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We argue that a cosmic neutrino background that carries non-zero leptoncharge develops gravitational instabilities. Fundamentally, these instabilitiesare related to the mixed gravity-lepton number anomaly. We have explicitlycomputed the gravitational Chern-Simons term which is generatedquantum-mechanically in the effective action in the presence of a lepton numberasymmetric neutrino background. The induced Chern-Simons term has a twofoldeffect: (i) gravitational waves propagating in such a neutrino backgroundexhibit birefringent behaviour leading to an enhancement/suppression of thegravitational wave amplitudes depending on the polarisation, where themagnitude of this effect is related to the size of the lepton asymmetry; (ii)Negative energy graviton modes are induced in the high frequency regime, whichleads to very fast vacuum decay producing, e.g., positive energy photons andnegative energy gravitons. From the constraint on the present radiation energydensity, we obtain an interesting bound on the lepton asymmetry of theuniverse.

Gravitational Instabilities of the Cosmic Neutrino Background with Non-zero Lepton Number