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Quasi-equilibrium binary black hole sequences for puncture data derived from helical Killing vector conditions

Wolfgang Tichy, Bernd Bruegmann

TL;DR

The paper presents a method to generate quasi-equilibrium binary black hole initial data in puncture form by enforcing approximate helical Killing vector conditions alongside constant puncture ADM masses. By solving for momentum and lapse parameters with Newton-Raphson iterations along a sequence of separations, the work yields sequences where the apparent horizon mass remains constant and the ISCO location matches previous effective potential results, while providing detailed parametric fits as functions of the orbital parameter $x=D/(2M)$. The approach extends prior KV-based ideas to puncture data and demonstrates consistency with PN predictions at large separations, offering a concrete, high-accuracy framework for initializing binary BH simulations. These results enhance the realism of initial data and facilitate precise studies of inspiral dynamics and gravitational wave generation.

Abstract

We construct a sequence of binary black hole puncture data derived under the assumptions (i) that the ADM mass of each puncture as measured in the asymptotically flat space at the puncture stays constant along the sequence, and (ii) that the orbits along the sequence are quasi-circular in the sense that several necessary conditions for the existence of a helical Killing vector are satisfied. These conditions are equality of ADM and Komar mass at infinity and equality of the ADM and a rescaled Komar mass at each puncture. In this paper we explicitly give results for the case of an equal mass black hole binary without spin, but our approach can also be applied in the general case. We find that up to numerical accuracy the apparent horizon mass also remains constant along the sequence and that the prediction for the innermost stable circular orbit is similar to what has been found with the effective potential method.

Quasi-equilibrium binary black hole sequences for puncture data derived from helical Killing vector conditions

TL;DR

The paper presents a method to generate quasi-equilibrium binary black hole initial data in puncture form by enforcing approximate helical Killing vector conditions alongside constant puncture ADM masses. By solving for momentum and lapse parameters with Newton-Raphson iterations along a sequence of separations, the work yields sequences where the apparent horizon mass remains constant and the ISCO location matches previous effective potential results, while providing detailed parametric fits as functions of the orbital parameter . The approach extends prior KV-based ideas to puncture data and demonstrates consistency with PN predictions at large separations, offering a concrete, high-accuracy framework for initializing binary BH simulations. These results enhance the realism of initial data and facilitate precise studies of inspiral dynamics and gravitational wave generation.

Abstract

We construct a sequence of binary black hole puncture data derived under the assumptions (i) that the ADM mass of each puncture as measured in the asymptotically flat space at the puncture stays constant along the sequence, and (ii) that the orbits along the sequence are quasi-circular in the sense that several necessary conditions for the existence of a helical Killing vector are satisfied. These conditions are equality of ADM and Komar mass at infinity and equality of the ADM and a rescaled Komar mass at each puncture. In this paper we explicitly give results for the case of an equal mass black hole binary without spin, but our approach can also be applied in the general case. We find that up to numerical accuracy the apparent horizon mass also remains constant along the sequence and that the prediction for the innermost stable circular orbit is similar to what has been found with the effective potential method.

Paper Structure

This paper contains 4 sections, 26 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quasi-circular sequences
  3. Discussion
  4. Computing masses at infinity on a finite grid

Figures (3)

  • Figure 1: The ADM mass at infinity, the total 2PN energy, and the apparent horizon mass as a function of angular velocity. The minimum in $M^{ADM}_{\infty}$ at $M \Omega_{min}=0.19\pm 0.03$ is marked by a dot.
  • Figure 2: The ADM angular momentum at infinity and the 2PN angular momentum as a function of angular velocity. The minimum in $J^{ADM}_{\infty}$ at $M \Omega_{min}=0.190\pm 0.015$ is marked by a dot.
  • Figure 3: $M^{ADM}_{\infty}$ computed using integral (\ref{['VolInt_approx0']}) over the numerical grid only, compared with $M^{ADM}_{\infty}$ computed using the extended volume integral (\ref{['VolInt_approx2']}) with $R_2 =10 R_1^2 /M$ for two different resolutions $h$. The extended integral (\ref{['VolInt_approx2']}) gives much more accurate results.