Simple Model of Complete Precessing Black-Hole-Binary Gravitational Waveforms

The construction of a model of the gravitational-wave (GW) signal from generic configurations of spinning-black-hole binaries, through inspiral, merger, and ringdown, is one of the most pressing theoretical problems in the buildup to the era of GW astronomy. We present the first such model in the frequency domain, PhenomP, which captures the basic phenomenology of the seven-dimensional parameter space of binary configurations with only three key physical parameters. Two of these (the binary’s mass ratio and an effective total spin parallel to the orbital angular momentum, which determines the inspiral rate) define an underlying nonprecessing-binary model. The nonprecessing-binary waveforms are then twisted up with approximate expressions for the precessional motion, which require only one additional physical parameter, an effective precession spin, ${\chi }_p$. All other parameters (total mass, sky location, orientation and polarization, and initial phase) can be specified trivially. The model is constructed in the frequency domain, which will be essential for efficient GW searches and source measurements. We have tested the model’s fidelity for GW applications by comparison against hybrid post-Newtonian-numerical-relativity waveforms at a variety of configurations—although we did not use these numerical simulations in the construction of the model. Our model can be used to develop GW searches, to study the implications for astrophysical measurements, and as a simple conceptual framework to form the basis of generic-binary waveform modeling in the advanced-detector era.

Figure 1

Fitting factors (FF) between PhenomP (solid lines) and PhenomC (dashed lines), averaged over binary orientations, as described in the text. Each color is one case: $q=3$, ${\chi }_p=0.75$ (black); $q=2$ double spin (red); $q=3$, ${\chi }_{\text{eff}}=-0.5$ (green); $q=3$, ${\chi }_{\text{eff}}=-0.125$ (blue). We see that in all cases PhenomP meets the 0.965 threshold for detection accuracy. Above $100M_{\odot }$, all of the curves are above 0.965.

Figure 2

Fitting factors (FF) between a $q=3$ highly precessing binary, and the nonprecessing PhenomC and precessing PhenomP models, as a function of binary orientation angles ($\theta$, $\varphi$); at $\theta =0$, an observer is oriented with the binary’s total angular momentum. $\mathrm{FF}<0.965$ for many orientations with PhenomC, while for PhenomP it is well above 0.965 for all orientations. See text for further details.