Milli-second pulsars with highly stable periods can be considered as very precise clocks and can be used for pulsar timing array (PTA) which attempts to detect nanoheltz gravitational waves (GWs) directly. Main sources of nanoheltz GWs are supermassive black hole (SMBH) binaries which have sub-pc-scale orbits. On the other hand, a SMBH binary which is in an earlier phase and has pc-scale orbit emits ultra-low-frequency ($\lesssim 10^{-9}\,\mathrm{Hz}$) GWs cannot be detected with the conventional methodology of PTA. Such binaries tend to obtain high eccentricity, possibly $\sim 0.9$. In this paper, we develop a formalism for extending constraints on GW amplitudes from single sources obtained by PTA toward ultra-low frequencies considering the waveform expected from an eccentric SMBH binary. GWs from an eccentric binaries are contributed from higher harmonics and, therefore, have a different waveform those from a circular binary. Furthermore, we apply our formalism to several hypothetical SMBH binaries at the center of nearby galaxies, including M87, using the constraints from NANOGrav's 11-year data set. For a hypothetical SMBH binary at the center of M87, the typical upper limit on the mass ratio is $0.16$ for eccentricity of $0.9$ and semi-major axis of $a=1~\mathrm{pc}$, assuming the binary phase to be the pericenter.
An Accurate Modeling of Nano-hertz Gravitational Wave Signal from Eccentric Supermassive Binary Black Holes: An Essential Step Toward a Robust Discovery
Effects of dark dipole radiation on eccentric supermassive black hole binary inspirals
School of Astronomy and Space Science, University of Chinese Academy of Sciences (UCAS)（中国科学院大学天文与空间科学学院）；School of Physics, Nanjing University（南京大学物理学院）
Dissecting the nanohertz gravitational wave sky: Frequency-correlated anisotropy induced by eccentric supermassive black hole binaries
Università degli Studi di Milano-Bicocca（米兰-比可卡大学）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比可卡分部）；INAF – Osservatorio Astronomico di Cagliari（意大利国家天体物理研究所卡利亚里天文台）；ASTRON, Netherlands Institute for Radio Astronomy（荷兰射电天文研究所）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；European Space Agency (ESA), European Space Research and Technology Centre (ESTEC)（欧洲空间局（ESA）欧洲空间研究与技术中心（ESTEC））
Electromagnetic Emission and Orbital Evolution of Eccentric Supermassive Black Hole Binaries in Retrograde Disks
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究院）；Niels Bohr International Academy, Niels Bohr Institute（尼尔斯·玻尔国际学院，尼尔斯·玻尔研究所）；Space Telescope Science Institute（太空望远镜科学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）；Columbia University（哥伦比亚大学）；New York University（纽约大学）
Targeted search for eccentric supermassive binary black holes in OJ 287 and nearby galaxy clusters with PPTA DR3
Institute for Frontier in Astronomy and Astrophysics & Faculty of Arts and Sciences, Beijing Normal University（北京师范大学前沿天文与天体物理研究所及文理学院）；School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；School of Physics and Astronomy, Monash University（蒙纳士大学物理与天文学院）；ARC Centre for Excellence for Gravitational Wave Discovery (OzGrav)（ARC引力波发现卓越中心（OzGrav））；Australia Telescope National Facility, CSIRO, Space & Astronomy（澳大利亚望远镜国家设施，CSIRO空间与天文部）；School of Physics, University of Melbourne（墨尔本大学物理学院）；Center for Gravitation, Cosmology and Astrophysics, Department of Physics, University of Wisconsin–Milwaukee（威斯康星大学密尔沃基分校物理系引力、宇宙学与天体物理中心）
Three-body encounters in black hole discs around a supermassive black hole: The disc velocity dispersion and the Keplerian tidal field determine the eccentricity and spin-orbit alignment of gravitational wave mergers
Constraining Orbital Eccentricity of a Supermassive Black Hole Binary Candidate PKS 2131-0211
Joint Astronomy Programme, Department of Physics, Indian Institute of Science（印度科学研究所物理系联合天文学项目）；Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research（塔塔基础研究院天文与天体物理学系）；Max-Planck-Institut für Radioastronomie（马克斯·普朗克射电天文研究所）；Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata（加尔各答印度科学教育研究所物理科学系）
The NANOGrav 12.5-year data set: A computationally efficient eccentric binary search pipeline and constraints on an eccentric supermassive binary candidate in 3C 66B
Statistics of the supermassive black hole gravitational wave background anisotropy
Laboratory of High Energy and Computational Physics, NICPB（高能计算物理实验室，NICPB）；Departament of Cybernetics, Tallinn University of Technology（塔林科技大学网络系）；Dipartimento di Fisica e Astronomia, Università degli Studi di Padova（帕多瓦大学物理与天文学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
Spoon-Feeding Giant Stars to Supermassive Black Holes: Episodic Mass Transfer from Evolving Stars and Their Contribution to the Quiescent Activity of Galactic Nuclei
Stellar binaries incident on supermassive black hole binaries: implications for double tidal disruption events, calcium-rich transients, and hypervelocity stars
