Einstein Telescope Advances Gravitational Wave Inference for up to Tens of Binaries

Researchers are now better equipped to analyse gravitational wave signals from binary black holes thanks to new modelling of ‘line-of-sight acceleration’ , a subtle effect caused by a third, distant companion. Kai Hendriks, Lorenz Zwick, and Pankaj Saini, all from the Center of Gravity at The Niels Bohr Institute, alongside János Takátsy and Johan Samsing et al, present a complete model capturing how this acceleration alters the observed gravitational wave patterns, improving upon existing methods. This work is significant because it allows scientists to potentially identify and characterise these three-body systems, offering a crucial test to distinguish between different formation pathways for binary black holes , specifically, whether they formed dynamically or within active galactic nuclei. Their analysis of past events, including GW190814 and recent O4a detections, currently finds no evidence of this acceleration, but anticipates that future detectors like the Einstein Telescope could reveal several such systems annually.

This breakthrough directly addresses a key challenge in GW astrophysics: discerning the formation pathways of BBHs and identifying the environmental effects that modify their signals. The work opens exciting possibilities for probing the environments surrounding merging black holes and unraveling the mysteries of their origins. Experiments show that the model accurately predicts the subtle shifts in GW phase caused by the gravitational influence of a third, distant body, a phenomenon known as LOSA.

The research establishes a method for extracting information about this third body, its mass and distance, from the observed GW signal, providing a unique window into the dynamics of three-body interactions. This is achieved through a novel approach to modelling the acceleration, incorporating the full curvature of the outer orbit and its projection onto the line of sight, which significantly improves the accuracy of the predicted dephasing. The team’s model provides a more realistic representation of the GW signal, enabling more precise parameter estimation and ultimately, a deeper understanding of the systems producing these signals. Interestingly, the previously reported LOSA signal in GW190814 disappeared when a longer data segment was employed in the analysis. This rigorous testing demonstrates the robustness of the new model and highlights the importance of careful data analysis in the search for subtle environmental effects.

LOSA Modelling and Parameter Inference with ET offers

The study employed circular waveform models, beginning with Fourier space Newtonian waveforms in the stationary phase approximation, expressed as h(f) = r 5 24π−2/3 Q D(z) (GMz) c3/2 5/6 f −7/6 exp [iψvac], where f represents the observer frame GW frequency and Mz is the red-shifted chirp mass. The GW phase, ψvac, was calculated using 2πftc −φc −π 4 − 3 128 πGMzf c3 −5/3, with phase and time at coalescence set to zero to simplify initial SNR estimates. Subsequently, total phase ψtot was determined by adding the vacuum phase to environmental effects, represented as ψtot = ψvac + ψEE. For detailed analysis of GW190814 and O4a events, scientists utilised the IMRPhenomXPHM waveform, incorporating the SpinTaylor extension for O4a detections to ensure consistency with the GWTC-4 catalogue.

Signal-to-noise ratio (SNR) calculations were performed using the noise weighted inner product ⟨h1, h2⟩= 2 Z ∞ 0 h1 h∗ 2 + h∗ 1 h2 Sn(f ′) df ′, where Sn defines the noise profile of the ET detector. A detectability criterion, δSNR2 ≡⟨∆h, ∆h⟩ C2, was established, comparing vacuum and perturbed waveforms (∆h = hvac −hEE) with a threshold of approximately 10 to ensure sufficient mismatch. The team further pioneered the concept of “delta, δSNR”, defined as ∆δSNR2 ≡⟨∆h, ∆h⟩ C2, with ∆h = hEE1 −hEE2, to assess the distinguishability between eccentric and generic dephasing prescriptions. This innovative approach involved comparing waveforms with eccentric dephasing (Eq0.13) to those with a generic dephasing term dψ = Af −13/3, matching the dephasing amplitude at a frequency of 2Hz to quantify the signal’s uniqueness.

LOSA unlocks tertiary mass and distance

The team measured the potential for distinguishing between different types of environmental effects within the same waveform, introducing the concept of “delta, δSNR” to survey large portions of parameter space. This metric quantifies how distinguishable a true eccentric signal is from its closest generic, non-eccentric counterpart. Tests prove the model’s ability to accurately predict phase shifts in GW signals, with the team employing Fourier space Newtonian waveforms in the stationary phase approximation; h(f) = r 5 24π−2/3 Q D(z) (GMz) c3/2 5/6 f −7/6 exp [iψvac], where f represents the observer frame GW frequency and Mz is the red-shifted chirp mass. The GW phase ψvac is defined as: ψvac = 2πftc −φc −π 4 − 3 128 πGMzf c3 −5/3, with the integration constant tc and phase offset φc both set to zero for simplified SNR estimates.

The total phase ψtot is then calculated as ψtot = ψvac + ψEE, incorporating additional dephasing prescriptions. Measurements confirm that the team employed the IMRPhenomXPHM waveform, alongside the SpinTaylor extension for O4a detections, to enable direct comparison with existing GWTC-4 catalogue data. The signal-to-noise ratio (SNR) was calculated using ⟨h1, h2⟩= 2 Z ∞ 0 h1 h∗ 2 + h∗ 1 h2 Sn(f ′) df ′, where Sn characterises the noise profile of the ET detector. A detectability criterion of δSNR2 ≡⟨∆h, ∆h⟩ C2, with ∆h = hvac −hEE, was used, employing a threshold of O(10) to ensure sufficient mismatch.

LOSA unlocks tertiary mass and distance

This new prescription incorporates curvature and projection effects, offering improvements over previous local-expansion-based methods. The study identifies two crucial conditions for distinguishable eccentric dephasing: a significantly eccentric outer orbit (eout ≳0.7) with a semi-major axis of 0.01 to 0.1 AU, and a merger occurring near pericentre. Acknowledging limitations, the authors note that accurate predictions depend on the poorly understood fraction of BBHs assembled through three-body interactions. Future research should focus on characterising the distributions of outer true anomaly, eccentricity, and semi-major axis from N-body simulations of three-body interactions. This will refine detection rate predictions and enhance understanding of the dynamical assembly of black holes in these systems.