Numerical Relativity Catalog: Simulated Gravitational Waves

• Numerical Relativity Catalog is a curated repository of gravitational waveforms, metadata, and simulation data capturing diverse binary merger phenomena.
• It employs state-of-the-art numerical relativity methods and robust extraction techniques to ensure reproducibility and cross-code consistency.
• The catalog underpins gravitational-wave data analysis, model calibration, and rigorous tests of general relativity through refined parameter estimation.

A Numerical Relativity Catalog is a curated, publicly accessible collection of gravitational waveforms and associated metadata generated from large-scale numerical simulations of Einstein’s equations, typically for binary black hole (BBH) or neutron star systems. These catalogs provide foundational resources for gravitational-wave data analysis, waveform modeling, code benchmarking, and the paper of relativistic astrophysical processes. They enable the calibration and validation of analytical and phenomenological models, support parameter estimation for gravitational wave observations, and serve as standardized datasets for comparison across numerical relativity (NR) codes and methodologies.

1. Scope and Content of Major Numerical Relativity Catalogs

Numerical relativity catalogs have progressively expanded to cover a broad swath of physical parameter space relevant for gravitational-wave astrophysics. Early catalogs, such as the NINJA-2 hybrid waveform set, initially focused on non-precessing BBHs with dominant $(\ell, m) = (2,2)$ modes and moderate ranges of mass ratio and spin (Ajith et al., 2012). Subsequent generations have greatly enriched this landscape:

Catalog/Group	Simulations (N)	Mass Ratio Range	Spin Configurations	Eccentricity Coverage
SXS (SpEC)	>2000	up to 10	Aligned, anti-aligned, precessing; $\chi \leq 0.98$	Low ($e\lesssim 10^{-4}$), some higher
RIT (multiple releases)	1881	$1/128 \leq q \leq 1$	Nonspinning, aligned, and precessing ($\chi\leq0.95$)	Eccentric orbits $0<e\leq1$ (Healy et al., 2022)
Georgia Tech / MAYA	635	up to 15	Nonspinning, aligned, precessing	$e>0.01$, including joint precessing/eccentric (Ferguson et al., 2023)
ICCUB (Barcelona)	128	$q=1$ (equal mass)	Equal, aligned spins, $\chi\in(-0.5,0.5)$	High ($e=0.62{-}0.79$), zoom-whirl (Trenado et al., 5 Sep 2025)

Key features across catalogs include coverage of both precessing and non-precessing binaries, a wide mass ratio range (from $q\sim 1/128$ to $q=15$), spin magnitudes approaching the theoretical Kerr limit, and systematic inclusion of moderate to high eccentricities and dynamical capture phenomena. Datasets typically provide strain $h^{(\ell,m)}$, Weyl scalar $\psi_4^{(\ell,m)}$, full metadata, and, in modern catalogs, initial conditions and parameter files for reproducibility.

2. Principles and Methodologies in Catalog Construction

The standard methodology in numerical relativity simulation and catalog construction consists of four main phases:

1. Initial Data Generation: Black hole or neutron star binaries are set up with specified mass ratios, spin vectors, and (for eccentric catalogs) prescribed initial momenta that tune the orbital eccentricity (Trenado et al., 5 Sep 2025). For hybrid catalogs (e.g., NINJA-2), a post-Newtonian (pN) portion is attached to the long numerical relativity segment to bridge the early and late inspiral.
2. Evolution and Waveform Extraction: The simulations solve the full Einstein equations in 3+1 formulation, employing codes such as SpEC, Einstein Toolkit, Maya, or MLBSSN (often with the BSSN or CCZ4 scheme) (Jani et al., 2016, Andrade et al., 2022, Trenado et al., 5 Sep 2025). Gravitational wave signals are extracted as the multipoles of the Weyl scalar $\psi_4^{(\ell,m)}$ at several finite radii.
3. Extrapolation to $r=\infty$ and Strain Computation: To remove gauge ambiguities and obtain physically meaningful observables, extracted waveforms are extrapolated to future null infinity, commonly through formulas such as

$$r \psi_4^{(\ell,m)} |_{\infty} = \alpha(r) \left[ r \psi_4^{NR}(t,r) - \beta(r,\ell) \int dt (r \psi_4^{NR}(t,r)) \right]$$

with $\alpha(r) = 1 - {2M}/{r}$, $$\beta(r, \ell) = {\left(\ell-1\right) \left(\ell+2\right)}/{2r}$$ (Trenado et al., 5 Sep 2025). The strain $h^{(\ell,m)}$ is then obtained by double time integration, frequently via fixed-frequency integration (FFI).

1. Metadata, Validation, and Dissemination: Each waveform is accompanied by metadata (initial/final parameters, trajectory, remnant properties), code parameter files, and resolution studies. Validation includes convergence checks, mismatch computations (e.g., $\mathcal{M}=1-\mathcal{O}$ with frequency-domain matches), and, for recent catalogs, anomaly detection via neural networks (Pereira et al., 2022).

3. Catalogs with Specialized Physical Coverage

Recent catalogs have addressed specific phenomena and previously underexplored sectors of parameter space:

• High-eccentricity mergers and dynamical capture catalogs (ICCUB): 128 simulations in $e \in (0.62, 0.79)$, systematically mapping zoom-whirl dynamics with multiple close encounters before merger (Trenado et al., 5 Sep 2025).
• Precessing single-spin binaries at high mass ratio: A catalog of 80 configurations systematically sampling $q = 1,2,4,8$ and $\chi_2 = 0.2-0.8$ with varying spin-misalignment angles, enabling model calibration for strong precession (Hamilton et al., 2023).
• Hybrid catalogs: The NINJA-2 project combined post-Newtonian and NR segments in hybrid waveforms, matching at an overlapping region via least-squares phase and amplitude fitting (Ajith et al., 2012). These are essential for low-mass systems and detector band coverage.

Such datasets are vital for calibrating and benchmarking models in parameter areas previously lacking sufficient numerical data, e.g., high mass ratio precessing binaries, dynamical captures, or extreme eccentricities.

4. Data Products and Reproducibility

Comprehensiveness and interoperability are ensured through:

• Waveform Modes: Most catalogs provide $(\ell,m)$ modes up to at least $(4,4)$, with more extended catalogs (e.g., Georgia Tech, MAYA) extending to $\ell = 6$ or $8$ (Jani et al., 2016, Ferguson et al., 2023).
• Strain and Multipole Output: Data includes time series for both $\psi_4$ and $h$, with explicit documentation of the integration methods (FFI, memory mode handling, etc.).
• Metadata and Parameter Files: Complete documentation, including initial state, relaxation parameters, extraction radii, and evolution settings, enables reproducibility of the entire pipeline (Trenado et al., 5 Sep 2025).
• Post-processing Tools: Python libraries and command-line tools (e.g., mayawaves, POWER) support batch waveform extraction, strain computation, extrapolation to infinity, and higher-mode analysis (Johnson et al., 2017, Ferguson et al., 2023).
• Quality Assurance: Recent practices include anomaly flagging with deep learning, removing simulations with nonphysical features such as lazy/asymptotic ringdowns or mode-mixing irregularities (Pereira et al., 2022).

5. Scientific Impact and Applications

Numerical relativity catalogs underpin the following areas:

• Parameter Estimation: Catalog data is essential for gravitational wave parameter estimation and model selection in LIGO/Virgo/KAGRA data analysis pipelines. NR-only approaches maximize accuracy by directly comparing full numerical solutions to observed signals without relying on semi-analytic waveform approximants, as illustrated in GW150914 analyses (Healy et al., 2019).
• Waveform Modeling and Calibration: Analytical models (e.g., Phenom, SEOBNR) and surrogates require dense NR sampling for calibration, especially in complex regimes (high-spin, high mass ratio, precessing, eccentric). Empirical correlations—such as those between the final spin $\alpha_f$, peak $(2,2)$ mode frequency $m\omega_{22}$, and radiated energy normalized by mass ratio—are derived from large catalogs to construct accurate remnant models (Healy et al., 2020, Healy et al., 2022).
• Testing General Relativity: Catalogs enable precision tests of relativistic dynamics in merger regimes (e.g., remnant recoil, tidal coupling, spin-orbit transfer) (Bachhar et al., 2023), and they facilitate systematic studies of the impact of physical phenomena such as higher harmonics, precession, and eccentricity on observable waveforms (Joshi et al., 2022, Huerta et al., 2019).
• Template Banks and Detection Algorithms: As detectors seek to observe increasingly exotic phenomena, catalogs with high eccentricity (Trenado et al., 5 Sep 2025), extreme mass ratio (Healy et al., 2022), or strong precession (Hamilton et al., 2023) inform the construction of template banks for unmodeled/burst searches, machine learning event classification, and improved matched-filter coverage.

6. Computational Advances and Future Directions

• Code Performance: The scaling of NR waveform production is being revolutionized by the adoption of GPU-accelerated solvers and auto-code-generation frameworks (e.g., NRPyEllipticGPU), enabling $16\times$ speedup in initial data generation and facilitating the production of large catalogs for next-generation detectors (CE, ET, LISA) (Tootle et al., 23 Jan 2025).
• Parameter Space Expansion: Ongoing efforts target systematic coverage of two-spin, higher-mass-ratio, and extreme-eccentricity domains. The ability to generate and post-process waveforms efficiently—together with standardized validation and metadata—will be decisive in meeting the computational demands of future observations (Healy et al., 2022, Tootle et al., 23 Jan 2025).
• Quality Control and Standardization: Deep learning approaches for anomaly detection and continuous benchmarking against “apples with apples” testbeds (e.g., direct error norm and Fourier analysis across code bases) are being integrated into catalog production, improving reliability and facilitating cross-catalog comparison (Daverio et al., 2018, Pereira et al., 2022).

7. Accessibility and Data Usage

All principal catalogs discussed provide open web interfaces or programmatic access (e.g., RIT http://ccrg.rit.edu/~RITCatalog, MAYA https://cgp.ph.utexas.edu/waveform, ICCUB https://data.icc.ub.edu/nrwaves/). Each data release includes (as standard):

• Extracted waveform multipoles (at $r\rightarrow\infty$), both as $\psi_4$ and $h$.
• Complete metadata with physical and numerical parameters.
• Parameter and configuration files for code-level reproducibility.
• Python libraries or scripts for loading, filtering, and further post-processing of the data.

The high level of documentation and reproducibility, combined with precise validation (e.g., mismatch errors $\leq 0.4\%$ (Hamilton et al., 2023)), underpins both theoretical studies and detector-facing data analysis in gravitational wave astrophysics.

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In summary, the Numerical Relativity Catalog constitutes the essential infrastructure for gravitational wave science, encompassing rigorously validated, publicly accessible simulated waveforms and metadata representing BBH (and, in extended catalogs, neutron star and mixed binaries) across the relevant astrophysical parameter space. Their continued expansion and refinement reflect ongoing advances in simulation techniques, computational hardware, and the demands of ground- and space-based gravitational wave observatories.