$Sp(4,\mathbb{Z})$ modular inflation

TL;DR

This paper explores inflationary models based on the Siegel modular group $Sp(4, ext{Z})$, extending the $SL(2, ext{Z})$ framework to three moduli, and constructs cosmological models compatible with observational data using genus 2 invariants.

Contribution

It introduces $Sp(4, ext{Z})$-based inflation models utilizing genus 2 invariants, expanding modular inflation to multiple moduli with realistic cosmological potentials.

Findings

Models produce plateau-like potentials consistent with Planck 2018 data.

Two-field inflation scenarios resemble E-model and T-model frameworks.

Modified polynomial $ ext{alpha}$-attractor models fit ACT and SPT data with larger spectral index.

Abstract

We investigate inflation models governed by the Siegel modular group $Sp(4,\mathbb{Z})$. The $Sp(4,\mathbb{Z})$ group extends the $SL(2,\mathbb{Z})$ framework from one modulus to three moduli while preserving the hyperbolic geometry of the K\"ahler potential, allowing for the construction of cosmological $\alpha$-attractor models. In this context, we use genus $g=2$ absolute invariants to construct inflationary potentials within specific subspaces of the Siegel moduli space. These models are driven by the imaginary components of the moduli $\tau$ and naturally yield plateau-like potentials consistent with Planck 2018 observations in large field limit. We employ two-dimensional complex subspaces to realize E-model and T-model like two-field inflation scenarios. We explore the subspace of complex dimension one to construct a modified polynomial $\alpha$-attractor model, which can accommodate the larger spectral index $n_s$ favored by recent ACT and SPT data, particularly in the larger $N$ regime.