Reducing the spectral index in F-term hybrid inflation through a complementary modular inflation

We consider two-stage inflationary models in which a superheavy scale F-term hybrid inflation is followed by an intermediate scale modular inflation. We confront these models with the restrictions on the power spectrum PR of curvature perturbations and the spectral index ns implied by the recent data within the power-law cosmological model with cold dark matter and a cosmological constant. We show that these restrictions can be met provided that the number of e-foldings NHI∗ suffered by the pivot scale k∗=0.002/Mpc during hybrid inflation is appropriately restricted. The additional e-foldings required for solving the horizon and flatness problems can be naturally generated by the subsequent modular inflation. For central values of PR and ns, we find that, in the case of standard hybrid inflation, the values obtained for the grand unification scale are close to its supersymmetric value MGUT=2.86×1016GeV, the relevant coupling constant is relatively large (≈0.005-0.14), and 10≲NHI∗≲21.7. In the case of shifted [smooth] hybrid inflation, the grand unification scale can be identified with MGUT provided that NHI∗≃21 [NHI∗≃18].