We investigate the second order gravitational waves induced by the primordial scalar and tensor perturbations during a radiation-dominated era. The explicit expressions of the power spectra of the second order GWs are presented. We calculate the energy density spectra of the second order GWs for the monochromatic primordial power spectra ${\mathcal{P}}_{\zeta }=A_{\zeta }{k}_{*}\delta (k-k_{*})$ and ${\mathcal{P}}_h=A_h{k}_{*}\delta (k-k_{*})$. For large $k$ $(k>k_{*})$, the primordial tensor perturbation will affect the total power spectrum of second order GWs significantly. For $f_{*}=1.3\times {10}^{-3}$ and $A_{\zeta }=0.02$, the effects of the primordial tensor perturbation will lead to an around 100% increase of the signal-to-noise ratio (SNR) for LISA observations if the tensor-to-scalar ratio $r=A_h/A_{\zeta }>0.4$.

• Received 15 September 2022
• Accepted 22 February 2023

DOI:https://doi.org/10.1103/PhysRevD.107.063510