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 of modular slow-roll inflation are small field models in which the inflaton moves about a Planck distance from an extremum of the potential. We then focus on models in which the inflaton is the bosonic component of a single (complex) chiral superfield and explain the generic difficulties in designing small field models of modular inflation. We then show that if the Kaehler potential (KP) 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 KP are large enough so it can be approximated by a canonical KP in the vicinity of the extremum, then viable slow-roll inflation is possible. In this case, several parameters have to be tuned to a fraction of a percent. We give a prescription for designing successful small field supergravity models of inflation when the KP is canonical and calculate the slow-roll parameters from the superpotential parameters. Our results strengthen the case for models in which the moduli slowly roll about a Planck distance from a relatively high scale extremum that is located in the vicinity of the central region of moduli space units. 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. 
Ido Ben-Dayan, Ram Brustein, Senarath P. de Alwis