Gravitational Instabilities of the Cosmic Neutrino Background with
  Non-zero Lepton Number
  We argue that a cosmic neutrino background that carries non-zero lepton
charge develops gravitational instabilities. Fundamentally, these instabilities
are related to the mixed gravity-lepton number anomaly. We have explicitly
computed the gravitational Chern-Simons term which is generated
quantum-mechanically in the effective action in the presence of a lepton number
asymmetric neutrino background. The induced Chern-Simons term has a twofold
effect: (i) gravitational waves propagating in such a neutrino background
exhibit birefringent behaviour leading to an enhancement/suppression of the
gravitational wave amplitudes depending on the polarisation, where the
magnitude of this effect is related to the size of the lepton asymmetry; (ii)
Negative energy graviton modes are induced in the high frequency regime, which
leads to very fast vacuum decay producing, e.g., positive energy photons and
negative energy gravitons. From the constraint on the present radiation energy
density, we obtain an interesting bound on the lepton asymmetry of the
universe.
