  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.
