Selected simulations performed with the Spectral Einstein Code

The following table lists some binary-black-hole (BBH) cases that we have evolved with the SXS collaboration's Spectral Einstein Code (SpEC), grouped into non-spinning, non-precessing and generic configurations. For the initial spin parameters S1/M12 and S2/M22, the z direction is parallel to the orbital angular momentum. In the "Merger" column, "Yes" means the simulation has evolved until the merged hole settles into its final state; otherwise the simulation has stopped about one orbit before merger. Click on the numbers in the "References" column to view the corresponding arXiv eprints.

N1 1 0 0 16 Yes 1, 2, 3, 4, 5 a
N2 2 0 0 15 Yes 4, 5 a, b
N3 3 0 0 15 - 4, 5 a, b
N4 4 0 0 15 - 5 a
N5 6 0 0 8 Yes 5 a
N6 1 0 0 17 -    
S1 1 0 0.5ez 15 -   c
S2 3 0 0.5ez 14 -   c
S3 1 -0.4ez -0.4ez 11 Yes 5, 6 a
S4 1 0.4ez 0.4ez 15 Yes 5 a
S5 1 0.1ez 0.1ez 18 -    
S6 1 0.2ez 0.2ez 19 -    
S7 1 0.3ez 0.3ez 19 -    
S8 1 0.5ez 0 15 -    
G1 1 -0.5ex -0.5ez 6.5 -   precessing
G2 2 0.2(ez-ex)/√2 -0.4(ez+ey)/√2 8.5 Yes 5, See below generic
G3 5 0.24ex-0.19ey-0.148ez 0 6 -   generic, d
G4 2.9622 0.24ex-0.19ey-0.148ez -0.158ex+0.370ey-0.045ez 6.25 -   generic, d
G5 2 -0.349ex-0.446ey+0.565ez 0 3 -   generic, |S1/M12|=0.8, d


  1. We and collaborators are currently using this simulation’s numerical-relativity waveform to underpin the effective-one-body parametrized analytical fit (Buonanno et. al.) and also the phenomenological parametrized analytical fit (Chen, Hannam, Husa, Parmeswaran) for future LIGO data analysis.
  2. An earlier 8-orbit version of this run has been used for calibrating EOB waveforms to numerical-relativity waveforms in Buonanno et al [Phys. Rev. D 79, 124028 (2009), arXiv:0902.0790].
  3. These ongoing runs are expected to proceed through about 15 orbits before merger.
  4. These simulations are being used as test cases for automating the process of generating initial data, interpolating onto the evolution grid, and evolving through inspiral, merger, and ringdown.

Selected figures for Case G2

2 0.2 (ez - ex)/√2 -0.4 (ez + ey)/√2 8.25

The following plots illustrate some of the features of case G2, which is an evolution of generic binary-black-hole initial data. The initial masses and spins for this simulation are summarized in the above table. The total number of orbits for this simulation is also shown.

Horizon trajectories during inspiral

The trajectories of hole 1 (blue) and hole 2 (red). Specifically, the centers of the apparent horizons in the asymptotically inertial coordinates of the simulation. The z=0 plane is shaded translucent gray; portions of the trajectories below z=0 then appear darker than portions of the curves that are above z=0.

GI Lev0 Trajectories


Spin precession during inspiral

The evolution of the spin directions for hole 1 (blue) and hole 2 (red). The tips of unit vectors (in the asymptotically inertial coordinates used in the simulation) trace out paths on the unit sphere (translucent gray).

GI Lev0 SpinPrecession


Horizon masses and spins during inspiral

The mass ratio and dimensionless spins are nearly constant throughout the 8.25 orbits of the inspiral shown here (with the merger occurring shortly thereafter). The axes are normalized by the total Arnowitt–Deser–Misner (ADM) mass of the system.

GI Lev0 MassAndSpin


Proper separation of the two holes' apparent horizons during inspiral

The separation of the holes decreases by about a factor of two throughout the simulation.

GI Lev0 ProperSep


Individual and common horizons

The individual and common apparent horizons are shown just after the common horizon forms. The wireframe mesh is colored by the value of the lapse.

gi casekmerger

Four Areas of Science


Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run twice as fast as that.

The Red Queen in Lewis Carroll's Through the Looking Glass

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About SXS

The SXS project is a collaborative research effort involving multiple institutions. Our goal is the simulation of black holes and other extreme spacetimes to gain a better understanding of Relativity, and the physics of exotic objects in the distant cosmos.

The SXS project is supported by Canada Research Chairs, CFI, CIfAR, Compute Canada, Max Planck Society, NASA, NSERC, the NSF, Ontario MEDI, the Sherman Fairchild Foundation, and XSEDE.