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We revisit the calculation of third order \acp{SIGW} and extend it from amonochromatic primordial power spectrum to a more general log-normal one. Weinvestigate the impact of third order SIGWs on \ac{SNR} of \ac{LISA} and\ac{PTA} observations, and find that third order SIGWs significantly contributeto the total energy density spectrum of \acp{GW} in high-frequency region. Fora primordial power spectrum amplitude of $A_{\zeta}=10^{-2}\sim 10^{-1}$, theeffects of third order SIGWs lead to a $40\%$ to $400\%$ increase in the SNRfor LISA. Additionally, our PTA data analysis reveals that third order SIGWsdiminish both the amplitude $A_{\zeta}$ and the peak frequency $f_*$ of theprimordial power spectrum.

Probing scalar induced gravitational waves with PTA and LISA: The Importance of third order correction