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Parameter control for binary black hole initial data

Iago B. Mendes, Nils L. Vu, Oliver Long, Harald P. Pfeiffer, Robert Owen

TL;DR

This work tackles the challenge of precisely steering binary black hole initial data to desired physical parameters by employing the extended conformal thin-sandwich (XCTS) formalism with a parameter-control loop. The authors extend SpECTRE’s capabilities by applying Broyden's method to all controlled quantities, enabling direct targeting of horizon masses and spins as well as asymptotic quantities like ${E}_{\text{ADM}}$ and ${J}^{\text{ADM}}$, thereby enabling both bound inspirals and hyperbolic encounters. Their results demonstrate robust convergence across extreme configurations (high spins up to ${\chi}=0.9999$, mass ratios up to ${q=50}$, separations up to ${D_0=1000M}$) and reveal improved efficiency for hyperbolic cases compared with SpEC, due to the comprehensive Jacobian updates. By providing open-source access to this control scheme, the work offers a scalable, accurate tool for BBH simulations that supports advanced waveform modeling and scattering studies for next-generation gravitational-wave detectors.

Abstract

When numerically solving Einstein's equations for binary black holes (BBH), we must find initial data on a three-dimensional spatial slice by solving constraint equations. The construction of initial data is a multi-step process, in which one first chooses freely specifiable data that define a conformal background and impose boundary conditions. Then, one numerically solves elliptic equations and calculates physical properties such as horizon masses, spins, and asymptotic quantities from the solution. To achieve desired properties, one adjusts the free data in an iterative ``control'' loop. Previous methods for these iterative adjustments rely on Newtonian approximations and do not allow the direct control of total energy and angular momentum of the system, which becomes particularly important in the study of hyperbolic encounters of black holes. Using the $\texttt{SpECTRE}$ code, we present a novel parameter control procedure that benefits from Broyden's method in all controlled quantities. We use this control scheme to minimize drifts in bound orbits and to enable the construction of hyperbolic encounters. We see that the activation of off-diagonal terms in the control Jacobian gives us better efficiency when compared to the simpler implementation in the Spectral Einstein Code ($\texttt{SpEC}$). We demonstrate robustness of the method across extreme configurations, including spin magnitudes up to $χ= 0.9999$, mass ratios up to $q = 50$, and initial separations up to $D_0 = 1000M$. Given the open-source nature of $\texttt{SpECTRE}$, this is the first time a parameter control scheme for constructing bound and unbound BBH initial data is available to the numerical-relativity community.

Parameter control for binary black hole initial data

TL;DR

This work tackles the challenge of precisely steering binary black hole initial data to desired physical parameters by employing the extended conformal thin-sandwich (XCTS) formalism with a parameter-control loop. The authors extend SpECTRE’s capabilities by applying Broyden's method to all controlled quantities, enabling direct targeting of horizon masses and spins as well as asymptotic quantities like and , thereby enabling both bound inspirals and hyperbolic encounters. Their results demonstrate robust convergence across extreme configurations (high spins up to , mass ratios up to , separations up to ) and reveal improved efficiency for hyperbolic cases compared with SpEC, due to the comprehensive Jacobian updates. By providing open-source access to this control scheme, the work offers a scalable, accurate tool for BBH simulations that supports advanced waveform modeling and scattering studies for next-generation gravitational-wave detectors.

Abstract

When numerically solving Einstein's equations for binary black holes (BBH), we must find initial data on a three-dimensional spatial slice by solving constraint equations. The construction of initial data is a multi-step process, in which one first chooses freely specifiable data that define a conformal background and impose boundary conditions. Then, one numerically solves elliptic equations and calculates physical properties such as horizon masses, spins, and asymptotic quantities from the solution. To achieve desired properties, one adjusts the free data in an iterative ``control'' loop. Previous methods for these iterative adjustments rely on Newtonian approximations and do not allow the direct control of total energy and angular momentum of the system, which becomes particularly important in the study of hyperbolic encounters of black holes. Using the code, we present a novel parameter control procedure that benefits from Broyden's method in all controlled quantities. We use this control scheme to minimize drifts in bound orbits and to enable the construction of hyperbolic encounters. We see that the activation of off-diagonal terms in the control Jacobian gives us better efficiency when compared to the simpler implementation in the Spectral Einstein Code (). We demonstrate robustness of the method across extreme configurations, including spin magnitudes up to , mass ratios up to , and initial separations up to . Given the open-source nature of , this is the first time a parameter control scheme for constructing bound and unbound BBH initial data is available to the numerical-relativity community.

Paper Structure

This paper contains 14 sections, 54 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Free data
  4. Physical parameters
  5. Control of bound orbits
  6. Control of hyperbolic encounters
  7. Comparison with SpEC
  8. Results
  9. Convergence of physical parameters
  10. Bound orbits
  11. Hyperbolic encounters
  12. Conclusion
  13. ADM integrals
  14. Center-of-mass integral

Figures (10)

  • Figure 1: Schematic representation of the BBH free data. The solid circles represent the two black hole excisions ($S_A$ and $S_B$), while the dashed ellipse represents the outer boundary of the computational domain ($S_\infty$). "$\boldsymbol+$" and "$\boldsymbol\times$" indicate the origin and the Newtonian center of mass, respectively. The gray free data are not explicitly used in the SpECTRE control scheme.
  • Figure 2: Histogram of the number of control iterations needed for all configurations in Tables \ref{['tab:bound-orbits']} and \ref{['tab:hyperbolic-encounters']}.
  • Figure 3: Resolution convergence of physical parameters for the runs in Tables \ref{['tab:bound-orbits']} and \ref{['tab:hyperbolic-encounters']}. SpECTRE data are shown in solid lines, while SpEC data are shown in dashed lines. The circle marks indicate the resolution chosen for the runs in Figs. \ref{['fig:control-bound']}--\ref{['fig:control-hyperbolic']}. The difference of any parameter $x$ with its highest resolution value is represented as $\Delta x = x - x|_{\max N}$.
  • Figure 4: Comparison of control loops in SpEC (dashed) and SpECTRE (solid) for bound orbits. The top panels show results for black hole masses and spins, whereas the bottom panels show the asymptotic quantities.
  • Figure 5: Final control Jacobian of the q10 case.
  • ...and 5 more figures