F-term hybrid inflation followed by a Peccei-Quinn phase transition

We consider a cosmological setup, based on renormalizable superpotential terms, in which a super-heavy scale F-term hybrid inflation is followed by a Peccei-Quinn phase transition, resolving the strong CP and mu problems of the minimal supersymmetric standard model. We show that the field which triggers the Peccei-Quinn phase transition can remain after inflation well above the Peccei-Quinn scale thanks to (i) its participation in the supergravity and logarithmic corrections during the inflationary stage and (ii) the high reheat temperature after the same period. As a consequence, its presence influences drastically the inflationary dynamics and the universe suffers a second period of reheating after the Peccei-Quinn phase transition. Confronting our inflationary predictions with the current observational data, we find that, for about the central value of the spectral index, the grand unification scale can be identified with its supersymmetric value for the relevant coupling constant kappa similar or equal to 0.002 and, more or less, natural values, +/-(0.01-0.1), for the remaining parameters. On the other hand, the final reheat temperature after the Peccei-Quinn phase transition turns out to be low enough so as the gravitino problem is avoided.