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

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 sourced during inflation is evaluated only by integrating the anomaly equation. How these two contributions are distributed has not been seriously investigated. Meanwhile, a dominance of vacuum contribution is observed in some specific types of Bianchi spacetime with parity-violating gravitational fields, whose application to cosmology is not straightforward. One may wonder whether such a vacuum dominance takes place also in the system with chiral gravitational waves 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 resembles the one considered in the gravitational leptogenesis scenario. This approach allows us 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 conjecture that level-crossing does not seem to be efficient while the charge accumulation in the vacuum likely takes place. Phenomenological implication is also discussed in the context of gravitational leptogenesis.