Abstract
Axion dark matter is interesting as it allows a natural coupling to the gravitational Chern-Simons term. In the presence of an axion background, the gravitational Chern-Simons term produces parity violating effects in the gravitational sector, in particular on the propagation of gravitational waves. Previously, it has been shown that the coherent oscillation of the axion field leads to a parametric amplification of gravitational waves with a specific frequency. In this paper, we focus on the parity violating effects of the Chern-Simons coupling and show the occurrence of gravitational birefringence. We also find deviation from the speed of light of the velocity of the gravitational waves. We give constraints on the axion-Chern-Simons coupling constant and the abundance of axion dark matter from the observation of GW170817 and GRB170817A.