Primordial black holes and third order scalar induced gravitational waves

The process of primordial black hole (PBH) formation is inevitably accompanied by scalar induced gravitational waves (SIGWs). The strong correlation between PBH and SIGW signals may offer a promising approach to detecting PBHs in upcoming gravitational wave experiments, such as the Laser Interferometer Space Antenna (LISA). We investigate third order SIGWs during a radiation-dominated era in the case of the monochromatic primordial power spectrum \begin{document}$ \mathcal{P}_{\zeta}=A_{\zeta}k_*\delta\left(k-k_*\right) $\end{document}. For LISA observations, the relationships between the signal-to-noise ratio (SNR) and monochromatic primordial power spectrum are studied systematically, revealing that the effects of third order SIGWs extend the cutoff frequency from \begin{document}$ 2f_* $\end{document} to \begin{document}$ 3f_* $\end{document} and lead to an approximately 200% increase in the SNR for the frequency band from \begin{document}$ 10^{-5} $\end{document}Hz to \begin{document}$ 1.6\times 10^{-3} $\end{document}Hz, corresponding to PBHs with masses in the range \begin{document}$4\times 10^{-12}M_{\odot} \sim 10^{-7}M_{\odot}$\end{document}. We find that there is a critical value, \begin{document}$ A_*=1.76\times 10^{-2} $\end{document}, for the amplitude of the monochromatic primordial power spectra, such that when \begin{document}$ A_{\zeta}>A_* $\end{document}, the energy density of third order SIGWs is larger than that of second order SIGWs.