A unified view of anisotropies in the astrophysical gravitational wave background

In the literature different approaches have been proposed to compute the anisotropies of the astrophysical gravitational wave background. The different expressions derived, although starting from our work Cusin, Pitrou, Uzan, Phys.Rev.D96, 103019 (2017) [1], seem to differ. This article compares the various theoretical expressions proposed so far and provides a separate derivation based on a Boltzmann approach. We show that all the theoretical formula in the literature are equivalent and boil down to the one of Ref. [1] when a proper matching of terms and integration by parts are performed. The difference between the various predictions presented for anisotropies in a cosmological context can only lie in the astrophysical modeling of sources, and neither in the theory nor in the cosmological description of the large scale structures. Finally we comment on the gauge invariance of expressions.