Can binary mergers produce maximally spinning black holes?

Gravitational waves carry away both energy and angular momentum as binary black holes inspiral and merge. The relative efficiency with which they are radiated determines whether the final black hole of mass $M_f$ and spin $S_f$ saturates the Kerr limit (${\chi }_f\equiv S_f/M_f^2\le 1$). Extrapolating from the test-particle limit, we propose expressions for $S_f$ and $M_f$ for mergers with initial spins aligned or anti-aligned with the orbital angular momentum. We predict the the final spin at plunge for equal-mass nonspinning binaries to better than 1%, and that equal-mass maximally spinning aligned mergers lead to nearly maximally spinning final black holes (${\chi }_f\simeq 0.9988$). We also find black holes can always be spun up by aligned mergers provided the mass ratio is small enough.

Figure 1

The final spin parameter ${\chi }_f$ for mergers with equal initial spin parameters (${\chi }_1={\chi }_2$) and mass ratio $\nu$. Each curve gives a different value of ${\chi }_1$, which can be read off the y-axis as ${\chi }_f={\chi }_1$ for $\nu =0$. Curves in blue (solid) denote our predictions, red (short-dashed) those of BKL, and green (long-dashed) those of HB. Top panel: the highly spinning regime (${\chi }_1\ge 0.95$). Bottom panel: aligned spins (${\chi }_1\ge 0$).

Figure 2

The final spin parameter ${\chi }_f$ as a function of $\nu$ for the merger of BBHs with equal and nearly maximal initial spin parameters (${\chi }_1={\chi }_2>0.99$). The red (short-dashed) curve shows the maximum initial spin (${\chi }_1\simeq 0.9916$) for which this function monotonically increases with $\nu$. The blue (long-dashed) curve shows the maximum initial spin (${\chi }_1\simeq 0.9987$) for which black holes are spun up by equal-mass mergers.

Figure 3

Top panel: The final spin parameter ${\chi }_f$ as a function of ${\chi }_i$ for the merger of BBHs with equal and nearly maximal initial spin parameters (${\chi }_1={\chi }_2={\chi }_i$). The blue (long-dashed) curve corresponds to equal-mass BBHs, the red (solid) curve to mergers with mass ratio ${\nu }_{\max }$ that maximizes ${\chi }_f(\nu )$, and the black (short-dashed) curve to the reference ${\chi }_f={\chi }_i$. Bottom panel: The value ${\nu }_{\max }({\chi }_i)$ described above.

Figure 4

Top panel: The final mass $M_f(\nu )$ for initally nonspinning mergers. The red (short-dashed) curve is the BKL approximation, while the blue (solid) curve is that of Eq. (3). The square, triangle, and circular points are three numerical estimates of $M_f$ [16]. Bottom panel: The final spin ${\chi }_f(\nu )$ for nonspinning mergers. The red (short-dashed) curve is the BKL prediction; the blue (solid) curve is that of Eq. (4). The black points are three estimates of the final spins from [16], while the green Xs are simulated final spins from [17].

Figure 5

Top panel: The final mass $M_f({\chi }_i)$ for equal-mass, equal-spin mergers. The red (short-dashed) curve is the BKL prediction, while the blue (solid) curve is that of Eq. (3). Square points are simulations from 21; the purple X is from 22. Bottom panel: The final spin parameter ${\chi }_f(\nu )$ for these mergers. The red (short-dashed) curve is the BKL prediction, the blue (solid) curve is that of Eq. (4). Black squares, purple Xs, and green triangles are from 18, 21, 22.

Figure 6

Final spin ${\chi }_f$ as a function of ${\chi }_1$ for the four sequences of equal-mass, unequal-spin simulations presented in 18. The value of ${\chi }_2$ for each sequence is listed on each panel, and we have exchanged ${\chi }_1$ and ${\chi }_2$ in the upper right panel for convenient presentation. Blue (solid) curves are our predictions, red (dashed) curves those of BKL.