The Washington University Gravity Group hosted a dynamic and exciting meeting in St. Louis entitled ``Numerical Relativity meets 3PN: A Workshop'', from February 8-11, 2007. In attendance were many of the leading researchers in the fields of numerical relativity, post-Newtonian theory, and gravitational-wave data analysis. The purpose of the meeting was to bring these researchers together in an effort to stimulate progress at the important interface between these fields. The workshop was organized with a set of invited talks each morning, designed to probe the central questions and problems with ample time for discussions. Each afternoon started with a small number of short contributed talks followed by an open ended working/discussion session focused on a different topic each afternoon. The workshop program can be found at http://nrm3pn.wustl.edu/. Below, we summarize the highlights of the meeting and attempt to give proper credit to individuals and the groups they represent. We apologize for any errors or omission, and for the fact that we cannot mention everyone who contributed to this very successful meeting.
During the workshop all of the major research groups performing simulations of black-hole binary collisions presented their most recent results [Pretorius (Alberta/Princeton); Lousto (UT Brownsville); Baker (NASA Goddard); Scheel (Cornell/Caltech); Hannam, Husa (Jena/AEI); Laguna (Penn State)], along with results from groups simulating NS-NS collisions [Suen (WU)] and NS-BH collisions [Faber(UIUC)]). But the focus was on the BH-BH inspiral phase which can be compared with PN models. Prior to the workshop, various groups had obtained simulations with as many as 8 orbits (16 gravitational-wave cycles) prior to merger. These simulations showed good agreement with analytic PN results at 3.5PN order [Baker; Buonanno (UMD)]. More specifically, standard PN models (expanded form of the balance equations) and the effective-one-body (EOB) model at 3.5PN match well with the inspiral waveforms obtained using the generalized harmonic and moving puncture codes. While the initial agreement is good, longer and more accurate simulations will determine how significant the dephasing is between the numerical and analytic predictions.
A significant problem with all of the simulations is that
the initial data
used by most groups place the binary in a ``quasi-circular'' orbit with
an effective eccentricity that modulates both the amplitude and the
phase of the gravitational wave. Scheel
showed preliminary results from the Cornell/Caltech group
for an inspiral waveform emitted by a
non-spinning equal-mass binary where the eccentricity had been reduced
to 
. The trajectory of a non-spinning equal-mass black-hole
binary moving along an adiabatic inspiral is unique, and this was the
first time such an evolution had been simulated. Interestingly,
the evolution was obtained without using PN initial data.
One of the more important aspects of the group discussions concerned how to quantify the differences between gravitational-wave signals, whether the differences are between NR and PN waveforms, or between waveforms created by different codes or with different initial data. Much of this discussion was guided by the data-analysis community [Sathyaprakash (Cardiff); Owen, Finn (PSU)]. There were also discussions of which kind of comparisons should be pursued in the future. Should one compare trajectories in the same gauge and coordinates, or focus on more invariant quantities?
While the talks and discussion naturally focused largely on the inspiral phase, other aspects of black-hole binary collisions were discussed. Some of the most surprising and intriguing new results presented at the workshop concerned the recoil velocity of the final black hole following binary coalescence [Lousto; Sperhake (Jena/AEI); Laguna]. The first two numerical relativity groups showed that for special spin configurations the kick can be unexpectedly large. Analytic studies aimed at understanding how the kick builds up during the inspiral, merger and ringdown phases were presented.
The ringdown signal following coalescence certainly contains valuable information and it is possible to extract a few of the quasi-normal modes of Kerr from these ringdown waveforms. Results were shown both for equal-mass [Buonanno] and unequal-mass binary systems [Berti (WU); Tiglio (LSU)]. Most of the presentation and discussion concerned new techniques to improve the fits.
The transition region connecting the end of the inspiral through the merger to the ring down was also considered. Combining an EOB waveform with a superposition of a fundamental quasi-normal mode plus two overtones by matching them at the light ring [Buonanno], it was shown that the ringdown phase could be modeled reasonably well. However, we lack an analytic description of how the quasi-normal modes are excited, whether non-linearities and/or mode-mixing are important. Understanding these issues is needed to improve the analytic matching of the inspiral to ring down.
There were long discussions on data analysis issues. Considering the current numerical results and studies, it seems that simple modifications of existing PN and resummed-PN template banks, guided by NR results, should lead to high matching performances with numerical waveforms and should be used for signal detection. However, for parameter estimation, we need to improve the current PN template banks. There were also discussions on if and by how much one should expand the template bank if one thinks that the real waveforms deviate from PN or NR predictions.
Finally, diagnostics of NR results through PN tools [Cook (Wake Forest); Will (WU)] and initial-data issues were discussed. New approaches for implementing PN initial data were presented [Blanchet (IAP Paris); Tichy (FL Atlantic)]. A central goal of these approaches is to incorporate a physically realistic gravitational-wave signal into the initial data. It is hoped that this will help one to understand and perhaps eliminate the initial burst of unphysical radiation seen in current evolutions. In spite of the initial burst of unphysical radiation, the current families of numerical initial data are proving to be very effective. Considering that one can reduce the eccentricity and get the unique adiabatic inspiral, there were discussions as to whether or not one needs to improve the initial data. There were no definite conclusions, but it seems that the current data may be sufficient for binaries on circular orbits. However, more work needs to be done with spinning and precessing binaries, and eccentric binary systems.
The meeting ended with a panel discussion. The moderator was Sam Finn, with Alessandra Buonanno, Greg Cook and Pablo Laguna serving as panelists. Finn started the session with introductory remarks about the theme of the NRm3PN meeting, namely how NR meets 3PN in understanding dynamics, nonlinear theory, applications to data analysis, and astrophysics. Next, each of the panelists was asked to recall the most interesting, unexpected, exciting or novel thing learned at the workshop. Buonanno responded that the work presenting the comparison of PN and NR waveforms was very encouraging and that the work presented by the Caltech/Cornell group, which succeeded in reducing the eccentricity to negligible values, showed, for the first time, the true, unique inspiral for an equal-mass binary on a quasi-circular orbit. Both Cook and Laguna pointed to the results presented by several groups on kicks as being extremely interesting, with Laguna pointing out that they also provide validation of the computational infrastructure. The panel session ended with Finn posing several forward looking questions: 1) What is the single-most important question that needs to be resolved for NR to become useful for data analysis? Buonanno's answer was to generate longer and more accurate waveforms for spinning, precessing binary systems; to understand more in detail how the ringdown modes are excited at the end of the inspiral phase; 2) What is the most tantalizing indication of unexpected, novel or surprising behavior associated with non-linear gravity? Cook responded that in his opinion it would be the lack of a clearly distinguishable dynamical plunge prior to merger; 3) What are the questions arising from the recent results on gravitational recoil? Laguna pointed out that current results have only probed a very small fraction of the parameter space. The time is right for a systematic collaborative exploration involving several groups.
The next installment of ``NRm3PN'' will be June 12 - 14, 2008 at the University of Jena, as part of a conference recognizing Gerhard Schäfer's 60 birthday.