Starobinsky Inflation from String Theory?

Starobinsky inflation is currently one of the best models concerning agreement with cosmological data. Despite this observational success, it is still lacking a robust embedding into a UV complete theory. Previous efforts to derive Starobinsky inflation from string theory have been based on the derivation of higher derivative curvature terms from the low-energy limit of ten-dimensional string theory. This approach is however known to fail due to the difficulty to tame the effect of contributions proportional to the Ricci scalar to a power larger than two. In this paper we investigate an alternative attempt which exploits instead the ubiquitous presence of scalar fields in string compactifications combined with the fact that Starobinsky inflation can be recast as Einstein gravity coupled to a scalar field with a precise potential and conformal coupling to matter fermions. We focus in particular on type IIB K\"ahler moduli since they have shown to lead to exponential potentials with a Starobinsky-like plateau. We consider three classes of moduli with a different topological origin: the volume modulus, bulk fibre moduli, and blow-up modes. The only modulus with the correct coupling to matter is the volume mode but its potential does not feature any plateau at large field values. Fibre moduli admit instead a potential very similar to Starobinsky inflation with a natural suppression of higher curvature corrections, but they cannot reproduce the correct conformal coupling to matter. Blow-up modes have both a wrong potential and a wrong coupling. Our analysis implies therefore that embedding Starobinsky inflation into string theory seems rather hard. Finally, it provides a detailed derivation of the coupling to matter of fibre moduli which could be used as a way to discriminate Starobinsky from fibre inflation.