1 Summary↩︎

Gravitational waves are ripples in space-time caused by the motion of massive objects. One of the most astrophysically important sources of gravitational radiation is caused by two orbiting compact objects, such as black holes and neutron stars, that slowly inspiral and merge. The motion of these massive objects generates gravitational waves that radiate to the far field where gravitational-wave detectors can observe them. Complicated partial or ordinary differential equations govern the entire process.

Traditionally, the dynamics of compact binary systems and the emitted gravitational waves have been computed by expensive simulation codes that can take days to months to run. A key simulation output is the gravitational wave signal for a particular set of parameter values describing the system, such as the black holes’ masses and spins. The computed signal is required for a diverse range of multiple-query applications, such as template bank generation for searches, parameter estimation, mock data analysis, studies of model bias, and tests of general relativity, to name a few. In such settings, the high-fidelity signal computed from differential equations is often too slow to be directly used.

Surrogate models offer a practical way to dramatically accelerate model evaluation while retaining the high-fidelity accuracy of the expensive simulation code; an example is shown in Fig. [fig:gws]. Surrogate models can be constructed in various ways, but what separates these models from other modeling frameworks is that they are primarily data-driven. Given a training set of gravitational waveform data sampling the parameter space, a model is built by following three steps:

1. Feature extraction: the waveform is decomposed into data pieces that are simple to model,

2. Dimensionality reduction: each data piece is approximated by a low-dimensional vector space, which reduces the degrees of freedom we need to model, and

3. Regression: fitting and regression techniques are applied to the low-dimensional representation of each data piece over the parameter space defining the model.

These model-building steps result in an HDF5 file defining the surrogate model’s data and structure, which is stored on Zenodo. The GWSurrogate package provides access to these models through its catalog interface, and all available models and their associated HDF5 files can be found in gwsurrogate.catalog.list(). For a recent overview of surrogate modeling as used in gravitational wave astrophysics, please see Section 5 of Afshordi et al. ().

The development of GWSurrogate is hosted on and distributed through both and . Quick start guides are found on the project’s while model-specific documentation is described through a collection of model-specific . Automated testing is run on .

2 Statement of need↩︎

GWSurrogate is a Python package first introduced in 2013 to provide an intuitive interface for working with gravitational wave surrogate models. Specifically, GWSurrogate gravitational wave models provide evaluation of \[h_{\tt S}(t, \theta, \phi;\Lambda) = \sum^{\infty}_{\ell=2} \sum_{m=-\ell}^{\ell} h_{\tt S}^{\ell m}(t;\Lambda) ~^{-2}Y_{\ell m}(\theta, \phi) \,,\] where \(^{-2}Y_{\ell m}\) are the spin\(=-2\) weighted spherical harmonics and \(\Lambda\) describes the model’s parameterization. The surrogate model provides fast evaluations for the modes, \(h_{\tt S}^{\ell m}\). As described more fully in the documentation, the high-level API allows users direct access to the modes \(\{h_{\tt S}^{\ell m}(t)\}\) (as a Python dictionary) or assembles the sum \(h_{\tt S}(t, \theta, \phi)\) at a particular location \((\theta, \phi)\). The models implemented in GWSurrogate are intended to be used in production data analysis efforts. As such,

• computationally expensive operations (e.g., interpolation onto uniform time grids) are implemented by wrapping low-level C code for speed, whereas GWSurrogate provides a user-friendly interface to the high-level waveform evaluation API,

• models implemented in GWSurrogate follow the waveform convention choices of the LIGO-Virgo-Kagra collaboration, thus ensuring that downstream data analysis codes can use GWSurrogate models without needing to worry about different conventions, and

• GWSurrogate models can be directly evaluated in either physical units (often used in data analysis studies) and dimensionless units (often used in theoretical studies) where all dimensioned quantities are expressed in terms of the system’s total mass.

Currently, there are 16 supported surrogate models (; , ; ; ; ; ; ; ; ; , ), with additional models under development (; ). These models vary in their duration, included physical effects (e.g. nonlinear memory, tidal forces, harmonic modes retained, eccentricity, mass ratio extent, precession effects, etc), and underlying solution method (e.g. Effective One Body, numerical relativity, and black hole perturbation theory). Details about all models can be found by doing gwsurrogate.catalog.list(verbose=True), while the GWSurrogate summarizes the state-of-the-art models for each particular problem. Certain models allow for additional functionality such as returning the dynamics of the binary black hole. These special features are described further in model-specific .

Several other software packages are available for waveform generation, including tools for effective-one-body models (; ), ringdown signals (; ; ), extreme-mass-ratio inspiral systems through the Black Hole Perturbation Toolkit’s FastEMRIWaveforms and BHPTNRSurrogate packages (), and the Ripple framework that enables specialized acceleration techniques using JAX (). Among these, LALSuite () stands out as the most comprehensive, offering the largest collection of waveform models via its LALSimulation subpackage, which includes Python bindings and the new Python-based gwsignal waveform generator. While GWSurrogate shares similarities with LALSuite in providing a variety of models, it differs by exclusively focusing on surrogate models. Notably, GWSurrogate includes many state-of-the-art numerical relativity models that are only available through its library, whereas LALSuite offers a broader but less specialized collection.

3 Acknowledgements↩︎

We acknowledge our many close collaborators for their contribution to the development of surrogate models. We further acknowledge the community of GWSurrogate users who have contributed pull requests and opened issues, including Kevin Barkett, Mike Boyle, Collin Capano, Dwyer Deighan, Raffi Enficiaud, Oliver Jennrich, Gaurav Khanna, Duncan Macleod, Alex Nitz, Seth Olsen, Swati Singh, and Avi Vajpeyi. GWSurrogate has been developed over the past 10 years with continued support from the National Science Foundation, most recently through NSF grants PHY-2110496, PHY-2309301, DMS-2309609, and AST-2407454. This work was partly supported by UMass Dartmouth’s Marine and Undersea Technology (MUST) research program funded by the Office of Naval Research (ONR) under grant no. N00014-23-1-2141. This work was also supported in part by the Sherman Fairchild Foundation, by NSF Grants PHY-2207342 and OAC-2209655 at Cornell, and by NSF Grants PHY-2309211, PHY-2309231, and OAC-2209656 at Caltech.

References↩︎