Recent progress in binary black hole simulations

Thomas Baumgarte, Bowdoin College

tbaumgar-at-bowdoin.edu

tbaumgar-at-bowdoin.edu

The past year has seen dramatic progress in numerical relativity
simulations of binary black holes. A number of groups have reported
significant advances and are now able to model the binary inspiral,
coalescence and merger together with the emitted gravitational wave
signal.

Simulating binary black holes has been a long-standing problem because it poses a number of ``grand challenges''. An incomplete list of these challenges includes the following

- Einstein's equations form a complicated, coupled set of non-linear PDEs, and it is far from clear which form of these equations is most suitable for numerical implementation.
- Somewhat related is the coordinate freedom, and the question what coordinate (or gauge) conditions lead to a non-pathological evolution.
- Black holes contain singularities, which can have very unfortunate consequences for numerical simulations.
- The individual black holes are much smaller than the wavelength of the emitted gravitational radiation. The resulting range in length-scales is difficult to accommodate in numerical simulations.

Pretorius (2005) first announced his new results at a numerical relativity workshop at the Banff International Research Station. Departing from numerical relativity convention he does not integrate a ``3+1'' decomposition of Einstein's equations that separates spatial and timelike parts, but instead discretizes the four-dimensional spacetime equations and their second derivatives directly. As suggested by a number of previous authors he introduces gauge source functions , in terms of which Einstein's equations reduce to wave equations for the components of the spacetime metric. In this formalism the coordinates are fixed through the gauge source functions (instead of the lapse and shift in 3+1 formalisms). Pretorius chooses to satisfy a somewhat ad-hoc wave equation and (which is related to spatial harmonic coordinates ). Pretorius uses black hole excision, whereby the black hole interior is removed from the computational grid. This is justified since the event horizon disconnects the interior causally from from the exterior. He also uses adaptive mesh refinement (AMR), which automatically allocates additional gridpoints in regions where they are needed to resolve small scale structures.

Several other features of his code are worth pointing out. The spatial coordinates are compactified, so that physically correct outer boundaries can be imposed at spatial infinity. He also introduces some numerical dissipation to control high-frequency instabilities, and added some ``constraint-damping'' terms that proved crucial for simulations of black holes.

With this code Pretorius has been able to simulate - without encountering a numerical instability - the inspiral and coalescence of black hole binaries through merger until late stages of the ring-down as the remnant settles into equilibrium. This is remarkable progress indeed. Figure 1 shows the trajectory of an inspiraling black hole binary and a gravitational waveform from his very recent calculation that adopts the state-of-the-art initial data of Cook and Pfeiffer (2004).

Following Pretorius' success four other groups (Campanelli *et.al. *
(2005), Baker *et.al. *(2005), Diener *et.al. *(2005) and most recently
Herrmann *et.al. *(2006)) have announced significant progress in their
binary black hole simulations. All of these calculations have several
features in common. They all use finite-difference implementations of
the BSSN equations^{1}, which
are based on a 3+1 formalism in contrast to Pretorius'
four-dimensional approach. They also use very similar gauge
conditions, namely ``1 + log'' slicing for the lapse, and a ``driver''
implementation of ``Gamma-freezing''. Finally they all use
``puncture'' initial data, which are constructed by absorbing the
singular terms in the black hole interior into an analytical
expression and solving for regular corrections.

The simulations of Campanelli *et.al. *(2005) and Baker *et.al. *(2005)
demonstrate that standard numerical relativity codes can handle binary
black holes with only very minor modifications, potentially opening
the field to a number of other groups. Herrmann *et.al. *(2006), for
example, adopt a technique very similar to that of Baker *et.al. *(2005).
While all other simulations focus on equal-mass black holes they
consider unequal-mass black holes with mass ratios
ranging from unity to 0.54. Their calculation represents a first step
toward analyzing the effect of binary parameters - including mass
ratios and black hole spins - on the gravitational waveforms in fully
dynamical simulations (even if the astrophysical relevance of their
initial data is somewhat limited). They also see evidence for
gravitational radiation recoil leading to a remnant ``kick'' (compare
the right panel of Figure 3).

**References:**

Alcubierre, M., Brügmann, B., Diener, P., Guzmán, F. S., Hawke, I.,
Hawley, S., Herrmann, F., Koppitz, M., Pollney, D., Seidel, E., &
Thornburg, J., 2005, Phys. Rev. D **72** 044004.

Baker, J. G., Centrella, J., Choi, D.-I., Koppitz, M., & van Meter, J.,
2005, submitted (also
gr-qc/0511103).

Baker, J. G., Brügmann, B., Campanelli, M., Lousto, C. O. & Takahashi, R.,
2001, Phys. Rev. Lett. **87**, 121103.

Brügmann, B., Tichy, W., & Jansen, N., 2004, Phys. Rev. Lett. **92**,
211101.

Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y.,
2005, submitted (also
gr-qc/0511048).

Cook, G. B., & Pfeiffer, H. P., 2004, Phys. Rev. D **70**, 104016.

Diener, P., Herrmann, F., Pollney, D., Schnetter, E., Seidel, E.,
Takahashi, R., Thornburg, J., & Ventrella, J., 2005, submitted (also
gr-qc/0512108).

Herrmann, F., Shoemaker, D., & Laguna, P., submitted (also
gr-qc/0601026).

Pretorius, F., 2005, Phys. Rev. Lett **95**, 121101.

Jorge Pullin 2006-02-28