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We study the second-order scalar and density perturbations generated by Gaussian curvature perturbations and primordial gravitational waves in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second-order scalar perturbations. We show that the primordial gravitational waves might affect second-order energy density perturbation δ (2) = δ ρ (2) / ρ (0) significantly. The effects of primordial gravitational waves are studied in terms of different kinds of primordial power spectra. [ABSTRACT FROM AUTHOR]

The damping effect of the free-streaming neutrinos on the second order gravitational waves is investigated in detail. We solve the Boltzmann equation and give the anisotropic stress induced by neutrinos to second order. The first order tensor and its coupling with scalar perturbations induced gravitational waves are considered. We give the analytic equations of the damping kernel functions and finally obtain the energy density spectrum. The results show that the free-streaming neutrinos suppress the density spectrum significantly for low frequency gravitational waves and enlarge the logarithmic slope n in the infrared region ( k ≪ k ∗ ) of the spectrum. For the spectrum of k ∗ ∼ 10 - 7 Hz , the damping effect in the range of k < k ∗ is significant. The combined 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 ∼ 10 - 9 Hz , which may be probed by the pulsar timing arrays (PTA) in the future. [ABSTRACT FROM AUTHOR]