Towards models of gravitational waveforms from generic binaries: II. Modelling precession effects with a single effective precession parameter

Gravitational waves (GWs) emitted by generic black-hole binaries show a rich structure that directly reflects the complex dynamics introduced by the precession of the orbital plane, which poses a real challenge to the development of generic waveform models. Recent progress in modelling these signals relies on an approximate decoupling between the nonprecessing secular inspiral and a precession-induced rotation. However, the latter depends in general on all physical parameters of the binary which makes modelling efforts as well as understanding parameter-estimation prospects prohibitively complex. Here we show that the dominant precession effects can be captured by a reduced set of spin parameters. Specifically, we introduce a single effective precession spin parameter, ${\chi }_p$, which is defined from the spin components that lie in the orbital plane at some (arbitrary) instant during the inspiral. We test the efficacy of this parameter by considering binary inspiral configurations specified by the physical parameters of a corresponding nonprecessing-binary configuration (total mass, mass ratio, and spin components (anti)parallel to the orbital angular momentum), plus the effective precession spin applied to the larger black hole. We show that for an overwhelming majority of random precessing configurations, the precession dynamics during the inspiral are well approximated by our equivalent configurations. Our results suggest that in the comparable-mass regime waveform models with only three spin parameters faithfully represent generic waveforms, which has practical implications for the prospects of GW searches, parameter estimation and the numerical exploration of the precessing-binary parameter space.

Figure 1

The ${\widehat{J}}_0$-aligned source frame of a precessing binary. $\theta$ denotes the angle between the line of sight (radiation propagation direction) and the total angular momentum; $\overrightarrow{S}={\overrightarrow{S}}_1+{\overrightarrow{S}}_2$ is the total spin.

Figure 2

Magnitude of the GW strain $h$ computed with all $\ell =2$ modes for a precessing binary, where the total angular momentum ${\widehat{J}}_0$ is aligned with the line of sight (top blue curve) and for the arbitrary orientation $(\theta ,\varphi )=({60}^{\circ },{113}^{\circ })$ (lower red curve). The binary’s parameters are $q=3$, ${\overrightarrow{\chi }}_1=(1,0,0)$ and ${\overrightarrow{\chi }}_2=(0.8,0,0.6)$. While only weak amplitude modulations are visible along ${\widehat{J}}_0$, we observe strong modulations for the arbitrary orientations.

Figure 3

Evolution of the two spins parallel to the orbital angular momentum. The top red graph shows the evolution of the parallel spin of the smaller black hole, $S_{1\Vert }$, and the lower blue curve that of the parallel spin of the larger black hole, $S_{2\Vert }$ for the case described in the text. The two horizontal lines indicate the mean value of each parallel spin with ${\overline{S}}_{1\Vert }=0.015$ and ${\overline{S}}_{2\Vert }=-0.045$.

Figure 4

The top panel shows the evolution of $S_{1\perp }$ as a function of time, and the bottom panel shows the evolution of $S_{2\perp }$. Similar to the parallel spin magnitudes, the in-plane spin magnitudes oscillate around some mean values, which are ${\overline{S}}_{1\perp }=0.030$ and ${\overline{S}}_{2\perp }=0.479$, respectively (horizontal lines).

Figure 5

Magnitude of the leading-order precession term $\Vert (A_1{\overrightarrow{S}}_1+A_2{\overrightarrow{S}}_2)\times \widehat{L}\Vert /(A_2{m}_2^2)$ (blue), its true mean $\overline{\Vert (A_1{\overrightarrow{S}}_1+A_2{\overrightarrow{S}}_2)\times \widehat{L}\Vert /(A_2{m}_2^2)}=0.845$ (lower red horizontal line) and its approximation ${\chi }_p=0.85$ as determined from Eq. (3.4) (upper green horizontal line).

Figure 6

The top panel shows $\alpha (t)$ for the generic configuration $\{q=3,{\overrightarrow{\chi }}_1=(0.4,-0.2,0.3),{\overrightarrow{\chi }}_2=(0.75,0.4,-0.1)\}$ (red) and the corresponding configuration utilizing ${\chi }_p$ given by $\{q=3,{\overrightarrow{\chi }}_1=(0.,0.,0.3),{\overrightarrow{\chi }}_2=(0.85,0.,-0.1)\}$ (blue, dashed). Since the two curves are not distinguishable over that time scale, the inset shows the difference $\mathrm{\Delta }\alpha$ as a function of time. The bottom panel compares the evolution of the opening angle of the precession cone $\iota (t)$. Both graphs reveal that the approximation discards the spin-spin couplings in the plane and therefore nutation effects (the visible oscillations).

Figure 7

The top panel shows $\alpha (t)$ for the case $\{q=3,{\overrightarrow{\chi }}_1=(0.38,0.319,-0.079)$, ${\overrightarrow{\chi }}_2=(-0.036,-0.036,-0.012)\}$ (top solid, red curve) and the corresponding configuration using ${\chi }_p$ given by $\{q=3,{\overrightarrow{\chi }}_1=(0.,0.,-0.079)$, ${\overrightarrow{\chi }}_2=(0.143,0.,-0.012)\}$ (bottom dashed, blue curve); the bottom panel compares the evolution of the opening angle of the precession cone $\iota (t)$. Both graphs highlight that in this case ${\chi }_p$ does not capture the precession of the system correctly.

Figure 8

The panel shows the precession term $\Vert A_1{\overrightarrow{S}}_{1\perp }+A_2{\overrightarrow{S}}_{2\perp }\Vert /(A_2{m}_2^2)$ and its mean as a function of time for four different mass ratios $q$: equal-mass (blue, solid), $q=1.1$ (purple, dashed), $q=1.2$ (green, dotted) and $q=1.5$ (red, dot-dashed). The mean value for $q=1$ is 1.175. For this spin configuration, however, Eq. (3.4) yields ${\chi }_p=0.85$ (indicated by the dashed horizontal line). We see that at a small mass ratio of $q=1.2$, ${\chi }_p$ is already a good estimator of the average precession.

Figure 9

The left panel shows the match contours for the extremal case with ${\overrightarrow{\chi }}_2=(1,0,0)$ and varying in-plane orientation of ${\overrightarrow{\chi }}_1$, while the right panel shows the contour for ${\overrightarrow{\chi }}_1=(1,0,0)$ and varying orientation of ${\overrightarrow{\chi }}_2$ as a function of the binary orientation $\theta$. The red dots mark the actual points at which the matches are evaluated.

Figure 10

The left panel shows the match for ${\overrightarrow{\chi }}_1=({\chi }_{1x},0,0)$ and ${\overrightarrow{\chi }}_2=(1,0,0)$ against the appropriate reduced-parameter waveforms as a function of the binary orientation; the right panel shows the match for ${\overrightarrow{\chi }}_1=(1,0,0)$ and ${\overrightarrow{\chi }}_2=({\chi }_{2x},0,0)$ against the appropriate reduced-parameter template waveforms. The red dots mark the actual configurations used to obtain the contours.

Figure 11

The panel shows the match contours for three different binary inclinations (0°,36° and 90°) for the configurations where ${\overrightarrow{\chi }}_2=(0.8,0,-0.6)$ and ${\overrightarrow{\chi }}_1=({\chi }_{1x},0,{\chi }_{1z})$. Each red dot represents one particular choice of $({\overrightarrow{\chi }}_1,{\overrightarrow{\chi }}_2)$. We find that the matches drop with increasing value of $|{\chi }_{1x}|$ and decrease overall with increasing inclination $\theta$.

Figure 12

The left panel shows the cumulative distribution function for all matches for the mass ratios $q=1$ (blue, solid), $q=3$ (green, dashed) and $q=10$ (purple, dot-dashed). The red vertical line indicates a match of $\mathcal{M}=0.965$. In the right panel the distribution is weighted according to the signal strength to represent the fraction of actually detectable signals.

Figure 13

The left panel shows the PN evolution of the precession angle $\alpha$ for the transitional precession case described in the text (red, dot-dashed) as well as $\alpha (t)$ for the corresponding reduced-parameter template (blue, dashed). The right panel compares the two precession cone opening angles. It is clear from those graphs that the mapping does not faithfully reproduce transitional precession. The green (solid) curves show the angles for a reduced-parameter system, where the precession is associated with the smaller black hole $m_1$, which appear to be closer to the angles in the generic system (red, dot-dashed).

Figure 14

The four panels show the matches for the case depicted in Fig. 6 with a series of reduced-parameter configurations with varying ${\chi }_p$ for four different pairs of binary orientation and signal polarization $(\theta ,{\psi }_S)$; these are as follows from the top left to the bottom right: $(0,\pi /8)$, $(0,\pi /4)$, $(\pi /4,\pi /8)$ and $(\pi /2,\pi /8)$. The red vertical line indicates the theoretical ${\chi }_p$ value; the black horizontal line in the lower two panels indicates the threshold of $\mathcal{M}=0.965$. We find a strong dependence of the match on the value of ${\chi }_p$ for growing inclinations, where waveform modulations become more pronounced. Moreover, the theoretical ${\chi }_p$ value is very close to the value yielding the maximal match.

Figure 15

Comparison of the ${\chi }_p$ model with alternative parameter reductions. Our $q=3$ results from Fig. 12 are repeated in the blue curve. Also shown are the PTF parametrization (green, dashed), the single parallel spin ${\chi }_{\text{eff}}$ applied to the larger black hole only (purple, dot-dashed) and ${\chi }_{\mathrm{eff}}$ as parallel spin on both black holes (orange, dotted). See text for more details.