Forecasting Supermassive Black Hole Binary Gravitational Wave Probes: Prospects for Future Pulsar Timing Array and Space-Borne Detectors

TL;DR

The paper develops a population-frame framework for forecasting SMBHB detections across future GW observatories by anchoring the SMBHB population in observed dual AGN fractions and the AGN X-ray luminosity function. It formalizes the detectable-source count from the intrinsic merger-rate density via $d^4 n/(dMdzdqd\log f)$ and employs a detailed S/N analysis for both PTA and space-borne detectors, including realistic noise models and two initial-separation prescriptions for space missions. The main results show that SKA-PTA could yield its first individual SMBHB detections within a few years, with $\sim 10^{2}$--$10^{3}$ detections after a decade, while space-borne detectors are expected to observe $\sim 1$--$20$ events per year (potentially up to $\sim 20$ with mass-dependent separations), predominantly at $z\lesssim 2$ and in the mass ranges $M_{\rm tot} \sim 10^{5}$--$10^{6} M_\odot$ (LISA/Taiji) and up to $\sim 10^{8} M_\odot$ (TianQin). The analysis also highlights that including pulsar red noise and SGWB reduces detectable counts by about an order of magnitude, underscoring the need for advanced data-analysis pipelines. Overall, the work provides quantitative benchmarks for multi-band and multi-messenger SMBHB campaigns and informs planning for next-generation GW astronomy.

Abstract

We present a comprehensive framework for predicting the detection prospects of supermassive black hole binaries (SMBHBs) by future gravitational wave (GW) observatories, examining both space-borne detectors (LISA, Taiji, TianQin) and next-generation pulsar timing array (PTA) combined with the Square Kilometre Array (SKA-PTA). Leveraging dual active galactic nucleus (AGN) fractions and AGN X-ray luminosity functions, we systematically evaluate the detectable SMBHB populations with a detection threshold of signal-to-noise ratio $\geq 5$ for each GW observatory. Our analysis reveals that space-borne detectors are expected to identify approximately $\sim 1 \text{--} 2$ to $\sim 20$ events per year, depending on the SMBHB orbital evolution prescriptions. On the other hand, SKA-PTA demonstrates the potential to reach the first GW detection from individual SMBHBs within a few years of observation and achieve detectable GW source counts of $10^2 \text{--} 10^3$ after about 10 years, depending on PTA configurations. These facilities will significantly improve SMBHB detectability and enable characterization of their properties across different frequency bands.

Figures (6)

• Figure 1: Contours of averaged S/N for LISA (left side) and SKA-PTA (right side) as functions of redshift and total binary mass $M_{\mathrm{tot}} = (1+q)M$, assuming equal-mass binaries ($q=1$). Shaded regions indicate detectable parameter space where the S/N exceeds the detection threshold ($\rho_{\mathrm{crit}} = 5$). For SKA-PTA, we consider an observational duration of $T_{\mathrm{obs}} = 10$ yr with the fiducial parameter set ($\sigma_t, N_{\mathrm{pl}}, \Delta t$) = ($30$ ns, $500$, $0.02$ yr).
• Figure 2: The detection rate of detectable GW source counts as a function of PTA observation time. The solid-blue curve indicates the fiducial parameter case in both panels. The gray horizontal lines correspond to the source counts unity. Left: Change the number of MSPs with $N_{\mathrm{pl}} = 1000$ (green dashed), 500 (blue solid), 100 (orange dotted), and fix $\sigma_t = 30$ ns. Right: Change the RMS timing residuals with $\sigma_t = 10$ ns (green dashed), 30 ns (blue solid), 50 ns (orange dotted), and fix $N_{\mathrm{pl}} = 500$.
• Figure 3: The detection number of GW sources per year from each redshift range, assuming fiducial parameter set. The dashed orange, dash-dotted green, dotted red, dash-dotted purple, dashed brown, and solid blue represent the number of sources at $0 < z < 1, 1 < z < 2, 2 < z < 3, 3 < z < 4, 4< z < 5$, and the total detection number, respectively.
• Figure 4: Left: Cumulative count rate per year as a function of the redshift for different space-borne GW observatories. The black solid curve indicates the LISA detection rate with mass-dependent separation prescription, while the dash-dotted blue, dashed-orange, and dotted-green curves represent the detection rate by LISA, TianQin, and Taiji assuming a fiducial initial separation case of $a_0$=10 kpc, respectively. The gray horizontal line corresponds to the count rate unity. Right: Same as the left figure, but showing it as a function of the primary BH mass.
• Figure 5: The detection number of GW sources per year against different noise configurations. The blue solid curve indicates the noise model, which only contains white noise, the same as in Figure \ref{['fig:PTA_cumulative']} (WN Only; our baseline model), while the orange dashed curve denotes the noise containing sampled red noise and SGWB in addition to white noise (WN + RN + SGWB). Here $A_{\mathrm{SGWB}} = 2.4 \times 10^{-15}$NANOGrav:2023gor and $\mathrm{log_{10}}A_{{\rm RN},i} \in [-17,-13]$ ,$\gamma_{\mathrm{RN},i} \in [1,5]$Babak:2024yhu are adopted for SGWB and red noise, respectively.