  Since the first LIGO/Virgo detection, Gravitational Waves (GWs) have been
very promising as a new complementary probe to understand our Universe. One of
the next challenges of GW search is the detection and characterization of the
stochastic gravitational wave background (SGWB), that is expected to open a
window on the very early Universe (cosmological background) and to provide us
new information on astrophysical source populations (astrophysical background).
One way to characterize the SGWB and to extract information about its origin is
through the cross-correlation with other cosmological probes. To this aim, in
this paper, we explore the cross-correlation between the astrophysical
background anisotropies and the Cosmic Microwave Background (CMB) ones. Such a
signal is sensitive to primordial non-Gaussianity (nG) through the GW bias.
Thus, we study the capability of next generation space-based interferometers to
detect such a cross-correlation signal and to constrain primordial nG.
