Surrogate model for gravitational wave signals from comparable and large-mass-ratio black hole binaries

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Surrogate model for gravitational wave signals from comparable and large-mass-ratio black hole binaries

Nur E. M. Rifat, Scott E. Field, Gaurav Khanna, and Vijay Varma

Phys. Rev. D 101, 081502(R) – Published 22 April 2020

Abstract

Gravitational wave signals from compact astrophysical sources such as those observed by LIGO and Virgo require a high-accuracy, theory-based waveform model for the analysis of the recorded signal. Current inspiral-merger-ringdown models are calibrated only up to moderate mass ratios, thereby limiting their applicability to signals from high-mass-ratio binary systems. We present EMRISur1dq1e4, a reduced-order surrogate model for gravitational waveforms of $13 500 M$ in duration and including several harmonic modes for nonspinning black hole binary systems with mass ratios varying from 3 to 10000, thus vastly expanding the parameter range beyond the current models. This surrogate model is trained on waveform data generated by point-particle black hole perturbation theory (ppBHPT) both for large-mass-ratio and comparable mass-ratio binaries. We observe that the gravitational waveforms generated through a simple application of ppBHPT to the comparable mass-ratio cases agree surprisingly well with those from full numerical relativity after a rescaling of the ppBHPT’s total mass parameter. This observation and the EMRISur1dq1e4 surrogate model will enable data analysis studies in the high-mass-ratio regime, including potential intermediate-mass-ratio signals from LIGO/Virgo and extreme-mass-ratio events of interest to the future space-based observatory LISA.

Received 28 October 2019
Accepted 2 April 2020

DOI:https://doi.org/10.1103/PhysRevD.101.081502

Physics Subject Headings (PhySH)

Classical black holes General relativity equations & solutions Gravitational wave sources

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Nur E. M. Rifat¹, Scott E. Field², Gaurav Khanna¹, and Vijay Varma³

¹Department of Physics and Center for Scientific Computing and Visualization Research, University of Massachusetts, Dartmouth, Massachusetts 02747, USA
²Department of Mathematics and Center for Scientific Computing and Visualization Research, University of Massachusetts, Dartmouth, Massachusetts 02747, USA
³Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125, USA

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Vol. 101, Iss. 8 — 15 April 2020

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Images

Figure 1
The bottom subfigure depicts each mode’s relative error by comparing the surrogate model and the training data. The relative error is computed from Eq. (3) with $α = 1$ and using the appropriate mode in place of where it says “ $h^{22}$ .” The penultimate subfigure shows the numerical truncation errors (orange plus) estimate the quality of the training waveform data by comparing two numerical simulations of increasing resolution. We also compared the full surrogate (including all 11 harmonic modes) and the training data (black circles) and a leave-one-out cross-validation (LOOCV) trial surrogate and the training data (blue triangles). The largest LOOCV error is for $q = 150$ , for which the $h_{+}$ polarization’s quadrupole mode is shown in the top subfigure. The second subfigure reports on the error in the amplitude and phase for this case; our full surrogate, trained on the entire dataset of 37 waveforms, is more accurate than the LOOCV diagnostics shown.
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Figure 2
At each empirical interpolation node we build surrogate models for amplitude, $A^{2, 2} (q) \approx A_{S}^{2, 2} (q)$ , and phase $ϕ^{2, 2} (q) \approx ϕ_{S}^{2, 2} (q)$ , across the training region. One of the key methodological improvements pursued here is modeling the training data after performing a logarithmic transformation (red asterisk) of the independent variable ${\ln (q), A^{2, 2} (q)}$ and ${\ln (q), ϕ^{2, 2} (q)}$ . Here we show the LOOCV model error at $q = 100$ , which leads to nearly 2 orders of magnitude improvement as compared with no transformation (blue triangle). Data for the other harmonic modes and empirical interpolation node data show similar improvement.
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Figure 3
Waveform difference between numerical relativity and ppBHPT waveforms before and after rescaling the ppBHPT’s total mass parameter. Bottom panel: These two figures focus on the case $q = 8$ . Before rescaling the NR (solid blue curve) and ppBHPT (dashed red curve) waveforms are noticeably different in both their amplitude and phasing and have an L2 difference of $7.03 \times 10^{- 1}$ . We find the optimum value of the rescaling parameter to be $α = 0.85837$ and modify the ppBHPT waveform according to Eq. (2). After rescaling the NR and modified ppBHPT waveforms demonstrate remarkable agreement with one another. Top panel: We repeat this comparison procedure for mass ratios $3 \leq q \leq 10$ , where NR data are available, and compute both the optimal scaling factor and the L2 difference between waveforms (NR vs scaled ppBHPT) for each case. In all cases the differences before rescaling are order unity, while the agreement between the rescaled ppBHPT and NR waveforms is $\approx 1 %$ . The dotted line refers to a naive value of $α = 1 / (1 + 1 / q)$ set by including the mass of the smaller black hole as part of the background space-time.
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Figure 4
Frequency-domain mismatch between NR and rescaled ppBHPT waveforms using all available modes. The mismatch computation uses an advanced LIGO design sensitivity curve [98], an upper frequency of 4096 Hz, and a variable lower frequency set to the (2,2) mode’s initial instantaneous frequency. We show a result for an inclination angle of 0, while nonzero inclinations are typically a factor of $\sim 1.5$ times larger. As expected, the ppBHPT and NR waveforms show better agreement at larger mass ratios.
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