This paper introduces a single, self-consistent model that treats both the diffuse gravitational‑wave background (GWB) and the brightest individual supermassive black‑hole binaries (SMBHBs) at the same time. The authors propose a useful detection number called the characteristic number of sources, Nc, that captures whether the background comes from many weak sources or a few bright ones. They apply the model to a simulated NANOGrav 15‑year data set that copies real data properties and report concrete limits and detection probabilities for isolated sources.

The researchers built a probabilistic model for the total gravitational‑wave strain as a sum of a stochastic background and one or more individual continuous waves (CW), the latter coming from the brightest SMBHBs. Pulsar Timing Arrays (PTAs) like NANOGrav look for nanohertz gravitational waves by monitoring tiny timing shifts in the signals from millisecond pulsars. In the model, bright binaries are treated as directional, deterministic CW signals, while the remaining binaries form an isotropic, random background. The model returns probability distributions (PDFs) for the background and for the brightest source, and those PDFs are used together in a hierarchical Bayesian analysis.

A central quantity is Nc, the characteristic number of sources that would produce the observed characteristic strain at a reference frequency. The authors choose a reference frequency of one cycle per year (fref = yr−1) and note that Nc changes with frequency roughly as f−11/3. When Nc is large, the GWB behaves like a Gaussian random process and follows the familiar power‑law strain spectrum. When Nc is small, random fluctuations from discrete sources become pronounced and individual sources are more likely to stand out.

Applying their method to the simulated NANOGrav 15‑year data, the team derives direct astrophysical limits on the strain of individually resolvable SMBHBs. They find that 21 out of 114 SMBHB candidates reported from active galactic nucleus observations are in tension with the simulated NANOGrav observations under their joint model. By contrast, the original NANOGrav upper‑limit analysis would place only one candidate in tension. Using the same simulated data, they estimate the probability of detecting gravitational waves from an isolated SMBHB in the 15‑year data at signal‑to‑noise ratio SNR = 5 to be about 2%. Projecting to an expected 20‑year NANOGrav data set raises that to about 5%, and the chance of seeing an outlier with SNR ≈ 2 in the 20‑year data is estimated at about 40%.