Resolving Individual Signals in the Presence of Stochastic Background in Future Pulsar Timing Arrays

TL;DR

The paper addresses the challenge of detecting and characterizing individual CGWs in the presence of a stochastic GWB in future PTAs like SKA. It revisits the $\mathcal{F}$-statistic for single-source searches and introduces a GWB-informed noise model that treats unresolved GWs as SGWB, enabling more accurate recovery of CGW sky position and amplitude when the SGWB is strong. Through simple and realistic mock datasets based on SMBH populations, the study demonstrates that neglecting the SGWB biases CGW parameter estimates, while incorporating the SGWB into the noise model yields improved localization and strain accuracy, particularly in SGWB-dominated regimes. The findings support developing a robust, SGWB-aware $\mathcal{F}$-statistic pipeline for SKA-era PTAs and outline future steps toward jointly inferring CGW and GWB parameters, multi-source scenarios, and scalable computation.

Abstract

Recent pulsar timing array (PTA) observations have reported evidence of a gravitational wave background (GWB). If supermassive black holes (SMBHs) are indeed the primary source of this signal, future PTA observations, such as those from the Square Kilometer Array (SKA), are expected to simultaneously capture multiple continuous gravitational waves (CGWs) emitted by bright individual SMBH binaries alongside a gravitational wave background (GWB). To address this anticipated scenario in the SKA era, we revisit the F-statistic, a detection method for single source signals in PTA datasets, and introduce a new modeling that accounts for unresolved GWs as a stochastic GWB. Here, we applied this improved F-statistic to the mock datasets that include both CGW and GWB and evaluated how accurately F-statistic can identify the parameters of CGW. As a result, we demonstrate that our approach can successfully improve the estimation of the sky position and the amplitude of CGW, particularly when the GWB is dominant over white noise. This work serves as an initial step toward developing an efficient and robust algorithm based on the F-statistic for future PTA observations.

Figures (9)

• Figure 1: Results of case 1. Predicted parameter values obtained from 100 realizations by applying the standard $\mathcal{F}$-statistic to mock datasets: one dataset includes white noise and a single CGW (blue circles with error bars or downward arrows), while the other includes white noise, a single CGW, and GWB (red diamonds with error bars or downward arrows). The left panel illustrates the deviation of the source position from the true position of the injected CGW source, evaluated at different noise levels. Each point with a downward arrow means that the deviation of the angle is within the indicated value for $68\%$ of the realizations. The right panel shows the predicted GW strain amplitude of the injected CGW source for different noise levels. Each point and error bar shows the median and the $16\%$-$84\%$ region of each distribution, respectively. The gray dashed line represents the true GW strain amplitude of the injected signal.
• Figure 2: Results of case 2. Predicted parameter values obtained from 100 realizations by applying the standard $\mathcal{F}$-statistic (blue circle with error bars or downward arrows) and GWB-informed $\mathcal{F}$-statistic (red diamond median with error bars or downward arrows). The mock datasets include white noise, a single CGW, and GWB. The left panel shows the deviation of the source position from the true position of the injected CGW source, evaluated at different noise levels. Each point with a downward arrow means that the deviation of the angle is within the indicated value for $68\%$ of the realizations. The right panel shows the estimated GW strain amplitude of the injected CGW source for different noise levels. The gray line represents the true GW strain of the injected signal. Each point and error bar shows the median and the $16\%$-$84\%$ region of each distribution, respectively.
• Figure 3: An example of the skymap of the reduced log-likelihood ratio $\mathcal{F}$ is shown for the standard (left panel) and GWB-informed (right panel) $\mathcal{F}$-statistics in case 2, which includes both the GWB and white noise with $\sigma=10$ ns. The stars and the green square represent the sky positions of pulsars and the injected CGW source, respectively. The blue circle (in the right panel) and the orange diamond (in the left panel) show the positions with the maximum value of $\mathcal{F}=\mathcal{F}_\mathrm{max}$ These heatmaps show the magnitude of $-2\mathcal{F}$ at each position within the region $-2\mathcal{F}\leq-2\mathcal{F}_\mathrm{max}+10$, which corresponds to $\Delta\chi^2 \leq 10$
• Figure 4: The GW spectrum is numerically computed using the SMBH population from the Agnostic Model. Five different colours represent five different realizations. The circles indicate the loudest GW in each frequency bin, and the lines represent the assembly of unresolved GW. The Gray dashed line shows the power-law spectrum with $\alpha_g=-2/3$, which is normalised to match the estimate from the latest NANOGrav's observation 2023ApJ...951L...8A.
• Figure 5: Results of Hard search ($N_\mathrm{p}=30$, $\sigma=100$$\mathrm{nHz}$). The cumulative error distribution of the predictions was statistically evaluated over 100 realizations. The left panel shows the direction error of the injected CGW source, and the right panel shows the error of the GW strain. The blue dashed line and the orange solid line represent the results estimated by the standard and GWB-informed $\mathcal{F}$-statistic, respectively. The $68\%$ values obtained for the two modellings are indicated at the bottom right of each figure.