A 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 build-up 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 non-precessing-binary model. The non-precessing-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 polarisation, 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 modelling in the advanced-detector era.

Introduction.-Theimminent commissioning of second-generation laser-interferometric gravitationalwave detectors will bring us closer to the era of gravitational-wave (GW) astronomy, which carries the potential to revolutionize our understanding of astrophysics, fundamental physics, and cosmology [1].Among the most promising GW sources are the inspiral and merger of black-hole binaries.Detection and interpretation of these signals requires analytic models that capture the phenomenology of all likely binary configurations; most of these will include complex precession effects due to the black-hole spins.However, most of the current models of the two black holes' inspiral, their merger, and the ringdown of the final black hole, consider only configurations where the black-hole spins are aligned with the binary's orbital angular momentum, i.e., they do not model precession effects.The binary's early inspiral can be modelled with analytic post-Newtonian (PN) calculations, but the late inspiral and merger require 3D numerical solutions of the full nonlinear Einstein equations.These expensive numerical relativity (NR) calculations must span a parameter space of binary configurations that covers, for non-eccentric inspiral, seven dimensions: the mass ratio of the binary, and the components of each black hole's spin vector; the total mass of the system is an overall scale factor.Previous work on phenomenological models of non-precessing binaries suggests that we require at least four simulations in each direction of parameter space that we intend to model [2][3][4].This implies that we need 4 7 ≈ 16, 000 numerical simulations to model the full parameter space, which is unfeasible in the near fu-ture.We must identify approximations and degeneracies that make the task tractable.
In recent work we identified an approximate mapping between inspiral waveforms from generic binaries, and those from a two-dimensional parameter space of nonprecessing binaries [5].This approximation holds because precession has little effect on the inspiral rate, and so precession effects approximately decouple from the overall inspiral, which can be described by a nonprecessing-binary model, neglecting the effect of breaking equatorial symmetry, which is responsible for large recoils [6].We further proposed that, given a model for the precessional motion of a binary, we could construct an approximate waveform by "twisting up" the appropriate non-precessing-binary waveform with the precessional motion.This technique was recently adopted to produce simple frequency-domain PN inspiral waveforms [7].It was more recently suggested that this mapping also holds through merger and ringdown [8].In this work we take this idea further, in two crucial ways.First, we use highorder PN expressions for the precession angles to twist up a phenomenological inspiral-merger-ringdown model for non-precessing binaries [4], and incorporate precession effects into the estimate of the final black-hole spin and the ringdown model, providing the first frequencydomain inspiral-merger-ringdown model of generic binaries.(Frequency-domain models are essential for both efficient GW searches and parameter estimation.)Second, we make use of a single parameter that captures the basic precession phenomenology for generic binary configurations [9].Our final model has only three dimensionless physical parameters, the two parameters of our previous non-precessing models (the mass ratio q = m 2 /m 1 ≥ 1, and an effective inspiral spin, χ eff , which characterizes the rate of inspiral); plus one additional parameter, an effective precession spin, χ p .We describe this parameterization in more detail below; its effectiveness in capturing the phenomenology of the inspiral across the full eight-dimensional parameter space is demonstrated in Ref. [9].Our evaluation of its fidelity for GW applications when including merger and ringdown by comparison against hybrid PN-NR waveforms constitutes our core quantitative result.
The purpose of this model is to (a) facilitate the development of computationally efficient generic-binary searches, (b) provide a starting point to investigate the parameter-estimation possibilities (and limitations) of generic-binary observations in second-generation detectors, and their astrophysical implications, and (c) as a simple framework for the construction of more refined models calibrated to NR simulations.If the dominant parameter space of binary simulations can be reduced to three dimensions (mass ratio, effective inspiral spin, effective precession spin), it may be feasible to produce a sufficient number of NR waveforms (∼100) to calibrate the model well before advanced detectors reach design sensitivity in 2018-20 [10].The model can be further refined, based on the results of these studies.As such, this model provides a practical road map to model generic binaries to meet the needs of GW astronomy over the next decade.This model will be provided in the LAL data analysis software, to facilitate the development and testing of search and parameter estimation pipelines [11].
Model.-We start from the frequency-domain model [4] ("PhenomC") of non-precessing waveforms, because it includes the standard state-of-the-art inspiral phase, TaylorF2.This model describes the ( = 2, m = |2|) modes of the waveform, with h(f ) = A(f )e iψ(f ) , where A(f ) and ψ(f ) are given in Ref. [4].Based on the approximate mapping identified in Ref. [5], for a given generic binary we start with the non-precessing waveform given by the parameters (M, η, χ eff ), where η = q/(1 + q) 2 and χ eff = (m 1 χ 1 + m 2 χ 2 )/M ; χ 1 and χ 2 are the components of the dimensionless spins (χ i = S i • L/m 2 i ) projected along the Newtonian orbital angular momentum L. The direction of Ĵ is approximately constant throughout the evolution, as angular-momentum loss via GWs is predominantly along Ĵ , with emission orthogonal to Ĵ averaging out due to the precession of L around Ĵ [12].We therefore assume that the final spin is in the same direction as Ĵ through the inspiral, and update the PhenomC final spin magnitude estimate to account for precession, using Ref. [13], with only one black hole spinning.
We then twist up the non-precessing model, i.e., we approximate the = 2 modes of a precessing binary waveform in the time domain by rotating the dominant modes of the corresponding non-precessing waveform [5,14] as where d nm denote the Wigner d-matrices.The angles α and ι that enter our model are defined as the spherical angles parametrizing the unit Newtonian orbital angular momentum L (see Fig. 1) in an inertial frame with ẑ = Ĵ .The third angle, defined from ˙ = α cos ι, parametrizes a rotation around L [15].During the inspiral phase, all of these angles vary slowly (on the precession timescale) with respect to the orbital timescale, which allows for a stationary-phase-approximation (SPA) transformation to the frequency domain (this was also noted in Ref. [7]).
Here, we use frequency domain PN expressions for these angles (valid for systems with only one spin in the orbital plane) to twist the entire non-precessing modes, formally continuing the SPA treatment through merger and ringdown.Although we do not expect these expressions, or the approximation of slowly varying precession angles, to be valid through merger and ringdown, in practice we find that they mimic to reasonable accuracy the phenomenology of our PN-NR hybrids and lead to small mismatches even for high masses.The output of our model is the two polarizations h P +,× (M f ; η, χ eff , χ p , θ, φ).The inclination ι is simply the angle between the binary's total angular momentum, Ĵ , and L so that cos ι = L • Ĵ = L • J /|J |.In practice we find that the accuracy in ι, which enters only in amplitude factors for the contributions in (1), is not critical and that it is sufficient to include only non spinning corrections in J beyond the total spin contribution at leading order.The precession angle α is computed using the expression for α obtained in [16] (see Eqs (4.10a) and (4.8)) by plugging in the highest order (next-to-next-to-leading in spin-orbit) ex-pressions available for the quantities entering the formula [17], PN re-expanding and averaging over the orientation of the spin in the orbital plane.Fig. 1 shows an example of one precession cycle for the q = 3 configuration discussed below.In this example our PN expression for α deviates from that of the hybrid waveforms by ∼π rad near the end of the waveform, but this disagreement can be removed by modifying the physical parameters in the model, with a bias of ∼10% in the spin parameters.
The spin parameters in our model are χ eff and χ p .The effective inspiral spin χ eff was defined earlier.The angle expressions (α, ι), require some choice for the distribution of spins across the two black holes, and for our implementation we let χ 1 = 0,χ 2 = (M/m 2 )χ eff , i.e., all of the spin is on the larger black hole.To ensure physical spins of χ ≤ 1 for each black hole, we could also choose χ 1 = χ 2 = χ eff .The implications of these choices for detection and parameter estimation will be explored in future work; in the cases we study here, we see that our model is likely to perform well for GW detection.The in-plane spin χ p is associated with the larger black hole.
We expect our model to capture the basic phenomenology of generic two-spin systems, motivated by the following argument.For the effective precession spin, if S 1⊥ and S 2⊥ are the magnitudes of the projections of the two spins in the orbital plane, then, according to the PN precession equations [12,18], the precession rate at leading order will be proportional to (A 1 S 1⊥ + A 2 S 2⊥ ) when the vectors S 1⊥ and S 2⊥ are parallel, and by (A 1 S 1⊥ −A 2 S 2⊥ ) when they point in opposite directions, where A i = 2 + (3m 3−i )/(2m i ).During the inspiral, to first approximation the average precession rate is simply the maximum of these two spin contributions, and we can define S p = max(A 1 S 1⊥ , A 2 S 2⊥ )/A 2 , and expect that applying an in-plane spin of χ p = S p /m 2 2 to the larger black hole will mimic the main precession effects of the full two-spin system.The full system will exhibit additional oscillations in the precession angles (see e.g., Fig. 4 of Ref. [19], and Ref. [9]), but we do not expect these effects to be detectable in most GW observations.These two parameters, χ eff and χ p , can be mapped to a range of physically allowable individual black-hole spins.
Results.-The most reliable way to test our model is to compare against hybrid PN (inspiral) and NR (mergerringdown) waveforms.But to do that across the full generic-binary parameter space would require the same number of waveforms as needed to construct a sevendimensional generic model, which is the computationally prohibitive task that we wished to avoid in the first place.In practice all we can do is identify what we expect to be challenging points in the parameter space.In this work we restrict ourselves to binaries with mass ratios q ≤ 3, because that is the mass ratio to which the underlying PhenomC model was calibrated to spinning-binary waveforms.We construct four hybrids at mass ratios 1, 2 and 3, for a variety of spin choices; the numerical simulations Fitting factors (FF) between a q = 3 highlyprecessing binary, and the non-precessing PhenomC and precessing PhenomP models, as a function of binary orientation angles (θ, φ); at θ = 0 an observer is oriented with the binary's total angular momentum.FF < 0.97 for many orientations with PhenomC, while for PhenomP it is well above 0.97 for almost all orientations.See text for further details.
were produced with the BAM code [20], and hybrids were constructed by the method described in Ref. [5], and also in the inertial frame of the NR waveforms.Among those are a q = 3 case where the larger black hole has a spin of χ 2 = 0.75 in the orbital plane (this leads to strong precession effects), and two double-spin q = 2 cases, which test our assumption that we can consider only a weighted average of the spins when constructing χ p .
As is standard in GW analysis, we calculate the noiseweighted inner product between our source waveform (in this case the hybrid), and a model (either the original non-precessing PhenomC model, or our new precessing "PhenomP" model).We use the current expectation for the design sensitivity of advanced LIGO [21], with a low-frequency cutoff of 20Hz.This inner product is maximised with respect to the parameters of the model, including the physical parameters and the binary orien-tation and polarization.This optimised inner product is called the "fitting factor"; its value indicates how well the signal can be found in detector data, and the bias between the best-fit model parameters, and the true source parameters, give us an indication of the errors in a GW measurement.We have computed fitting factors using PhenomC and PhenomP for total source masses between 20 M and 200 M and as functions of binary orientations.As an example, Fig. 2 shows results for the q = 3 high precession configuration at 50 M , which proved to be the most challenging to our model.Results are similar for lower masses, while for higher masses fitting factors improve at the expense of parameter accuracy.The standard requirement for GW searches is that the fitting factor be above 0.97, corresponding to a loss of no more than 10% of sources in a search (disregarding additional loss due to a discrete template bank).Comparing the two panels of Fig. 2 we see that while the fitting factors for PhenomC are above 0.97 only for near-optimal orientations (from which the precession has only a small effect on the signal), they are above 0.97 for almost all orientations with the PhenomP model.We leave a full study of parameter biases to future work, but our results suggest that a measurement of χ p reliably identifies precession.
Discussion.-We have presented the first frequencydomain inspiral-merger-ringdown model for the GW signal from precessing-black-hole binaries.Incorporating a series of insights from our previous work, our model is constructed by a straightforward transformation of a non-precessing-binary model, in this case PhenomC; in practice any workable non-precessing model could be used instead.The current model did not require any precessing-binary numerical simulations in its construction, although in the future we plan to use extensive simulations to refine the model, based on tests of the model's accuracy for GW searches and parameter estimation.Finally, we are able to model the essential phenomenology of the seven-dimensional parameter space of binary configurations with a model that requires only three physical parameters.This will simplify the model's incorporation into search and parameter estimation pipelines, as well as making tractable the problem of producing enough numerical simulations to produce a model of sufficient accuracy for GW astronomy with advanced detectors.
Our ability to model generic waveforms with only two spin parameters implies strong degeneracies that will make it difficult to identify the individual black-hole spins.This may well be the reality of GW observations with second-generation detectors, for which 80% of signals will be at signal-to-noise ratios between 10 and 20, in which the subtle double-spin effects on the waveform may be difficult to identify.These are important issues that deserve further attention in future work.
The current model is valid only in the region of parameter space for which PhenomC was calibrated (q ≤ 3, |χ eff | ≤ 0.75).More challenging precession cases are ex-pected at higher mass ratios and spins (e.g., transitional precession), and the ability of our prescription to model those configurations will need to be tested when refined non-precessing-binary models become available.

FIG. 1 :
FIG.1:The precession angles (ι, α) of the Newtonian orbital angular momentum L about the total angular momentum Ĵ .The figure also indicates one early precession cycle for a q = 3 hybrid PN-NR waveform (red), and the results from the PN formulas used in the PhenomP model (black).The different curve lengths indicate the dephasing in α; see text.
FIG. 2:Fitting factors (FF) between a q = 3 highlyprecessing binary, and the non-precessing PhenomC and precessing PhenomP models, as a function of binary orientation angles (θ, φ); at θ = 0 an observer is oriented with the binary's total angular momentum.FF < 0.97 for many orientations with PhenomC, while for PhenomP it is well above 0.97 for almost all orientations.See text for further details.