The General Relativistic Two Body Problem and the Effective One Body Formalism

Abstract

A new analytical approach to the motion and radiation of (comparable mass) binary systems has been introduced in 1999 under the name of Effective One Body (EOB) formalism. We review the basic elements of this formalism, and discuss some of its recent developments. Several recent comparisons between EOB predictions and Numerical Relativity (NR) simulations have shown the aptitude of the EOB formalism to provide accurate descriptions of the dynamics and radiation of various binary systems (comprising black holes or neutron stars) in regimes that are inaccessible to other analytical approaches (such as the last orbits and the merger of comparable mass black holes). In synergy with NR simulations, post-Newtonian (PN) theory and Gravitational Self-Force (GSF) computations, the EOB formalism is likely to provide an efficient way of computing the very many accurate template waveforms that are needed for Gravitational Wave (GW) data analysis purposes.

Figures (4)

• Figure 1: Sketch of the correspondence between the quantized energy levels of the real and effective conservative dynamics. $n$ denotes the 'principal quantum number' ($n = n_r + \ell + 1$, with $n_r = 0,1,\ldots$ denoting the number of nodes in the radial function), while $\ell$ denotes the (relative) orbital angular momentum $({\bm L}^2 = \ell (\ell + 1) \, \hbar^2)$. Though the EOB method is purely classical, it is conceptually useful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables $I_R$ and $J$ to motivate the identification between $n$ and $\ell$ in the two dynamics.
• Figure 2: This figure illustrates the comparison (made in Refs. DamourNagar2009DamourNagar2011) between the (NR-completed) EOB waveform (Zerilli-normalized quadrupolar ($\ell=m=2$) metric waveform (\ref{['eqn18']}) with parameter-free radiation reaction (\ref{['eq:RR_new']}) and with $a_5=0$, $a_6=-20$) and one of the most accurate numerical relativity waveform (equal-mass case) nowadays available Scheel2009. The phase difference between the two is $\Delta\phi\leq\pm 0.01$ radians during the entire inspiral and plunge, which is at the level of the numerical error.
• Figure 3: Close up around merger of the waveforms of Fig. \ref{['fig:waveform']}. Note the excellent agreement between both modulus and phasing also during the ringdown phase.
• Figure 4: Comparison (made in Damour2012) between various analytical estimates of the energy-angular momentum functional relation and its numerical-relativity estimate (equal-mass case). The standard "Taylor-expanded" 3PN $E(j)$ curve shows the largest deviation from NR results, especially at low $j$'s, while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB $E(j)$ curves agree remarkably well with the NR one.