  The initial conditions on the anisotropies of the stochastic
gravitational-wave background of cosmological origin (CGWB) largely depend on
the mechanism that generates the gravitational waves. Since the CGWB is
expected to be non-thermal, the computation of the initial conditions could be
more challenging w.r.t. the Cosmic Microwave Background (CMB), whose
interactions with other particles in the early Universe lead to a blackbody
spectrum. In this paper, we show that the initial conditions for the
cosmological background generated by quantum fluctuations of the metric during
inflation deviate from adiabaticity. These primordial gravitational waves are
indeed generated by quantum fluctuations of two independent degrees of freedom
(the two polarization states of the gravitons). Furthermore, the CGWB plays a
negligible role in the Einstein's equations, because its energy density is
subdominant w.r.t. ordinary matter. Therefore, the only possible way to compute
the initial conditions for inflationary gravitons is to perturb the
energy-momentum tensor of the gravitational field defined in terms of the
small-scale tensor perturbation of the metric. This new and self-consistent
approach shows that a large, non-adiabatic initial condition is present even
during the single-field inflation. Such a contribution enhances the total
angular power spectrum of the CGWB compared to the standard adiabatic case,
increasing also the sensitivity of the anisotropies to the presence of
relativistic and decoupled particles in the early Universe. In this work we
have also proved that our findings are quite general and apply to both
single-field inflation and other scenarios in which the CGWB is generated by
the quantum fluctuations of the metric, like the curvaton.
