Efficient multi-timescale dynamics of precessing black-hole binaries

We present analytical and numerical progress on black-hole binary spin precession at second post-Newtonian order using multitimescale methods. In addition to the commonly used effective spin which acts as a constant of motion, we exploit the weighted spin difference and show that such reparametrization cures the coordinate singularity that affected the previous formulation for the case of equal-mass binaries. The dynamics on the precession timescale is written down in closed form in both coprecessing and inertial frames. Radiation reaction can then be introduced in a quasiadiabatic fashion such that, at least for binaries on quasicircular orbits, gravitational inspirals reduce to solving a single ordinary differential equation. We provide a broad review of the resulting phenomenology and rewrite the relevant physics in terms of the newly adopted parametrization. This includes the spin-orbit resonances, the up-down instability, spin propagation at past time infinity, and new precession estimators to be used in gravitational-wave astronomy. Our findings are implemented in version 2 of the public Python module precession. Performing a precession-averaged post-Newtonian evolution from/to arbitrarily large separation takes $\lesssim 0.1\mathrm{s}$ on a single off-the-shelf processor—a $50\times$ speedup compared to our previous implementation. This allows for a wide variety of applications including propagating gravitational-wave posterior samples as well as population-synthesis predictions of astrophysical nature.

Figure 1

Magnitude of the total spin $S$ as a function of the weighted spin difference $\delta \chi$. We consider a set of binaries with ${\chi }_1={\chi }_2=0.9$, ${\chi }_{\mathrm{eff}}=0.3$, $\tilde{\kappa }=0.5$, and a range of values of $q\in (0,1]$ as indicated in the color bar. The dotted black curves mark the locations of the two turning points $\delta {\chi }_{\pm }$. The solid black line indicates the $q=1$ limiting case where the $S$ parametrization becomes degenerate. The $\delta {\chi }_{\pm }$ configurations with $q=0$, 1 are indicated with round markers.

Figure 2

The left panel shows the normalized cubic equation $\mathrm{\Sigma }/\overline{\sigma }$ for the derivative $(\mathrm{d}\delta \chi /\mathrm{d}t{)}^2$ as reported in Eq. (28). Spin precession takes place in the bounded and positive intervals highlighted with the shaded areas. The left and right edges of these intervals correspond to $\delta {\chi }_{-}$ and $\delta {\chi }_{+}$, respectively. The spurious, largest root (which is absent for $q=1$) corresponds to $\delta {\chi }_3/(1-q)$. The right panel shows the discriminant $\mathrm{\Delta }$ of the $\mathrm{\Sigma }/\overline{\sigma }$ equation normalized to a positive coefficient $\overline{\delta }$; see Eq. (40). Spin precession can only take place in the bounded, positive intervals highlighted with the shaded areas. The edges of these intervals are the spin-orbit resonances ${\kappa }_{\pm }$. In both panels, we consider binaries with ${\chi }_1=0.9$, ${\chi }_2=0.9$, ${\chi }_{\mathrm{eff}}=0.5$, $r=15M$, and three values of the mass ratio $q=0.4$ (blue), 0.7 (orange), and 1 (green). For the left panel, we further impose $\tilde{\kappa }=0.5$. The corresponding values of $\kappa$ are indicated with dashed lines in the right panel.

Figure 3

Evolution of the weighted spin difference $\delta \chi$ as a function of time on the precession timescale (i.e., neglecting radiation reaction). All three binaries have the same values of $q=0.8$, ${\chi }_1=0.5$, ${\chi }_2=0.9$, ${\chi }_{\mathrm{eff}}=-0.1$, and $r=10M$, but three different values of $\tilde{\kappa }=0.3$ (blue, top panel), 0.5 (orange, middle panel), and 0.7 (green, bottom panel). Solid curves show the full solution which is provided in terms of Jacobi elliptic functions. Dotted curves show approximate solutions where we set $m=0$. Dashed curves show approximate solutions where we use standard trigonometric functions but keep the period of the oscillation as derived from the full solution.

Figure 4

Available region in the $\delta \chi -{\chi }_{\mathrm{eff}}$ parameter space for two sets of binaries with fixed values of $q$, ${\chi }_1$, ${\chi }_2$, and $r$. The black rectangles indicate the bounds from Eq. (45). Each edge corresponds to one of the four aligned conditions, $\cos {\theta }_{1,2}=\pm 1$. The rectangles themselves are inscribed in wider squares (gray dotted lines) corresponding to Eqs. (38) and (44). The parameters used in the left (right) panel have been chosen such that ${\chi }_1-q{\chi }_2>0$ (${\chi }_1-q{\chi }_2<0$), which sets the orientation of the rectangle. The white circles at the vertices correspond to the four aligned configurations up-up, up-down, down-up, and down-down (where “U” stands for up and “D” stands for down). The dashed red and blue curves indicate binaries with $\tilde{\kappa }=0$ and $\tilde{\kappa }=1$, respectively, which are the spin-orbit resonances. The gray areas indicate the parameter space available to binaries with $\tilde{\kappa }=0.5$. For these same binaries, the gray dashed curves mark the turning points $\delta {\chi }_{-}$ (left edge) and $\delta {\chi }_{+}$ (right edge).

Figure 5

Moments of the weighted spin difference $\delta \tilde{\chi }$ rescaled using the turning points and averaged over a precession cycle. Solid and dashed lines show $⟨\delta \tilde{\chi }⟩$ and $⟨\delta {\tilde{\chi }}^2⟩$, respectively, as a function of the elliptic parameter $m$; cf. Eqs. (59) and (60).

Figure 6

Representative integrations of the inspiral dynamics using a precession-averaged approach. The left panel shows the evolution of the asymptotic angular momentum $\kappa$. The right panel shows the same results rescaled to [0, 1] using $\tilde{\kappa }$; see Eq. (43). The gray curves represent binaries with $q=0.9$, ${\chi }_1=0.8$, ${\chi }_2=1$, ${\chi }_{\mathrm{eff}}=-0.2$, and a set of equally spaced initial values of $\tilde{\kappa }$. These are evolved from $r={10}^{4}M$ to $r=10M$. The spin-orbit resonances ${\kappa }_{\pm }$ are shown with red and blue dashed curves, respectively.

Figure 7

Relative (blue) and absolute (red) differences between the two precession estimators $⟨{\chi }_{\mathrm{p}}⟩$ and $\sqrt{⟨{\chi }_{\mathrm{p}}^2⟩}$. We consider a sample of BH binaries where $\sqrt{⟨{\chi }_{\mathrm{p}}^2⟩}$ is uniformly distributed in [0, 2], as obtained from reweighting a base distribution where $q$, ${\chi }_1$, and ${\chi }_2$ are uniformly distributed in [0.1, 1], $r=10M$, and spin directions are isotropic. Shaded areas encompass 90% of the binaries in each bin. Dashed lines mark the median error values.

Figure 8

Distribution of the CPU time (in seconds) required to perform precession-averaged evolutions. The blue histogram reports timings obtained with the version of the code presented in this paper (precession v2). The orange histogram reports timings obtained for the same BH binaries evolved with the version of the code presented in Ref. [5] (precession v1). The unit on the y-axis is arbitrary.

Figure 9

CPU time (in seconds) required to perform precession-averaged evolutions in bins of initial separation $r_i$ (top panel, green) and mass ratio $q$ (bottom panel, purple). Dashed (solid) lines indicate the median (90% interval) wall-clock time recorded across a broad population of sources.

Figure 10

Code profiling of precession v2. The length of each colored bar indicates the fraction of the total CPU time spent on a given operation, as indicated on the bottom x-axis. The top x-axis is rescaled to the mean CPU time per binary $\sim 0.064\mathrm{s}$. Performing a precession-averaged evolution requires an ODE integration (blue) and a resampling of the precessional phase at the final separation (yellow). In turn, the integrator requires multiple evaluations of the right-hand side (orange) and the ODE stepper (cyan). In turn, evaluating the right-hand side requires finding the roots of the cubic polynomial $\mathrm{\Sigma }(\delta \chi )$ (green), computing the coefficients ${\sigma }_i$ (red), and evaluating elliptic integrals (purple). Minor additional operations are marked in gray.