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The close-limit approximation and numerical simulations

For a subset of black hole collisions, where the black holes collide head-on, there exist as of today (Feb 1999) reasonably reliable full numerical simulations of the collision. Because of the lack of an inspiral phase, the waveform profiles of these kind of collisions are completely dominated by the ringdown of the final black hole [for a recent reference see Anninos, Brandt, and Walker [49] (ABW). Even for this simple case, there are some discrepancies between various numerical codes.

A separate approach to black hole collisions has been the close-limit approximation (see Khanna et al. gr-qc/9905081 for a description of the close-limit approximation applied to inspiralling black holes), which describes the merger of two black holes as a perturbation of a single black hole; the perturbation is based on a small parameter measuring the separation of the two black holes. This approximation has had an uncanny degree of success in replicating (at least for the head-on case) numerical estimates of the merger waveform, and it provides ``a little-bit more'' of the merger waveform than the quasinormal ringdown. It is useful to examine how good ringdown filters will work in detecting the more realistic waveforms produced by the close-limit approximation. In this addendum we will describe the use of both close limit and full nuemerical simulations in the GRASP package.

There are two data sets containing full numerical waveforms for head on collisions of two black holes released initially from rest. These correspond to the two codes described in ABW [49] for a moderate separation of the holes ($\mu_0=1.9$ in the ABW [49] notation, about 10 in terms of single black hole radii). These waveforms are shown in figure [*].

Figure: ABW waveforms with $\mu_0=1.9$.

These two waveforms correspond to the two codes described in ABW [49], based on two different set of coordinate systems. To a certain extent, they exhibit the limitations of the state of the art in numerical relativity: both waveforms represent the best effort by correct, ``convergent'' codes, and yet there is some disagreement among them. This disagreement can be settled by comparing with the close approximation (see ABW [49]). But it is also instructive to compare how bad the disagreement is from the point of view of a data analyst.

The fitting factor is defined by

\eta = \max_t r(t) = \max_t \alpha^{-1} \Re \int_0^\infty df e^{-2\pi ift}
\end{displaymath} (8.19.188)

\alpha^2 = \int_0^\infty df \frac{\Vert\tilde{a}(f)\Vert^2}...
\times\int_0^\infty df \frac{\Vert\tilde{b}(f)\Vert^2}{S(f)}
\end{displaymath} (8.19.189)

with $a(t)$ being the close-limit approximation waveform, $b(t)$ being the ringdown waveform, and $S(f)$ being the detector noise power spectrum. This is computed by the program corr (below). For the LIGO-I interferometer noise curve and a $200\,M_\odot$ black hole, the fitting factors are $91\%$ and $85\%$ for the full numerical waveforms. These factors represent the fraction of the signal-to-noise ratio that the ringdown filter will obtain relative to the optimal filter. As can be seen in the following figure, the moment at which the ABW [49] waveform looks most ``ringdown-like'' is not the moment at which the peak signal-to-noise ratio is obtained.

Figure: Pure ringdown fits to the ABW waveform. First: the correlation between the ABW waveform and a pure ringdown as a function of time. Second: pure ringdowns superimposed on the ABW waveform for the times of greatest correlation.

From the point of view of source modelling, the difference in head-on collisions between the full numerical waveforms and the close limit ones is not substantial.

Authors: Jolien Creighton ( and Jorge Pullin (

next up previous contents
Next: Inspiralling collisions Up: GRASP Routines: Black hole Previous: Function: qn_template_grid()   Contents
Bruce Allen 2000-11-19