  We analyze the cosmological implications of F-term hybrid inflation with a
subdominant Fayet--Iliopoulos D-term whose presence explicitly breaks a
D-parity in the inflaton-waterfall sector. This scenario of inflation, which is
called F_D-term hybrid model for brevity, can naturally predict lepton number
violation at the electroweak scale, by tying the mu-parameter of the MSSM to an
SO(3)-symmetric Majorana mass m_N, via the vacuum expectation value of the
inflaton field. We show how a negative Hubble-induced mass term in a
next-to-minimal extension of supergravity helps to accommodate the present CMB
data and considerably weaken the strict constraints on the theoretical
parameters, resulting from cosmic string effects on the power spectrum P_R. The
usual gravitino overabundance constraint may be significantly relaxed in this
model, once the enormous entropy release from the late decays of the ultraheavy
waterfall gauge particles is properly considered. As the Universe enters a
second thermalization phase involving a very low reheat temperature, which
might be as low as about 0.3 TeV, thermal electroweak-scale resonant
leptogenesis provides a viable mechanism for successful baryogenesis, while
thermal right-handed sneutrinos emerge as new possible candidates for solving
the cold dark matter problem. In addition, we discuss grand unified theory
realizations of F_D-term hybrid inflation devoid of cosmic strings and
monopoles, based on the complete breaking of an SU(2) subgroup. The F_D-term
hybrid model offers rich particle-physics phenomenology, which could be probed
at high-energy colliders, as well as in low-energy experiments of lepton
flavour or number violation.
