Models of modular inflation and their phenomenological consequences

We study models of modular inflation of the form expected to arise from low energy effective actions of superstring theories. We argue on general grounds that the most likely models are small field models in which the inflaton moves about a Planck distance from an extremum of the potential. We then explain the generic difficulties in designing small field models of supergravity modular inflation. We show that if the Kähler potential of the inflaton is logarithmic as in perturbative string theories, then it is not possible to satisfy the slow-roll conditions for any superpotential. We find that if the corrections to the Kähler potential are large enough that it can be approximated by a canonical Kähler potential in the vicinity of the extremum, then viable slow-roll inflation is possible and we give a prescription for designing such models. In this case, several parameters have to be tuned to a fraction of a per cent. Generic models of this class predict a red spectrum of scalar perturbations and negligible spectral index running. They also predict a characteristic suppression of tensor perturbations despite the high scale of inflation. Consequently, a detection of primordial tensor anisotropies or spectral index running in cosmic microwave background observations in the foreseeable future will rule out this entire class of modular inflation models.