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

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