Primordial black holes and third order scalar induced gravitational waves

The process of \acp{PBH} formation would be inevitably accompanied by \acp{SIGW}. This strong correlation between \acp{PBH} and \acp{SIGW} signals could be a promising approach to detecting \acp{PBH} in the upcoming \ac{GW} experiments, such as \ac{LISA}. We investigate the third order \acp{SIGW} during a \ac{RD} era in the case of a monochromatic primordial power spectrum $\mathcal{P}_{\zeta}=A_{\zeta}k_*\delta\left(k-k_*\right)$. For \ac{LISA} observations, the relations between \ac{SNR} and monochromatic primordial power spectrum are studied systematically. It shows that the effects of third order \acp{SIGW} extend the cutoff frequency from $2f_*$ to $3f_*$ and lead to about $200\%$ increase of the \ac{SNR} for frequency band from $10^{-5}$Hz to $1.6\times 10^{-3}$Hz corresponding to \acp{PBH} with mass range $4\times 10^{-12}M_{\odot} \sim 10^{-7}M_{\odot}$. We find that there exists a critical value $A_*=1.76\times 10^{-2}$ for the amplitude of the monochromatic primordial power spectra, such that when $A_{\zeta}>A_*$, the energy density of third order \acp{SIGW} will be larger than the energy density of second order \acp{SIGW}.