  We obtain the equation governing the evolution of the cosmological
gravitational-wave background, accounting for the presence of cosmic neutrinos,
up to second order in perturbation theory. In particular, we focus on the epoch
during radiation dominance, after neutrino decoupling, when neutrinos yield a
relevant contribution to the total energy density and behave as collisionless
ultra-relativistic particles. Besides recovering the standard damping effect
due to neutrinos, a new source term for gravitational waves is shown to arise
from the neutrino anisotropic stress tensor. The importance of such a source
term, so far completely disregarded in the literature, is related to the high
velocity dispersion of neutrinos in the considered epoch; its computation
requires solving the full second-order Boltzmann equation for collisionless
neutrinos.
