Controlled Penumbral Inflation from Monodromic Valleys

Abstract

Realizing controlled, single-clock inflation in string theory is fundamentally obstructed by the backreaction of heavy moduli. We show that in the \emph{penumbra} -- the near-boundary regime of complex-structure moduli space where asymptotic symmetries are partially broken -- this obstruction can be exactly quantified. We derive a covariant control theorem demonstrating that local branch data dictate whether a monodromic valley supports a controlled inflationary plateau, thereby isolating the first controlled penumbral inflationary window. The result turns the penumbra from a geometric regime into a dynamical filter. In the axion-saxion effective theory, a branch-displacing odd term generates a plateau when $\hat{I}”\equiv p+2\hat{I}½-q>0$, while covariant single-clock control further requires $p<2$, or $p=2$ with $12A_p{m}^2/(d{V}_0)\gg 1$ over the observational window. This splits penumbral valleys into no plateau, uncontrolled plateau, and controlled plateau before global completion is attempted. We identify a minimal analytic family with a closed-form valley and an invariant attractor equation for the full two-field dynamics, providing the first exactly solvable penumbral realization that remains predictive under the next penumbral order. The controlled corridor targets $r\sim 10^{-3}$ with the correlated running $\hat{I}{\pm }_s\simeq -r/2$ for the $d=q=1$ benchmark, providing a falsifiable target for LiteBIRD/CMB-S4.

5 pages +7 pages supplementry material , any comments are highly appreciated and welcomed, added discussion