Modular Invariant Starobinsky Inflation and the Species Scale

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.