Statistics of the supermassive black hole gravitational wave background anisotropy

We study the statistical properties of the anisotropy in the gravitational wave (GW) background originating from supermassive black hole (SMBH) binaries. We derive the distribution of the GW anisotropy power spectrum coefficients, $C_{l\ge 1}\mathrm{/}{C}_0$, in scenarios including environmental effects and eccentricities of the SMBH binaries. Although the mean of $C_{l\ge 1}\mathrm{/}{C}_0$ is the same for all multipoles, we show that their distributions vary, with the low $l$ distributions being the widest. We also find a strong correlation between spectral fluctuations and the anisotropy in the GW signal. We show that the GW anisotropy can break the degeneracy between the scenarios including environmental effects or eccentricity. In particular, we find that existing NANOGrav constraints on GW anisotropy begin to constrain SMBH scenarios with strong environmental effects.