On the Complementarity of Pulsar Timing and Space Laser Interferometry for the Individual Detection of Supermassive Black Hole Binaries

Gravitational waves coming from supermassive black hole binaries (SMBHBs) are targeted by both the Pulsar Timing Array (PTA) and Space Laser Interferometry (SLI). The possibility of a single SMBHB being tracked first by PTA, through inspiral, and later by SLI, up to merger and ring-down, has been previously suggested. Although the bounding parameters are drawn by the current PTA or the upcoming Square Kilometer Array (SKA), and by the New Gravitational Observatory (NGO), derived from the Laser Interferometer Space Antenna (LISA), this paper also addresses sequential detection beyond specific project constraints. We consider PTA-SKA, which is sensitive from 10^-9 to p × 10^-7 Hz (p = 4, 8), and SLI, which operates from s × 10^-5 up to 1 Hz (s = 1, 3). An SMBHB in the range of 2 × 10⁸-2 × 10⁹ M _⊙ (the masses are normalized to a (1 + z) factor, the redshift lying between z = 0.2 and z = 1.5) moves from the PTA-SKA to the SLI band over a period ranging from two months to fifty years. By combining three supermassive black hole (SMBH)-host relations with three accretion prescriptions, nine astrophysical scenarios are formed. They are then related to three levels of pulsar timing residuals (50, 5, 1 ns), generating 27 cases. For residuals of 1 ns, sequential detection probability will never be better than 4.7 × 10^-4 yr^-2 or 3.3 × 10^-6 yr^-2 (per year to merger and per year of survey), according to the best and worst astrophysical scenarios, respectively; put differently this means one sequential detection every 46 or 550 years for an equivalent maximum time to merger and duration of the survey. The chances of sequential detection are further reduced by increasing values of the s parameter (they vanish for s = 10) and of the SLI noise, and by decreasing values of the remnant spin. The spread in the predictions diminishes when timing precision is improved or the SLI low-frequency cutoff is lowered. So while transit times and the SLI signal-to-noise ratio (S/N) may be adequate, the likelihood of sequential detection is severely hampered by the current estimates on the number—just a handful—of individual inspirals observable by PTA-SKA, and to a lesser extent by the wide gap between the pulsar timing and space interferometry bands, and by the severe requirements on pulsar timing residuals. Optimization of future operational scenarios for SKA and SLI is briefly dealt with, since a detection of even a single event would be of paramount importance for the understanding of SMBHBs and of the astrophysical processes connected to their formation and evolution.