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 P_R of curvature perturbations and the spectral index n_s 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 N_HI* 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 P_R and n_s, we find that, in the case of standard hybrid inflation, the values obtained for the grand unification scale are close to its supersymmetric value M_GUT=2.86 x 10^16 GeV, the relevant coupling constant is relatively large (0.005-0.14), and N_HI* between about 10 and 21.7. In the case of shifted [smooth] hybrid inflation, the grand unification scale can be identified with M_GUT provided that N_HI*=21 [N_HI*=18].