Modular Invariant Starobinsky Inflation and the Species Scale

TL;DR

This paper proposes a modular invariant inflaton potential based on the species scale, leading to a model resembling Starobinsky inflation with a direct link between the number of species and inflation duration.

Contribution

It introduces a novel automorphic invariant form for the inflaton potential tied to the species scale, extending Starobinsky inflation to modular invariant frameworks.

Findings

Potential resembles Starobinsky model at large moduli

Number of e-folds proportional to the number of species

Inflation parameters align with observational data

Abstract

Potentials in cosmological inflation often involve scalars with trans-Planckian ranges. As a result, towers of states become massless and their presence pushes the fundamental scale not to coincide with $M_{\rm P}$ but rather with the $species\, scale$, $\Lambda$. This scale transforms as an automorphic form of the theory's duality symmetries. We propose that the inflaton potential should be 1) an automorphic invariant form, non-singular over all moduli space, 2) depending only on $\Lambda$ and its field derivatives, and 3) approaching constant values in the large moduli to ensure a long period of inflation. These conditions lead to the proposal $V \sim \lambda(\phi, \phi^*)$, with $\lambda = G^{i\bar{j}} (\partial_i \Lambda)(\partial_{\bar{j}} \Lambda) / \Lambda^2$, determining the 'species scale convex hull'. For a single elliptic complex modulus with $SL(2, Z)$ symmetry, this results in an inflaton potential $V \simeq (\text{Im} \tau)^2 |\tilde{G}_2|^2 / N^2$, with $N \simeq -\log(\text{Im} \tau |\eta(\tau)|^4)$, where $\eta$ is the Dedekind function and $\tilde{G}_2$ the Eisenstein modular form of weight 2. Surprisingly, this potential at large modulus resembles that of the Starobinsky model. We compute inflation parameters yielding results similar to Starobinsky's, but extended to modular invariant expressions. Interestingly, the number of e-folds is proportional to the number of species in the tower, $N_e \simeq N$, and $\epsilon \simeq \Lambda^4$ at large moduli, suggesting that the tower of states plays an important role in the inflation process.