Periastron advance in spinning black hole binaries: comparing effective-one-body and numerical relativity

We compute the periastron advance using the effective-one-body formalism for binary black holes moving on quasicircular orbits and having spins collinear with the orbital angular momentum. We compare the predictions with the periastron advance recently computed in accurate numerical-relativity simulations and find remarkable agreement for a wide range of spins and mass ratios. These results do not use any numerical-relativity calibration of the effective-one-body model, and stem from two key ingredients in the effective-one-body Hamiltonian: (i) the mapping of the two-body dynamics of spinning particles onto the dynamics of an effective spinning particle in a (deformed) Kerr spacetime, fully symmetrized with respect to the two-body masses and spins, and (ii) the resummation, in the test-particle limit, of all post-Newtonian corrections linear in the spin of the particle. In fact, even when only the leading spin post-Newtonian corrections are included in the effective-one-body spinning Hamiltonian but all the test-particle corrections linear in the spin of the particle are resummed we find very good agreement with the numerical results (within the numerical error for equal-mass binaries and discrepancies of at most 1% for larger mass ratios). Furthermore, we specialize to the extreme mass-ratio limit and derive, using the equations of motion in the gravitational skeleton approach, analytical expressions for the periastron advance, the meridional Lense-Thirring precession and spin precession frequency in the case of a spinning particle on a nearly circular equatorial orbit in Kerr spacetime, including also terms quadratic in the spin.

Figure 1

Parameter choice for the binary geometry. The ${\mathit{e}}_3$ axis of the fixed frame is along the direction of the total angular momentum $\mathit{J}$. The orbital plane is perpendicular to $\mathit{k}=\mathit{L}/L$ and the vector $\mathit{i}$ points to the intersection of the orbital plane and the plane normal to ${\mathit{e}}_3$. The total spin $\mathit{\sigma }$ and $\mathit{L}$ subtend the angle ${\alpha }_{LS}$. The projection of ${\mathit{S}}_1$ onto the $(\mathit{i},{\widehat{e}}_Z^{\sigma }\times \mathit{i})$ plane defines the angle ${\varphi }_S$.

Figure 2

Aligned spins, equal masses. The EOB results are shown as red dashed lines. The solid blue curves are the fits to the NR data and the shaded region indicates the error estimate. The dash-dotted black curves are the PN predictions.

Figure 3

Antialigned spins, equal masses.

Figure 4

Varying the spin parameter. Binaries with equal spins and mass ratio $q=1$ at a fixed frequency $M{\Omega }_{\varphi }=0.02$. The red circles are the EOB result; the blue diamonds are the NR data points, together with their error bounds. The black squares are the PN predictions. The inset shows an enlargement for spins $\ge 0.6$.

Figure 5

Unequal masses. Results for a binary with mass ratio $q=3$ and a single spin ${\chi }_1=\mp 0.5$.

Figure 6

Varying the mass ratio. Results for binaries with a single spin ${\chi }_1=\mp 0.5$ at $M{\Omega }_{\varphi }=0.02$.

Figure 7

Effect of varying the PN spin information included in the EOB model. This figure shows, for the case ${\chi }_1={\chi }_2=0.95$, $q=1$, the fractional differences in the PA prediction from the EOB model and the NR data when either we vary the PN order of the spin terms in the baseline EOB model or when we consider the EOB model of Refs. 16, 33, 37 in which the terms linear in the effective particle’s spin are not deformations of the results for a spinning particle in Kerr. All the EOB models employ the nonspinning 3PN terms. The gray shaded area indicates the uncertainty of the numerical data $K_{\mathrm{NR}}$.

Figure 8

Postgeodesic effects in Schwarzschild. Gravitational self-force and spin-dipole effects at $O(\nu )$ in the quantities $Q=(E({\Omega }_{\varphi }),K({\Omega }_{\varphi }))$ for a Schwarzschild background. Upper panel: Relative importance of conservative spin-dipole and SF effects for strictly ($Q=E$, solid lines) and nearly ($Q=K$, dashed lines) circular orbits. Lower panel: Combined $O(\nu )$ effect for maximally spinning particles with $s=1$. Solid lines are for $\mathrm{sgn}(S_{*}{L}_z)=+1$, dashed lines for $\mathrm{sgn}(S_{*}{L}_z)=-1$.