Reheating and dark radiation after fibre inflation

We study perturbative reheating at the end of fibre inflation where the inflaton is a closed string modulus with a Starobinsky-like potential. We first derive the spectral index n(s) and the tensor-to-scalar ratio r as a function of the number of efoldings and the parameter R which controls slow-roll breaking corrections. We then compute the inflaton couplings and decay rates into ultra-light bulk axions and visible sector fields on D7-branes wrapping the inflaton divisor. This leads to a reheating temperature of order 10(10) GeV which requires 52 efoldings. Ultra-light axions contribute to dark radiation even if Delta N-eff is almost negligible in the generic case where the visible sector D7-stack supports a non-zero gauge flux. If the parameter R is chosen to be small enough, n(s) similar or equal to 0.965 is then in perfect agreement with current observations while r turns out to be of order r similar or equal to 0.007. If instead the flux on the inflaton divisor is turned off, Delta N-eff less than or similar to 0.6 which, when used as a prior for Planck data, requires n(s) similar or equal to 0.99. After R is fi xed to obtain such a value of n(s), primordial gravity waves are larger since r similar or equal to 0.01.