Analytical Solution of Spinning, Eccentric Binary Black Hole Dynamics at the Second Post-Newtonian Order

Abstract

Recent gravitational wave (GW) detections showing signatures of eccentricity and spin precession underscore the need to model binary black holes (BBHs) possessing these features simultaneously. Most efforts over the past fifteen years to model spinning BBHs and their corresponding GWs have relied on heuristically twisting waveforms from non-precessing systems. This approach is based on empirical observations rather than first principles. This article aims to model the GWs from spinning and eccentric BBHs from a first-principles approach within general relativity and post-Newtonian (PN) approximation. Building on the already-existing 1.5 PN solution, we construct an analytical solution for the time evolution of the relative separation vector, the individual black hole spin vectors, and the orbital angular momentum vector at 2PN order for BBHs with arbitrary spins and eccentricity. Such a solution is not fully 2PN accurate in that the tiny orbital timescale fluctuations in the solutions for the spins are only leading 1.5PN order accurate, instead of 2PN. However, it is shown that our new 2PN solution is still an order of magnitude improvement over the earlier 1.5PN solution, underlining the sub-dominant nature of the neglected next-to-leading-order oscillations in the spin solutions.

Figures (3)

• Figure 1: Evolution of $\mathbf{S}_{1x}$ for an eccentric BBH with $e = 0.61$, $q \equiv m_2/m_1 = 0.5$, and $x_{\mathrm{PN}} = 2 \times 10^{-2}$. Initial angles: $\langle \mathbf{L}, \mathbf{S}_{1} \rangle = 32^\circ$, $\langle \mathbf{L}, \mathbf{S}_{2}\rangle = 82^\circ$, $\langle \mathbf{S}_{1}, \mathbf{S}_{2}\rangle = 54^\circ$. We compare the evolution under four different evolution schemes: 1.5PN solution with Newtonian $r$ injected (0PN QKP, blue dashed), the same analytical solution with 1PN $r$ injected (1PN QKP, orange dashed), 1.5PN numerical solution (green), and 2PN numerical solution (red). Left: Evolution over one slow precession period. Right: Two-orbit zoomed-in view into the time interval $[12\tau_{\mathrm{orb}}, 14\tau_{\mathrm{orb}}]$, highlighting the fast orbital-timescale oscillations.
• Figure 2: The solid frame represents the inertial frame (IF) $(\mathbf{x},\mathbf{y},\mathbf{z})$ centered on $\mathbf{J}$. The dashed frame is the non-inertial frame (NIF) $(\mathbf{i},\mathbf{j},\mathbf{k})$ centered on $\mathbf{L}$, obtained by rotating the IF first by angle $\theta_L$ around the $z$-axis, followed by a rotation of angle $\phi_L$ around the $x$-axis.
• Figure 3: Comparison of the spin vector $\mathbf{S}_1$ across approximate models against the full 2PN dynamics obtained by numerical integration (red). The $1.5$PN solution $(1.5)$ (blue), the hybrid model $(2,\mathrm{H})$ (orange), and the orbit-averaged 2PN solution $(2,\mathrm{A})$ (green) are shown. The hybrid model accurately recovers the leading-order fast orbital oscillations of the 2PN evolution while preserving the correct secular behavior; the orbit-averaged model reproduces only the secular evolution; and the $1.5$PN model captures the oscillations but misses the secular spin--spin effects. Results are shown for four systems with different mass ratios ($q = m_2/m_1$) and eccentricities, all sharing the random same spin orientations: $\kappa_1 = 68^\circ$, $\kappa_2 = 10^\circ$, $\gamma = 73^\circ$. The first three columns display the $x$-component of $\mathbf{S}_1$ over increasing timescales, from orbital periods to precession times, while the fourth column shows the difference between each approximate solution and the numerical 2PN result. The vertical axis of the fourth column is normalized by $x_{\rm PN} \sim 10^{-2}$: a value of $y = 0$ corresponds to a 2PN-level error ($\mathcal{O}(c^{-4})$), while $y = 2$ indicates the error has grown to 1PN order ($\mathcal{O}(c^{-2})$).