Probabilistic model for the gravitational wave signal from merging black holes

Open Access

Probabilistic model for the gravitational wave signal from merging black holes

Sebastian Khan

Phys. Rev. D 109, 104045 – Published 13 May 2024

Abstract

Parametrized models that predict the gravitational-wave (GW) signal from merging black holes are used to extract source properties from GW observations. The majority of research in this area has focused on developing methods capable of producing highly accurate, point estimate, predictions for the GW signal. A key element missing from every model used in the analysis of GW data is an estimate for how confident the model is in its prediction. This omission increases the risk of biased parameter estimation of source properties. Current strategies include running analyses with multiple models to measure systematic bias however, this fails to accurately reflect the true uncertainty in the models. In this work we develop a probabilistic extension to the phenomenological modeling workflow for nonspinning black holes and demonstrate that the model not only produces accurate point estimates for the GW signal but can be used to provide well-calibrated local estimates for its uncertainty. Our analysis highlights that there is a lack of numerical relativity (NR) simulations available at multiple resolutions which can be used to estimate their numerical error and implore the NR community to continue to improve their estimates for the error in NR solutions published. Waveform models that are not only accurate in their point-estimate predictions but also in their error estimates are a potential way to mitigate bias in GW parameter estimation of compact binaries due to unconfident waveform model extrapolations.

Received 25 March 2024
Accepted 19 April 2024

DOI:https://doi.org/10.1103/PhysRevD.109.104045

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Gravitational wave sources Gravitational waves

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Sebastian Khan

School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 109, Iss. 10 — 15 May 2024

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
GW signal from a mass ratio $8 ∶ 1$ BBH system compared with predictions from the PPM model. This NR simulation was not used to train the model. Top row: $h_{+}$ from the highest resolution NR simulation (black), the mean PPM model prediction (dashed orange) and the $90 th$ percentile width from 100 PPM samples (orange shaded region). Top left panel: shows the inspiral up to $t = 0 M$ . The level of accuracy and uncertainty are such that deviations between the NR and the model are barely visible on this scale. The top right panel shows the ringdown where uncertainty in the model is more noticeable. Bottom row: the phase error. The black lines are the two lower resolution NR simulations compared with the reference NR simulation. The orange dashed line is the PPM mean and the orange shaded regions show the 50th, 90th and 99th percentile width of the phase error distribution from 100 PPM samples. The match between the highest and lowest NR simulation is 0.9996. The match between highest resolution NR simulation and the PPM mean is ${0.99984}_{0.99936}^{0.99993}$ . Where the upper and lower bounds are the 5th and 95th percentile of the match between 100 PPM samples and the NR simulation.
Reuse & Permissions
Figure 2
Toy example to illustrate collocation method. We compare the least-squares approach (blue) and the collocation point approach (black/gray) when applied to the task of modeling the data (red). For the collocation point method with specify the value and the first derivative at $x = 0$ and $x = 1$ as well as a collocation point at $x = 0.5$ . To generate samples from the collocation point method we perturb the fitted collocation point values with a $N (0, 0.1)$ distribution.
Reuse & Permissions
Figure 3
Example GPR fit for the amplitude merger model. We show the data and fit for the collocation point at time $t = 0 M$ . The training and test set are shown as blue and orange points. The mean GP fit is the red line and the red shaded region is the $2 σ$ prediction interval.
Reuse & Permissions
Figure 4
The mismatch between various objects as a function of the mass ratio. The black error bars are an estimate of the NR uncertainty obtained by computing the mismatch between NR waveforms at the same mass ratio. The mismatch between PPM mean prediction and NR simulations in the training set (open circles) and the test set (filled circles), which are colored based on the NR code, represent the typical accuracy metric of waveform models. The solid blue curve is the median of the predicted mismatch distribution [Eq. (34)] and the shaded blue regions show the 50th, 90th and 99th percentile widths.
Reuse & Permissions
Figure 5
In each panel we compare our model with representative NR simulations at mass ratios $4 ∶ 1$ . $8 ∶ 1$ , $15 ∶ 1$ and $32 ∶ 1$ from the test set. The NR is shown as a solid black line. We generate 1000 samples and the mean prediction from our PPM model and optimally align them with the NR waveform over a time and phase shift. We show the minimum and maximum range of values from the 1000 samples as the shaded region and explicitly plot the mean as well as three samples.
Reuse & Permissions
Figure 6
Calibration score distribution as defined by Eq. (36). We show the median value and the 90% width of the predicted mismatch distribution for the train (blue) and test (orange) sets. We also show the individual results for each NR simulation compared with the median value of the predicted mismatch.
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology