Constraining the origin of the nanohertz gravitational-wave background by pulsar timing array observations of both the background and individual supermassive binary black holes

Reconstructed GWB spectra from the BBH population model constrained by the NANOGrav 15-year free-spectrum data (cyan and magenta lines). The grey violins are from \citet{NG23hd} and \cite{NG23constraint}, showing the square root of the cross-correlated timing residual power ($\rho$) in the left panel and the characteristic strain amplitude $h\rmc$ in the right panel, respectively, for the NANOGrav 15-year free-spectrum GWB posteriors obtained by assuming the HD-w/MP+DP+CURN model. The cyan line in each panel shows the reconstructed GWB spectra for the reference BBH population model, while the thin magenta lines show the resulting GWB spectra for 200 random draws from the posterior parameter distributions of $\talpha$, $\tomega$, and $\tepsilon$. Note that when constraining the BBH population model, only the leftmost five frequency-bin data are used, and the $-2/3$ power-law spectrum model has been assumed. See details in Section~\ref{sec:BBHmodel}.

Synthetic GWB strain spectra from cosmic BBHs generated from the reference BBH population model. The lower panel shows the GWB characteristic strain amplitude $h\rmc$ as a function of the GW frequency $f$ for 10 realizations of the cosmic BBHs (each one represented by a black curve with significant fluctuation at high frequencies). For comparison, the canonical $-2/3$ power-law strain spectrum for the same model is indicated by the cyan line. For each realization, the top contributor to $h\rmc$ within each NANOGrav frequency bin is marked by a red filled circle if that contributor dominates the frequency bin, while an open magenta circle if not. In the upper panel, the thin histogram shows the probability $P_{\rm dom}(f)$ that a given realization contains a dominant BBH source within a given NANOGrav frequency bin, and the thick dashed curve shows the cumulative probability $P_{\rm dom,cum}(f)$ that a given realization contains at least one dominant BBH source in all NANOGrav frequency bins left to a given frequency. Both probability distributions are evaluated based on $1000$ realizations generated from the reference BBH population model. The grey vertical lines in both panels denote the leftmost $14$ NANOGrav frequency bins. See details in Section~\ref{sec:detectability}.

Individual BBH sources in the $h_0$--$f$ plane. The lower panel shows the BBH sources in $10$ realizations generated from the reference BBH population model. For each realization, we record the top $30$ contributors to the synthetic GWB characteristic strain amplitude $h\rmc$ within each NANOGrav frequency bin (filled circles). Among them, those dominating the $h\rmc$ of their frequency bins are marked in red colors (they are the same sources as those red filled circles in Fig.~\ref{fig:GWBSynthesisHigh}). The green curve represents the 95\% upper limit on individual BBHs derived from the NANOGrav $15$-year data set \citep{NG23indv}, and the blue curve represents the sensitivity curve on individual sources for the adopted CPTA configuration assuming an S/N threshold of $3$. In the upper panel, the thin histogram shows the expected number of individual BBH sources detectable by CPTA in each NANOGrav frequency bin $\langle N\rangle(f)$, and the thick dashed curve shows the corresponding cumulative expected number of detectable sources $\langle N\rangle_{\rm cum}(f)$. Both quantities are the average evaluated based on $1000$ realizations of cosmic BBHs from the reference BBH population model. The grey vertical lines in each panel indicate the leftmost $14$ NANOGrav frequency bins, and the magenta vertical dashed lines denote the three frequencies explored by CPTA in \citet{CPTA23hd}. See details in Section~\ref{sec:detectability}.

Legends are the same as that for Fig.~\ref{fig:LoudConstrained}, except that the mock BBHs are generated from the BBH population model adopting the MBH-host galaxy relationship given by \citet{KH13}, which leads to a GWB with characteristic strain amplitude only a fraction $\sim 28\%$ of the detected GWB signal. See details in Section~\ref{sec:detectability}.

Expected constraints ($95\%$ upper limits) on the ratio of the BBH-induced GWB strain amplitude to the detected GWB strain amplitude $\calH_{\rm BBH}$, as a function of the CPTA observation time $T\obs$, if there is no detection of individual BBHs within $T\obs$. Red, green, and blue colors correspond to the cases in which the threshold $\snr$ for individual BBH detections are $\rhoth=3$, $5$, and $8$, respectively. The corresponding $\snr$ of the GWB detection by CPTA is shown by the top axis, and the BBH-induced characteristic strain amplitude at frequency $f=1\iyr$ is shown by the right axis. The vertical dotted line marks the $3.4\yr$ CPTA observation. See details in Section~\ref{sec:detectability}.