the early Universe, specifically for $oldsymbol{eta}$-attractor models, using perturbative techniques to analyze curvature perturbations.

Contribution

It develops an analytical framework for solving higher-order Hill equations in $oldsymbol{eta}$-attractor models, extending previous methods to include cubic and quartic potential terms.

Findings

Derived expressions for Floquet exponents of curvature perturbations.

Validated analytical solutions with numerical simulations.

Potential applications in predicting primordial black holes and gravitational waves.

Abstract

In this paper, for the first time, we obtain a new class of solutions for the Hill-type differential equations, which emerge in the preheating self-resonance phase of the expanding Universe. We study, in particular, the class of symmetric and asymmetric scalar field potentials coming from the so-called $\alpha$-attractor models of the early Universe cosmology. By making a series expansion of the potential and employing perturbative techniques we reformulate the Mukhanov-Sasaki equation, which captures the dynamics of the curvature perturbation in these models, into a Hill equation. This last includes higher-order terms that were never solved in the literature. Namely, those coming from the cubic and quartic contributions of the scalar field potential. Then, we derive the expressions for the Floquet exponents of the Mukhanov-Sasaki variable. Our analytical results are then compared with numerical computations, showing a good agreement and thus making this method valuable for obtaining theoretical predictions with new observational applications in the contexts of Primordial Black Holes and Scalar-Induced Gravitational Waves.