On the inefficiency of fermion level-crossing under the parity-violating spin-2 gravitational field

A bstract Gravitational chiral anomaly connects the topological charge of spacetime and the chirality of fermions. It has been known that the chirality is carried by the particles (or the excited states) and also by vacuum. While the gravitational anomaly equation has been applied to cosmology, distinction between these two contributions has been rarely discussed. In the study of gravitational leptogenesis, for example, lepton asymmetry associated with the chiral gravitational waves (GWs) sourced during inflation is evaluated only by integrating the anomaly equation. This approach, however, does not reveal how these two contributions are distributed in this scenario. Meanwhile, the dominance of vacuum contribution is observed in some specific types of Bianchi spacetime with parity-violating gravitational fields. One may wonder whether such a vacuum dominance takes place also in the system with chiral GWs around the flat background, which is more suitable for application to realistic cosmology. In this work, we apply an analogy between U(1) electromagnetism and the weak gravity to the spacetime that captures the characteristics of the one considered in the gravitational leptogenesis. With this approach, we try to obtain intuitive understanding of the fermion chirality generation under the parity-violating spin-2 gravitational field. By assuming the emergence of Landau level-like dispersion relation in our setup, we observe that spin-2 nature seems to make the level-crossing inefficient, indicating that the chrial charge is likely to accumulate in the vacuum. On this basis, phenomenological implications for gravitational leptogenesis are discussed.