Slow Roll during the Waterfall Regime: The Small Coupling Window for SUSY Hybrid Inflation

It has recently been pointed out that a substantial amount of e-folds can occur during the waterfall regime of hybrid inflation. Moreover, Kodama et.al. have derived analytic approximations for the trajectories of the inflaton and of the waterfall fields. Based on these, we derive here the consequences for F- and D-term SUSY hybrid inflation: A substantial amount of e-folds may occur in the waterfall regime, provided kappa << M^2/M_P^2, where kappa is the superpotential coupling, M the scale of symmetry breaking and M_P the reduced Planck mass. When this condition is amply fulfilled, a number of e-folds much larger than N_e\approx60 can occur in the waterfall regime and the scalar spectral index is then given by the expression found by Kodama et.al. n_s=1-4/N_e. This value may be increased up to unity, if only about N_e e-folds occur during the waterfall regime, such that the largest observable scale leaves the horizon close to the critical point of hybrid inflation, what can be achieved for kappa\approx10^(-13) and M\approx5x10^(12) GeV in F-term inflation. Imposing the normalization of the power spectrum leads to a lower bound on the scale of symmetry breaking.