Periastron advance in spinning black hole binaries: Gravitational self-force from numerical relativity

We study the general relativistic periastron advance in spinning black hole binaries on quasicircular orbits, with spins aligned or antialigned with the orbital angular momentum, using numerical-relativity simulations, the post-Newtonian approximation, and black hole perturbation theory. By imposing a symmetry by exchange of the bodies’ labels, we devise an improved version of the perturbative result and use it as the leading term of a new type of expansion in powers of the symmetric mass ratio. This allows us to measure, for the first time, the gravitational self-force effect on the periastron advance of a nonspinning particle orbiting a Kerr black hole of mass $M$ and spin $S=-0.5M^2$, down to separations of order $9M$. Comparing the predictions of our improved perturbative expansion with the exact results from numerical simulations of equal-mass and equal-spin binaries, we find a remarkable agreement over a wide range of spins and orbital separations.

Figure 1

Periastron advance extracted from numerical simulations. Upper panel: The dashed curves show $K_{\mathrm{NR}}({\Omega }_{\phi })/K_{\mathrm{Sch}}({\Omega }_{\phi })$ as computed from Eqs. (3, 4) using fitting intervals with two different widths $\varpi$. The solid lines show polynomial fits to $K_{\mathrm{NR}}/K_{\mathrm{Sch}}$. Lower panel: Residuals of the polynomial fits.

Figure 2

Relative uncertainty $\Delta K/K$ in the numerical-relativity periastron advance as a function of the eccentricity $e$ of the configuration. Shown are data for four black-hole binaries with different mass ratios $q=m_1/m_2$, one of them with a nonzero spin. Each symbol represents a separate numerical binary black hole evolution. The results shown here were computed at the orbital frequency $m{\Omega }_{\phi }=0.033$.

Figure 3

Effect of the choice of width $\varpi$ on the measured periastron advance. Shown are data for the three reference frequencies ${\Omega }_{e/m/l}$ and for two exemplary runs: $(q,{\chi }_1,{\chi }_2)=(1,0,0)$ and $(q,{\chi }_1,{\chi }_2)=(8,0.5,0)$. The symbols denote $K_{\mathrm{NR}}/K_{\mathrm{Sch}}$ as measured with width $\varpi$ indicated on the $x$ axis. The dotted lines denote fits indicating the extrapolation to zero width, $\varpi \to 0$. The number next to each dotted line indicates the fractional change in $K_{\mathrm{NR}}/K_{\mathrm{Sch}}$ between $\varpi =1.2$ and $\varpi \to 0$. For ease of plotting, the data for $q=8$ and $m{\Omega }_{\phi }=0.036$ has been shifted up by 0.1.

Figure 4

Fractional difference between the NR and PN predictions for the periastron advance $K$ as a function of spin, at different PN orders, for equal-mass black-hole binaries. We set $m{\Omega }_{\phi }=0.021$.

Figure 5

Fractional difference between the NR and PN predictions for $K$ for black-hole binaries with mass ratios $q\in \{1,1.5,3,5,8\}$ and spins $\chi \equiv {\chi }_1\in \{-0.5,0,0.5\}$ and ${\chi }_2=0$. We set $m{\Omega }_{\phi }=0.021$.

Figure 6

Periastron advance $K$ as a function of the orbital frequency $m{\Omega }_{\phi }$, for equal-mass binaries with equal spins ${\chi }_1={\chi }_2=0.9$ (top) and ${\chi }_1={\chi }_2=-0.9$ (bottom). The black dashed lines show the estimated numerical-relativity uncertainties.

Figure 7

The difference $\delta W=W_{\mathrm{NR}}-W_{\mathrm{SB}}$ as a function of the orbital frequency $m{\Omega }_{\phi }$, for nonspinning binaries with mass ratios $q=1$ (blue lines), 1.5 (red lines), 3 (green lines), 5 (orange lines), and 8 (cyan lines). The dashed lines show the estimated NR uncertainties.

Figure 8

The rescaled difference $\delta W/\nu$ as a function of $m{\Omega }_{\phi }$, for nonspinning binaries, including (top) and excluding (bottom) the uncertainties affecting the NR results.

Figure 9

The difference $\delta W=W_{\mathrm{NR}}-W_{\mathrm{SB}}$ as a function of the orbital frequency $m{\Omega }_{\phi }$, for spinning binaries with $({\chi }_1,{\chi }_2)=(-0.5,0)$ and mass ratios $q=1.5$ (red lines), 3 (green lines), 5 (orange lines), and 8 (cyan lines). The dashed lines show the estimated NR uncertainties.

Figure 10

The rescaled difference $\delta W/\nu$ as a function of $m{\Omega }_{\phi }$, for spinning binaries with $({\chi }_1,{\chi }_2)=(-0.5,0)$, including (top) and excluding (bottom) the uncertainties affecting the NR results.

Figure 11

Gravitational self-force correction $W_{\mathrm{GSF}}$ to the periastron advance of a nonspinning particle of mass $\mu$ orbiting a black hole of mass $M$ and spin $S\equiv \chi M^2$, as measured using NR simulations of black-hole binaries with mass ratios $q=1$, 1.5, 3, 5, 8. Also shown are the 3.5PN prediction (red lines) and the exact result for $\chi =0$ (blue lines).

Figure 12

The periastron advance $K$ as a function of the circular-orbit frequency $m{\Omega }_{\phi }$ for equal-mass, equal-spin configurations, as computed using NR simulations (black lines), post-Newtonian theory to 3.5PN order (cyan lines), and the improved perturbative expansion (35) to first order (red lines).