Spin stability of a binary black hole coalescence

We analyze the spin stability of a binary black hole coalescence when the binary system is described by the Post-Newtonian (PN) equations in the adiabatic regime. The main idea in this work is to make a massive exploration of the solution space in search of chaos. For that, we evolve the PN equations using a CUDA implementation of the RKF78 scheme and study the dynamical behavior of the system. Each initial spin configuration run in the GPU is composed by more than 80000 simulations. The chaos indicator used to characterize the degree of separation of two infinitesimally close trajectories is the Lyapunov exponent. We find zones in the solution space where the separation between nearby trajectories reaches several orders of magnitude bigger than the initial separation. Also, we note that the chaotic behavior can be observed in forward as well as in backward evolution.

Figure 1

This figure shows the architecture of Omega-CUDA; the gray blocks represent the folders where the files are located. For simplicity, some headers are not included in the diagram.

Figure 2

The figures show the same initial spin configurations with a mass $m_1=0.7$, $m_2=0.3$ in the left figure, and $m_1=0.5$, $m_2=0.5$ in the right one. In both plots, we “sweep” over $S_{1x}$ and $S_{1y}$, keeping the remaining components fixed.

Figure 3

These figures show the same previous initial configuration assuming $m_1=0.4$, $m_2=0.6$ and $m_1=0.3$, $m_2=0.7$, respectively.

Figure 4

These figures show the separation between two between two trajectories. The initial spin configurations are $S_1=[S_{1x},S_{2y},1]$, $S_2=[0,0,-1]$ for the right plot and $S_1=[S_{1x},S_{2y},-1]$, $S_2=[0,0,-1]$ for the left.

Figure 5

Plots for the spin configurations $S_1=[S_{1x},S_{2y},1]$, $S_2=[0,-1,-1]$ for the right plot and $S_1=[S_{1x},S_{2y},-1]$, $S_2=[0,-1,-1]$ for the left.

Figure 6

In the left figure, $S_1=[S_{1x},S_{2y},0]$, $S_2=[1,0,0]$, and the right figure is started with $S_1=[S_{1x},S_{2y},0]$, $S_2=[-1,0,0]$.

Figure 7

In both figures $S_2=0$, with the initial orientation of $S_{1z}$ aligned (left figure) or antialigned (right figure) with $L_z$.

Figure 8

Q coefficient for one spinning black hole. On the left, we plot the configuration $S_1=[S_{1x},S_{1y},0]$, $S_2=0$. On the right, we plot a rotation around the point (0, 0) for the previous configuration $S_{1z}=0$, where the perturbations are aligned with the $S_{1x},S_{1y}$ direction.

Figure 9

These plots show $Q$ vs $t$ and vs $|t|$ for the two configurations shown in the previous table. The left plot correspond to a forward integration of the i.c. 1 and 2 staring from $\omega =0.01$ to ${\omega }_{\max }$, while the right plot is a backward integration in time starting from the i.c. at $t_m$.

Figure 10

This figure shows $\ln (Q)$ vs $|t|$, the orange curve corresponds to the least-squares fit to the simulation data. The slope of this curve is equal to the principal Lyapunov exponent $0.25\times {10}^{-6}$.