Neutrino mass, dark matter, and inflation

We show that spontaneous breaking of global $B-L$ symmetry responsible for small neutrino masses via the seesaw mechanism provides a unified picture of hybrid inflation and dark matter if the scale of $B-L$ breaking is close to the GUT scale. The Majoron, which acquires a small mass due to Planck scale breaking of $B-L,$ is the dark matter candidate. The coupling of the Majoron field to the neutrino induces a small violation of CPT and Lorentz invariance at the present epoch. We discuss some of the phenomenological and cosmological implications of the model.