{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "# The Spinning Effective One-Body Constant Coefficients\n", "\n", "## Author: Tyler Knowles\n", "\n", "## This module documents the reduced spinning effective one-body constant coefficients: terms calibrated to numerical relativity simulations.\n", "\n", "\n", "**Notebook Status:** In progress \n", "\n", "**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]( https://git.ligo.org/lscsoft/lalsuite) that was reviewed and approved for LIGO parameter estimation by the LIGO Scientific Collaboration.\n", "\n", "\n", "## Introduction\n", "### The Physical System of Interest\n", "\n", "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](https://arxiv.org/abs/0912.3517) 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.\n", "\n", "The fits to numerical relativity rely on the following physical parameters.\n", "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](https://arxiv.org/abs/0912.3517) Equation (5.1))\n", " \\begin{equation*}\n", " \\eta = \\frac{ m_{1} m_{2} }{ \\left( m_{1} + m_{2} \\right)^{2} }.\n", " \\end{equation*}\n", "1. Let ${\\bf S}_{1}$, ${\\bf S}_{2}$ be the spins of the compact objects in units of FIXME. Combining [BB2010](https://arxiv.org/abs/0912.3517) Equations (5.2), (5.64), and (5.67) we have that the spin of the deformed Kerr background is\n", " \\begin{equation*}\n", " {\\bf S}_{\\rm Kerr} = {\\bf S}_{1} + {\\bf S}_{2}.\n", " \\end{equation*}\n", "From [BB2010](https://arxiv.org/abs/0912.3517) Equation (4.9), we then define the parameter $a$ which appears in metric potenials for the Kerr spacetime as\n", " \\begin{equation*}\n", " a = \\frac{ \\left\\lvert {\\bf S}_{\\rm Kerr} \\right\\rvert }{ M^{2} }\n", " \\end{equation*}\n", "where $M = m_{1} + m_{2}$.\n", "\n", "We also use the Euler–Mascheroni constant $\\gamma$, [hard-coded in LALSuite](https://lscsoft.docs.ligo.org/lalsuite/lal/group___l_a_l_constants__h.html) with the following value:\n", " \\begin{equation*}\n", " \\gamma = 0.577215664901532860606512090082402431\n", " \\end{equation*}\n", "\n", "Please note that throughout this notebook we adpot the following conventions:\n", "1. $c = G = 1$ where $c$ is the speed of light in a vacuum and $G$ is Newton's gravitational constant, and\n", "1. $m_{1} \\ge m_{2}$.\n", "\n", "### Citations\n", "Throughout this module, we refer to\n", "* [Buonanno, Chen, and Damour (2006)](https://arxiv.org/abs/gr-qc/0508067) as BCD2006.\n", "\n", "LALSuite line numbers are taken from Git commit bba40f2 (see [LALSuite's GitLab page](https://git.ligo.org/lscsoft/lalsuite))." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Constants of fit to numerical relativity for the spinning effective one-body formulation\n", "\n", "# Import necessary NumPy, SymPy, and SEOBNR modules\n", "import numpy as np\n", "\n", "# Compute fits to numerical relativity\n", "def compute_const_coeffs(eta, EMgamma, a):\n", "\n", " # Define frequently-used constants\n", " asq = a*a\n", " pisq = np.pi*np.pi\n", "\n", " # Define constants that determine the fitting parameter K\n", " # See the discussion in https://arxiv.org/pdf/1311.2544.pdf between Equations (3) and (4)\n", " K0 = 1.712\n", " K1 = -1.803949138004582\n", " K2 = -39.77229225266885\n", " K3 = 103.16588921239249\n", "\n", " # Compute the fitting parameter K\n", " # See https://arxiv.org/abs/0912.3517 Equation (5.67) and the discussion following Equation 6.9\n", " # as well as https://arxiv.org/pdf/1311.2544.pdf\n", " K = K0 + K1*eta + K2*eta*eta + K3*eta*eta*eta\n", "\n", " # Define more frequently-used constants\n", " EtaKm1 = eta*K - 1.\n", " EtaKm1sq = EtaKm1*EtaKm1\n", "\n", " # Compute the Post-Newtonian coefficients\n", " # See https://arxiv.org/abs/0912.3517 Equations (5.77) to (5.81) and\n", " # https://arxiv.org/pdf/1311.2544.pdf Equation (2)\n", " Delta0 = K*(EtaKm1 - 1.)\n", " Delta1 = -2.*(Delta0 + K)*EtaKm1\n", "\n", " Delta1sq = Delta1*Delta1\n", " Delta1cu = Delta1*Delta1sq\n", " Delta1ft = Delta1cu*Delta1\n", "\n", " Delta2 = 0.5*Delta1*(Delta1 - 4.*EtaKm1) - asq*EtaKm1sq*Delta0\n", "\n", " Delta2sq = Delta2*Delta2\n", "\n", " Delta3 = -Delta1cu/3. + Delta1*Delta2 + Delta1sq*EtaKm1 - 2.*(Delta2 - EtaKm1)*EtaKm1 - asq*Delta1*EtaKm1sq\n", " Delta4 = 1./12.*(6*asq*(Delta1sq - 2*Delta2)*EtaKm1sq + 3*Delta1ft - 8*EtaKm1*Delta1cu - 12*Delta2*Delta1sq\n", " + 12*(2*EtaKm1*Delta2 + Delta3)*Delta1 + 12*(94./3. - 41./32.*pisq)*EtaKm1sq\n", " + 6*(Delta2*Delta2 - 4*Delta3*EtaKm1))\n", " Delta5 = EtaKm1sq*(-4237./60. + 128./5.*EMgamma + 2275./512.*pisq - asq*(Delta1cu - 3.*Delta1*Delta2 + 3.*Delta3)/3.\n", " - (Delta1ft*Delta1 - 5.*Delta1cu*Delta2 + 5.*Delta1*Delta2sq + 5.*Delta1sq*Delta3\n", " - 5.*Delta2*Delta3 - 5.*Delta1*Delta4)/(5.*EtaKm1sq) + (Delta1ft - 4.*Delta1sq*Delta2\n", " + 2.*Delta2sq + 4.*Delta1*Delta3 - 4.*Delta4)/(2*EtaKm1) + (256./5.)*np.log(2))\n", " Delta5l = (64./5.)*EtaKm1sq\n", "\n", " #Add comment here\n", " dSO = -74.71 - 156.*eta + 627.5*eta*eta\n", " dSS = 8.127 - 154.2*eta + 830.8*eta*eta\n", "\n", " return K, Delta0, Delta1, Delta2, Delta3, Delta4, Delta5, Delta5l, dSO, dSS" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.3" } }, "nbformat": 4, "nbformat_minor": 2 }