The Third Order Scalar Induced Gravitational Waves

Since the gravitational waves were detected by LIGO and VIRGO, it has been promising that lots of information about the primordial Universe could be learned by further observations on stochastic gravitational waves background. The studies on gravitational waves induced by primordial curvature perturbations are of great interest. The aim of this paper is to investigate the third order induced gravitational waves. Based on the theory of cosmological perturbations, the first order scalar induces the second order scalar, vector and tensor perturbations. At the next iteration, the first order scalar, the second order scalar, vector and tensor perturbations all induce the third order tensor perturbations. We present the energy density spectrum of the third order gravitational waves for a monochromatic primordial power spectrum. The shape of the energy density spectrum of the third order gravitational waves is different from that of the second order scalar induced gravitational waves. And it is found that the third order gravitational waves sourced by the second order scalar perturbations dominate the energy density spectrum.