The detection and characterization of the stochastic gravitational wave background (SGWB) is one of the main goals of gravitational wave (GW) experiments. The observed SGWB will be the combination of GWs from cosmological (as predicted by many models describing the physics of the early universe) and astrophysical origins, which will arise from the superposition of GWs from unresolved sources whose signal is too faint to be detected. Therefore, it is important to have a proper modeling of the astrophysical SGWB (ASGWB) in order to disentangle the two signals; moreover, this will provide additional information on astrophysical properties of compact objects. Applying the cosmic rulers formalism, we compute the observed ASGWB angular power spectrum, hence using gauge-invariant quantities, accounting for all effects intervening between the source and the observer. These are the so-called projection effects, which include Kaiser, Doppler, and gravitational potentials effect. Our results show that these projection effects are the most important at the largest scales, and they contribute to up to tens of percent of the angular power spectrum amplitude, with the Kaiser term being the largest at all scales. While the exact impact of these results will depend on instrumental and astrophysical details, a precise theoretical modeling of the ASGWB will necessarily need to include all these projection effects.
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