The Spinning Effective One-Body Constant Coefficients¶

Author: Tyler Knowles¶

This module documents the reduced spinning effective one-body constant coefficients: terms calibrated to numerical relativity simulations.¶

Notebook Status: In progress

Validation Notes: This module is under active development -- do *not* use the resulting code for scientific applications. In the future, this module will be validated against the LALSuite SEOBNRv3/SEOBNRv3_opt code that was reviewed and approved for LIGO parameter estimation by the LIGO Scientific Collaboration.

Introduction¶

The Physical System of Interest¶

Consider two compact objects (e.g. black holes or neutron stars) with masses $m_{1}$, $m_{2}$ (in solar masses) and spin angular momenta ${\bf S}_{1}$, ${\bf S}_{2}$ in a binary system. The spinning effective one-body ("SEOB") Hamiltonian $H_{\rm real}$ (see BB2010 Equation (5.69)) describes the dynamics of this system. Certain terms of this $H_{\rm real}$ are calibrated to numerical relativity simulations, and we document those terms here.

The fits to numerical relativity rely on the following physical parameters.

1. Let $m_{1}$, $m_{2}$ be the masses of the compact objects in units of solar mass. We then define the symmetric mass ratio $\eta$ of the system to be (see discussion preceeding BB2010 Equation (5.1)) \begin{equation*} \eta = \frac{ m_{1} m_{2} }{ \left( m_{1} + m_{2} \right)^{2} }. \end{equation*}
2. Let ${\bf S}_{1}$, ${\bf S}_{2}$ be the spins of the compact objects in units of FIXME. Combining BB2010 Equations (5.2), (5.64), and (5.67) we have that the spin of the deformed Kerr background is \begin{equation*} {\bf S}_{\rm Kerr} = {\bf S}_{1} + {\bf S}_{2}. \end{equation*}

From BB2010 Equation (4.9), we then define the parameter $a$ which appears in metric potenials for the Kerr spacetime as \begin{equation*} a = \frac{ \left\lvert {\bf S}_{\rm Kerr} \right\rvert }{ M^{2} } \end{equation*} where $M = m_{1} + m_{2}$.

We also use the Euler–Mascheroni constant $\gamma$, hard-coded in LALSuite with the following value: \begin{equation*} \gamma = 0.577215664901532860606512090082402431 \end{equation*}

Please note that throughout this notebook we adpot the following conventions:

1. $c = G = 1$ where $c$ is the speed of light in a vacuum and $G$ is Newton's gravitational constant, and
2. $m_{1} \ge m_{2}$.

Citations¶

Throughout this module, we refer to

• Buonanno, Chen, and Damour (2006) as BCD2006.

LALSuite line numbers are taken from Git commit bba40f2 (see LALSuite's GitLab page).