Effective-one-body modelling of eccentric supermassive black hole binaries for Pulsar Timing Array

TL;DR

This work delivers an eccentric SMBHB gravitational-wave template tailored for Pulsar Timing Arrays by embedding an Effective-One-Body (EOB) dynamics framework with conservative dynamics to $2$PN order and dissipative terms up to higher PN orders. It provides both a fast approximate evolution and a full numerical integration path, and demonstrates that eccentric binaries exhibit significant orbital evolution and a rich harmonic structure in PTA residuals, necessitating a full, PN-consistent treatment to connect Earth- and pulsar-term signals. The model yields a detailed frequency-domain depiction of residuals via harmonics of the mean anomaly and precession phase, highlighting substantial pulsar-term power at low frequencies and the need to incorporate multiple harmonics for robust detection and parameter inference. The approach advances PTA GW searches by enabling accurate, efficient templates for eccentric SMBHBs, with implications for SMBH demographics in the local Universe and potential extension to LISA-band stellar-mass binaries.

Abstract

Pulsar Timing Arrays (PTAs) observations will detect gravitational waves (GWs) from the early inspiral phase of supermassive black hole binaries (SMBHBs) with orbital periods of weeks to years. Current PTA analyses generally assume circular binaries; however, dynamical interactions with the surrounding environment can prevent complete circularisation, allowing SMBHBs to retain appreciable eccentricities. In this work, we present a gravitational waveform model for eccentric binaries based on the Effective-One-Body (EOB) formalism, designed for continuous GW searches in PTA data. The model is accurate up to the second post-Newtonian (2PN) order for the conservative dynamics and up to post-leading order for the radiation-reaction terms. We provide both a numerically precise and a computationally efficient approximate implementation and evaluate the latter's accuracy against the full model over a broad range of eccentricities and initial orbital frequencies. Our results show that a substantial region of the parameter space exhibits pronounced orbital evolution, much stronger than in the circular case. We demonstrate the rich harmonic structure of timing residuals induced by eccentric GWs. Properly characterising eccentric binaries is an essential step toward detecting GWs in PTA data and interpreting the results, ultimately improving our understanding of the supermassive black hole population in the local Universe.

Figures (10)

• Figure 1: Two diagrams representing the eccentric binary properties and phases.
• Figure 2: Evolution of $\gamma(t)$ as a function of time computed over $20 yrs$. Different colors correspond to different frequencies, while initial eccentricity is fixed to $e_0 = 0.5$ and $\mathcal{M}_c = 10^{9.2}M_{\odot}$. Approximate solution $<\gamma(t)>$, is dashed and compared with the "exact" time integral of $\dot{\gamma} = \dot{\phi} - \dot{\xi}$.
• Figure 3: Evolution of $\xi(t)$ as a function of time computed over $20 yrs$. Different colors correspond to different initial eccentricities, while initial azimuthal orbital frequency is fixed to $F_{\text{orb}} = 7.5 \text{nHz}$ and $\mathcal{M}_c = 10^{9.2}M_{\odot}$.
• Figure 4: Discrepancy between numerical and approximate solution of $\xi(t)$ as a function of time computed over $20 yrs$. Different colors correspond to different orbital frequencies and upper panel is for $e_0 = 0.5$ and lower panel for $e_0 = 0.7$.
• Figure 5: Difference between numerical and approximate evolution of $e$ for a SMBHB with $\mathcal{M}_c = 10^{9.2}$ on a timespan of $20$ yrs, computed at different values of $e_0$ and $x_0$.