  Since WMAP and Planck some anomalous features appeared in the Cosmic
Microwave Background (CMB) large-angle anisotropy, the so-called anomalies. One
of these is the hemispherical power asymmetry, i.e. a difference in the average
power on the two hemispheres centered around (l, b) = (221, -20), which shows a
relatively high level of significance. Such an anomaly could be the signature
of a departure from statistical isotropy on large scales. Here we investigate
the physical origin of this anomaly using the Cosmological Gravitational Wave
Background (CGWB) detectable by future GW detectors. Indeed, the CGWB offers a
unique window to explore the early universe and we show that it can be used in
combination with CMB data to shed light on the statistical isotropy of our
universe. Specifically, we study the evolution of gravitons in the presence of
a modulating field in the scalar gravitational potentials accounting for the
hemispherical power asymmetry and we infer the amplitude of this modulating
field through a minimal variance estimator exploiting both constrained and
unconstrained realizations of the CGWB. We show that the addition of the CGWB
will allow an improvement in the assessment of the physical origin of the CMB
power asymmetry. Accounting for the expected performances of LISA and BBO, we
also show that the latter is expected to be signal-dominated on large-scales,
proving that the CGWB could be the keystone to assess the significance of this
anomaly.
