\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."> Primordial black holes and third order scalar induced gravitational waves -
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