Cross-correlating Astrophysical and Cosmological Gravitational Wave
  Backgrounds with the Cosmic Microwave Background
  General Relativity provides us with an extremely powerful tool to extract at
the same time astrophysical and cosmological information from the Stochastic
Gravitational Wave Backgrounds (SGWBs): the cross-correlation with other
cosmological tracers, since their anisotropies share a common origin and the
same perturbed geodesics. In this letter we explore the cross-correlation of
the cosmological and astrophysical SGWBs with Cosmic Microwave Background (CMB)
anisotropies, showing that future GW detectors, such as LISA or BBO, have the
ability to measure such cross-correlation signals. We also present, as a new
tool in this context, constrained realization maps of the SGWBs extracted from
the high-resolution CMB {\it Planck} maps. This technique allows, in the
low-noise regime, to faithfully reconstruct the expected SGWB map by starting
from CMB measurements.
