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The damping effect of the free-streaming neutrinos on the second ordergravitational waves is investigated in detail. We solve the Boltzmann equationand give the anisotropic stress induced by neutrinos to second order. The firstorder tensor and its coupling with scalar perturbations induced gravitationalwaves are considered. We give the analytic equations of the damping kernelfunctions and finally obtain the energy density spectrum. The results show thatthe free-streaming neutrinos suppress the density spectrum significantly forlow frequency gravitational waves and enlarge the logarithmic slope $n$ in theinfrared region ($k \ll k_*$) of the spectrum. For the spectrum of $k_*\sim10^{-7}$Hz, the damping effect in the range of $k<k_*$ is significant. Thecombined effect of the first and second order could reduce the amplitude by$30\%$ and make $n$ jump from $1.54$ to $1.63$ at $k\sim 10^{-9}$Hz, which maybe probed by the pulsar timing arrays (PTA) in the future.

Impact of the free-streaming neutrinos to the second order induced gravitational waves