An important prediction of inflation is the production of a primordial stochastic gravitational wave background. Observing this background is challenging due to the weakness of the signal and the simultaneous presence of an astrophysical background generated by many unresolved late-time sources. One possible way to distinguish between the two is to examine their anisotropies.
In this talk, I present two works on the study of gravitational wave anisotropies in the cosmological background arising from axion inflation, where the inflaton is a pseudo-Nambu-Goldstone boson coupled to gauge fields. In this scenario, tensor modes arise not only from the standard amplification of vacuum fluctuations present in any inflationary model, but also from the inverse decay process of the produced gauge fields. In the first work, I study the cross-correlation between gravitational wave anisotropies and CMB anisotropies, which arises because scalar and tensor perturbations share a common origin in this model. This correlation is quantified by the normalized primordial correlator between the curvature perturbation and the gravitational wave energy density and turns out to be of the order of 10^(-4)-10^(-2). The observability of this result depends on the amplitude of the anisotropies in the gravitational wave spectrum, encoded in the normalized correlator of the gravitational wave energy densities, which is the subject of the second work. This correlator can reach values of the order of 10^(-1), showing that axion inflation can indeed generate large anisotropies with the potential to be observed by gravitational wave detectors within a reasonable time frame.