Symmetry Breaking and Reheating after Inflation in No-Scale Flipped SU(5)

No-scale supergravity and the flipped SU(5)$\times$U(1) gauge group provide an ambitious prototype string-inspired scenario for physics below the string scale, which can accommodate the Starobinsky-like inflation favoured by observation when the inflaton is associated with one of the singlet fields associated with neutrino mass generation. During inflation, the vacuum remains in the unbroken GUT phase, and GUT symmetry breaking occurs later when a field with a flat direction (the flaton) acquires a vacuum expectation value. Inflaton decay and the reheating process depend crucially on GUT symmetry breaking, as decay channels open and close, depending on the value of the flaton vacuum expectation value. Here, we consider the simultaneous cosmological evolution of both the inflaton and flaton fields after inflation. We distinguish weak, moderate and strong reheating regimes, and calculate in each case the entropy produced as all fields settle to their global minima. These three reheating scenarios differ in the value of a Yukawa coupling that introduces mass mixing between the singlets and the ${\bf 10}$s of SU(5). The dynamics of the GUT transition has an important impact on the production of gravitinos, and we also discuss the pattern of neutrino masses we expect in each of the three cases. Finally, we use recent CMB limits on neutrino masses to constrain the reheating models, finding that neutrino masses and the cosmological baryon asymmetry can both be explained if the reheating is strong.