Boltzmann equations for astrophysical Stochastic Gravitational Wave Backgrounds scattering off of massive objects

The goal of this work is to present a set of coupled Boltzmann equations describing the intensity and polarisation Stokes parameters of the SGWB. Collision terms (as discussed e.g. in ref. [1]) which account for gravitational Compton scattering off of massive objects, are also included. This set resembles that for the CMB Stokes parameters, but the different spin nature of the gravitational radiation and the physics involved in the scattering process determine crucial differences. In the case of gravitational Compton scattering, due to the Rutherford angular dependence of the cross section, all the SGWB intensity multipoles of order ℓ are scattered out, therefore producing outgoing intensity anisotropies of any order ℓ if they are present in the incoming radiation. On the other hand, as already outlined in [1], SGWB linear polarisation modes can be expanded in a basis of spherical harmonics with m = ±4 and ℓ ≥ 4. This means that SGWB polarisation modes can be generated from unpolarised anisotropic radiation only with m = ±4, therefore requiring at least a hexadecapole anisotropy (ℓ ≥ 4) in the incoming intensity. Assuming a simplified toy model where scattering targets are localised in a small redshift range, we solve analytically the set of coupled Boltzmann equations to get explicit expressions for the intensity and polarisation angular power spectra. We confirm the contribution of the gravitational Compton scattering to the SGWB anisoptropies is extremely small for collisions with massive compact objects (BH and SMBH) in the frequency range of current and upcoming surveys. The system of coupled Boltzmann equations presented here provides a way to accurate estimate the total amount of anisotropies generated by multiple SGWB scattering processes off of massive objects, as well as the interplay between polarisation and intensity, during the GW propagation across the LSS of the universe. These results will be useful for the full treatment of the astrophysical SWGB anisotropies in view of upcoming gravitational waves observatories.