Neutrino number asymmetry and cosmological birefringence

We study a new type of effective interactions in terms of the $CPT$-even dimension-six Chern-Simons-like term, which could originate from superstring theory, to generate the cosmological birefringence. We use the neutrino number asymmetry to induce a sizable rotation polarization angle in the data of the cosmic microwave background radiation polarization. The combined effect of the new term and the neutrino asymmetry provides an alternative way to understand the birefringence.