The U(1)$_{ B-L}$ symmetry could be restored during inflation, since the BICEP2 results suggest a GUT-scale inflation with the Hubble parameter, $H_{ \rm inf} \simeq 10^{ 14}$ GeV, close to the U(1)$_{ B-L}$ breaking scale. We consider a scenario in which the ${ B-L}$ Higgs field dominates the Universe after inflation, and mainly decays into the U(1)$_{ B-L}$ gauge bosons, whose subsequent decays reheat the Universe. Interestingly, if one of the right-handed neutrinos is extremely light and behaves as dark radiation or hot dark matter, its abundance is determined by the $B-L$ charge assignment and the relativistic degree of freedom in plasma. We find that $\Delta N_{ \rm eff}$ takes discrete values between $0.188$ and $0.220$ in the standard model plus three right-handed neutrinos, depending on whether the decay into heavier right-handed neutrinos are kinematically accessible or not. In the fiveness U(1)$_{ \bf 5}$ case, we find that $\Delta N_{ \rm eff}$ takes discrete values between $0.313$ and $0.423$. The tension between BICEP2 and { \it Planck} can be partially relaxed by dark radiation.
Categories: hep-ph