Anisotropy of the astrophysical gravitational wave background: analytic expression of the angular power spectrum and correlation with cosmological observations

Unresolved sources of gravitational waves are at the origin of the existence of a stochastic gravitational wave background. While the computation of is mean density as a function of frequency in a homogeneous and isotropy universe is standard lore, the computation of its anisotropies requires to understand the coarse graining from local systems, to galactic scales and then to cosmology.  We derive an expression of the gravitational wave energy density valid in any general spacetime.  We then specialize to a perturbed Friedmann-Lemaˆıtre spacetime in order to determine the angular power spectrum of this stochastic background as well as its correlation with other cosmological probes, such the distribution of galaxies and weak lensing.