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Semianalytic calculation of the gravitational wave spectrum induced by curvature perturbations

The stochastic gravitational wave (GW) background is secondarily and inevitably induced by the primordial curvature perturbations beyond the first order of the cosmological perturbation theory. We analytically calculate the integration kernel of the power spectrum of the induced GWs, which is the universal part independent of the spectrum of the primordial curvature perturbations, in the radiation-dominated era and in the matter-dominated era. We derive fully analytic expressions of the GW spectrum when possible. As a minor update, we study the case of the top-hat function as the spectrum of the curvature perturbations. We also discuss generalization in the presence of multiple cosmological eras with different equations of state.