Probing past mergers of supermassive black holes with pulsar timing arrays: The role of pulsar terms

This paper investigates a previously unconsidered class of gravitational wave sources for Pulsar Timing Arrays (PTAs) known as zombie binaries. These are supermassive black hole binaries (SMBHBs) that merged before the commencement of timing observations, yet remain detectable through the pulsar term in their timing residual waveforms. Studying these systems offers a novel window for studying the most massive SMBHBs in our local universe.

The mechanism of zombie binaries

The signal produced by a gravitational wave source in PTAs consists of two distinct contributions: the Earth term, which corresponds to the integrated strain signal at Earth today, and the pulsar term, which is produced by the metric perturbation at the moment of pulse emission. Because of the kiloparsec-scale distances between monitored pulsars and Earth, there is a significant geometrical time delay in the pulsar term.

While an SMBHB that has already coalesced may have an Earth term that vanishes or lies outside the PTA band, its associated pulsar terms can still fall within the nanohertz frequency range. Consequently, PTAs can probe binaries that coalesced on timescales up to thousands of years in the past. These signals appear as quasi-monochromatic sine waves where the frequency and amplitude are specific to each pulsar based on their relative position and distance from Earth.

Modeling and detection efficiency

To estimate the likelihood of observing these systems, the research utilizes a Schechter-like parametric model for comoving SMBHB merger density. This approach accounts for current uncertainties in the SMBHB population by evaluating three distinct models:

• Model M1: Includes a large number of relatively low-mass binaries.

• Model M2: Favors a smaller number of higher-mass systems.

• Model M3: A third model used to test sensitivity across different population assumptions.

The detection efficiency is determined by simulating thousands of binaries with varying extrinsic parameters—such as sky location, inclination angle, and orbital phase—to calculate the probability that a zombie binary yields a Signal-to-Noise Ratio (SNR) exceeding a threshold of 3.

Observatory sensitivity

The study compares the capacity of past, current, and future PTA configurations to detect these signals. The results indicate that detection probability is highly dependent on the observatory's technical specifications:

• EPTA (DR2new): Lacks sufficient sensitivity, with a probability P(N z > 0) 3%.

• IPTA (DR3): Serves as a conservative baseline with a detection probability of approximately 22% to 23% for models M2 and M3.

• SKA: Expected to achieve sufficient sensitivity with a probability P(N z > 0) 90% for all population models, potentially detecting several bright zombie binaries.

Characteristics of detectable signals

The most detectable bright zombie binaries possess specific physical properties. These systems are generally characterized by:

• High Mass: They are typically "massive systems with M c,r > 10 9 M."

• Low Redshift: Most are located at z < 1.5.

• Optimal Frequency: Their signals reside in a sweet spot of approximately 10-20 nanohertz, situated between the high-frequency regime limited by timing precision and the low-frequency regime dominated by red noise.

Identifying these systems through their sky-dependent frequency signature across multiple pulsars could allow for the coherent reconstruction of signals, providing critical data on SMBHB dynamics during the final thousands of years before coalescence.

1. Temporal-Lag Signal Reconstruction (TLSR) Modules

• Improvement: Integrating a dual-term signal processing architecture into Recurrent Neural Networks (RNNs) or Transformers that explicitly models echo signals (the pulsar term equivalent). This involves training the model to correlate current state observations with delayed, spatially-offset signals that represent the system's historical state.

• Capability: In large-scale distributed computing or global IoT networks, the AI can reconstruct the history of a systemic event (e.g., a power surge or a network breach) by analyzing the time-delayed echoes of that event as they propagate through different nodes. This allows the system to identify the exact origin and timing of a past event that has already passed through certain sub-sectors but is still detectable via delayed signals.

2. Distributed Spatio-Temporal Correlation Transformers (DSTCT)

• Improvement: Implementing a multi-node attention mechanism that prioritizes geometric correlation patterns (similar to the Hellings-Downs quadrupolar correlation) over individual node signal strength. This moves away from local threshold-based detection toward a global, spatially-aware correlation metric.

• Capability: In cybersecurity and high-frequency trading, the AI can detect stealth attacks or market manipulations that are intentionally kept below the noise floor of any single sensor or node. By identifying specific, low-SNR spatial correlation patterns across the entire network, the system can differentiate between localized stochastic noise and a coordinated, systemic intrusion.

3. Chirp-Rate Parameterized State Estimators (CPSE)

• Improvement: Replacing static frequency-domain analysis with dynamic chirp modeling, where the AI predicts the evolution of a signal's frequency and amplitude based on its mass (the magnitude of the underlying systemic driver). This uses the 0-PN (Post-Newtonian) evolution logic to map current signal characteristics to a trajectory of coalescence.

• Capability: In industrial predictive maintenance and structural health monitoring, the AI can identify zombie components—machinery that has already entered a terminal failure trajectory (a merger of state) but whose degradation signal is currently in a low-frequency, non-critical band. The AI can predict the exact time of catastrophic failure by modeling the chirp (the rate of change in vibration or electrical frequency) rather than waiting for a threshold breach.

By monitoring the times of arrival of radio pulses from millisecond pulsars, Pulsar Timing Arrays (PTAs) serve as unique gravitational wave (GW) laboratories in the nanohertz band. To date, the primary astrophysical sources of GWs targeted in this frequency range have been inspiraling supermassive black hole binaries (SMBHBs) on circular and eccentric orbits. In this work, we demonstrate that, thanks to the so-called pulsar term in the timing residual waveform of GW signals, PTAs can probe individual SMBHBs that merged before timing observations began. We refer to the latter as zombie binaries. Using SMBHB population models consistent with current PTA constraints, we find that while the probability of detecting such systems in existing PTA datasets remains low, the Square Kilometer Array observatory is expected to achieve sufficient sensitivity to have a few zombie binaries with optimal matched filter signal-to-noise ratios exceeding 3 in its data. Although their confident identification might be challenging, this new class of PTA sources opens a novel window for studying the most massive SMBHBs in our local universe.