Probing scalar induced gravitational waves with PTA and LISA: the importance of third order correction

We revisit the calculation of third order scalar induced gravitational waves (SIGWs) and extend it from a monochromatic primordial power spectrum to a more general log-normal one. We investigate the impact of third order SIGWs on signal-to-noise ratio (SNR) of Laser Interferometer Space Antenna (LISA) and pulsar timing array (PTA) observations, and find that third order SIGWs significantly contribute to the total energy density spectrum of gravitational waves (GW) in high-frequency region. For a primordial power spectrum amplitude of A ζ = 10 -2 ∼ 10 -1 , the effects of third order SIGWs lead to a 40% to 400% increase in the SNR for LISA. Additionally, our PTA data analysis reveals that third order SIGWs diminish both the amplitude A ζ and the peak frequency f * of the primordial power spectrum.