Constraints on ultra-low-frequency gravitational waves from an eccentric supermassive black hole binary

Milli-second pulsars with highly stable periods can be considered as very precise clocks and used for pulsar timing array (PTA) which attempts to detect nanoheltz gravitational waves (GWs) directly. Main sources of GWs are supermassive black hole (SMBH) binaries have sub-pc-scale orbits. On the other hand, a SMBH binary is in an earlier phase has pc-scale orbit emits ultra-low-frequency ($\lesssim 10^{-9}\,\mathrm{Hz}$) cannot detected conventional methodology PTA. Such tend obtain high eccentricity, possibly $\sim 0.9$. In this paper, we develop formalism extending constraints on GW amplitudes from single obtained by PTA toward ultra-low frequencies considering waveform expected eccentric binary. contributed higher harmonics and, therefore, different those circular Furthermore, apply our several hypothetical at center nearby galaxies, including M87, using NANOGrav's 11-year data set. For typical upper limit mass ratio $0.16$ eccentricity $0.9$ semi-major axis $a=1~\mathrm{pc}$, assuming pericenter.