Comparing gravitational waves from nonprecessing and precessing black hole binaries in the corotating frame

Previous analytic and numerical calculations suggest that, at each instant, the emission from a precessing black-hole binary closely resembles the emission from a nonprecessing analog. In this paper we quantitatively explore the validity and limitations of that correspondence, extracting the radiation from a large collection of 224 generic black-hole binary merger simulations both in the simulation frame and in a corotating frame that tracks precession. To a first approximation, the corotating-frame waveforms resemble nonprecessing analogs, based on similarity over a band-limited frequency interval defined using a fiducial detector (here, advanced LIGO) and the source’s total mass $M$. By restricting attention to masses $M\in 100$, $1000M_{\odot }$, we ensure that our comparisons are sensitive only to our simulated late-time inspiral, merger, and ringdown signals. In this mass region, every one of our precessing simulations can be fit by some physically similar member of the IMRPhenomB phenomenological waveform family to better than 95%; most fit significantly better. The best-fit parameters at low and high mass correspond to natural physical limits: the pre-merger orbit and post-merger perturbed black hole. Our results suggest that physically motivated synthetic signals can be derived by viewing radiation from suitable nonprecessing binaries in a suitable nonintertial reference frame. While a good first approximation, precessing systems have degrees of freedom (i.e., the transverse spins) which a nonprecessing simulation cannot reproduce. We quantify the extent to which these missing degrees of freedom limit the utility of synthetic precessing signals for detection and parameter estimation.

Figure 1

Precessing binaries resemble nonprecessing binaries: Distribution of best fit complex overlap $P_{\max ,\mathrm{corot}}$ [Eq. (1)] between the IMRPhenomB and our simulated signals of mass $M$. The solid black curve shows the median-probability overlap $P_{\max }$ at each mass: half of our precessing simulations have a fitting factor above the black solid line at each mass; the dotted black curves show the 90% confidence interval, estimated from our set of 224 simulations. For comparison, the red (light gray) solid and dotted curves show the median fitting factor and 90% confidence interval estimated from 62 nonprecessing simulations; these simulations are not included in the previous list. All calculations are performed by comparing the two models’ (2, 2) modes, each in a suitable corotating frame. Our simulations have significantly different initial conditions and hence durations: many shorter simulations do not have enough data to reliably estimate the waveform and hence $P$ for $M\lesssim 250M_{\odot }$. Our longer simulations, including the nonprecessing simulations, are relatively well fit by the IMRPhenomB model down to $\simeq 100M_{\odot }$.

Figure 2

Best-fit nonprecessing parameters depend on precessing parameters and mass, and can differ from the simulation parameters: Top panel: For selected simulations studied in this work, a plot of the best-fitting effective spin ${\chi }_{\mathrm{PB}}$ versus the simulated binary’s mass. The best-fit effective spin changes as a function of mass: while the initial and final state separately resemble a nonprecessing system, no single nonprecessing system fits the whole time-dependent corotating-frame mode. Instead, the best-fitting parameters interpolate (sometimes discontinuously) between the initial and final state. Colors indicate Tq(1.5, 0.4, 60, 10) (blue or top curve), S(1, 0.2, 180, 6.2) (black), Sq(4, 0.6, 270, 6.2) (green or middle curve on left), and several instances of Sq(4, 0.6, 270, 9) using $h=M/180$, $M/160$, and $M/140$ resolutions; see Appendix  for details. Bottom panel: For all simulations used in this work, the ratio $M_{\mathrm{fit}}/M_{\mathrm{sim}}$ is between the fitted and simulated mass. As indicated by the shaded region, the best-fitting mass can differ by up to $\simeq 10\%$ from the simulated mass.

Figure 3

Synthetic precessing waveforms resemble precessing waveforms: Top panel: The distribution of $P_{\max ,\mathrm{sim}}$ [Eq. (2)], the match between ${\psi }_{4\mathrm{NR}2,2}$ and $R{\psi }_{4\mathrm{PB}}$, and a synthetic precessing signal derived by applying a known time-dependent rotation to a best-fitting IMRPhenomB waveform. This distribution is effectively identical to that shown in Fig. 1. Bottom panel: For the Sq(1.5, 0.6, 45) simulation a plot of the match $P_{\max ,\mathrm{sim}}$ and $P_{\max ,\mathrm{corot}}$ [Eq. (1)] is displayed. As this example illustrates, except for very high masses $P_{\max ,\mathrm{corot}}\simeq P_{\max ,\mathrm{sim}}$. To avoid over-weighting the single worst case, which has $P_{\max ,\mathrm{corot}}\simeq 0.95$ for all masses, we have explicitly eliminated all five resolutions of the strongly precessing Sq(4, 0.6, 270, 9) and the equivalent Tq(4, 0.6, 90, 9) from this comparison. By contrast, all resolutions and iterations are included in Fig. 1.

Figure 4

Corotating signal resembles nonprecessing signal: Comparison of $|r{R}^{-1}{\psi }_{4l,m}|$ (top panel) and $\arg R^{-1}{\psi }_{42,2}$ (bottom panel) for a nonspinning $q=4$ simulation (dotted) with the corotating waveform from the Sq(4, 0.6, 270, 9) simulation (thick). The colored curves correspond to the $(2,\pm 2)$ modes (red or top, solid), the $(2,\pm 1)$ modes (blue or bottom), and the (3, 3) mode (green or middle). A timeshift has been applied to align the two simulations. This comparison shows that several modes from the same nonprecessing simulation reproduce the corotating frame modes from another 18.

Figure 5

Precessing systems break reflection symmetry in the corotating frame: For the Sq(4, 0.6, 90, 9) simulation provided in Ref. 14, we plot the reflection-symmetric (solid) and reflection-antisymmetric (dotted) parts of the corotating-frame $l=2$ modes, for $m=\pm 2$ (red or top solid and dotted) and $\pm 1$ (blue or bottom solid and dotted); see Eq. (3). This figure suggests that precessing binaries radiate in a way unlike any nonprecessing binary, even one viewed in a noninertial frame, that this missing physics has a comparable effect to higher harmonics like the $(2,\pm 1)$ modes [e.g., dotted red (top) versus solid blue (bottom)], and that both reflection symmetry breaking and higher harmonics are required to accurately model the merger phase of generic precessing binaries.

Figure 6

Does it matter if we omit the transverse spin?: This figure demonstrates that transverse spins have a significant effect on the gravitational-wave signal, via reflection-symmetry breaking between the $(2,\pm 2)$ modes. For the special-purpose $V(a,\theta ,\varphi )$ family of precessing simulations described in the text and Appendix , points show the match $P_{\max ,\mathrm{corot}}$ versus the angle $\varphi$, for the fixed angle $\theta ={34}^o$. For comparison, this figure uses $\psi$ to indicate the match $P_{\max ,\mathrm{NR}}$ [Eq. (7)] evaluated between the corotating-frame (2, 2) mode for one member of that family [$V(0.6,34,240)$] and all others, directly demonstrating how much these signals differ from one another. For another comparison, this figure uses $a$ to indicate the match $P_{\max ,\mathrm{NR},\mathrm{a}}$ [Eq. (6)] between the reflection-symmetric part of the corotating-frame (2, 2) ($a_{2,2}$) of that same simulation and all other angles $\varphi$. The strong variability in the first set ($\psi$) and lack of variability in the second ($a$) suggests that reflection asymmetry is the dominant source of variation between different simulations. This figure illustrates strong, generic correlations between the precise magnitude and direction of the transverse spins, the reflection asymmetry between the $(2,\pm 2)$ modes in the corotating frame, and the degree of similarity between the (2, 2) corotating-frame mode and conventional nonspinning approximations.

Figure 7

Corotating modes evolve nearly in phase prior to merger: Phases of corotating modes in an absolute scale (top panel) and scaled in proportion to their angular order ${\varphi }_{l,m}/|m|$ (bottom panel), for the Tq(2, 0.6, 90) simulation. To a good approximation all evolve in phase prior to merger. After merger, all angular harmonics shown continue to evolve nearly in phase, except the (2, 1) mode. In the top panel, from top to bottom, the modes are (2, 1); (2, 2) and (3, 2); (4, 3) and (3, 3); and (4, 4).

Figure 8

Low-mass limit: recovering the mass ratio: At low masses and therefore early times, the best-fit corotating-frame model reflects the initial binary. These panels show comparisons between the best-fit IMRPhenomB parameter ${\eta }_{\mathrm{fit}}$ and the physical binary mass ratio $\eta$, using best-fit values at $M=100M_{\odot }$; compare to Fig. 9. This figure explicitly excludes all of our shorter simulations ($d<7M$). Green (light) points are nonprecessing spin-aligned systems, black points are long-duration simulations from the Sq series, and blue (remaining) points are long-duration simulations from the Tq series. A solid blue line at ${\eta }_{\mathrm{sim}}={\eta }_{\mathrm{fit}}(100M_{\odot })$ is shown to guide the eye. (An identical color scheme is adopted in the subsequent Fig. 9.)

Figure 9

Low mass limit: recovering the effective spin: At low masses and therefore early times, the best-fit corotating-frame model reflects the initial binary. These panels show comparisons between the best-fit IMRPhenomB parameter ${\chi }_{\mathrm{eff}}$ and ${\chi }_{\mathrm{PB}}$ [Eq. (9)], using best-fit values at $M=300M_{\odot }$. Different point shadings, indicate the Sq series (black), the Tq series (blue), and the T series (red). Different point shadings, indicate the Sq series (black), precessing binaries in the Tq series (blue), aligned spin binaries from the Tq series (green), and the short simulations of the T series (red), Eq series (purple), and Lq series (dark red). A solid blue line at ${\chi }_{\mathrm{PB},\mathrm{sim}}={\eta }_{\mathrm{PB},\mathrm{fit}}(100M_{\odot })$ is shown to guide the eye. Though provided at high mass ($M=300M_{\odot }$) to ensure that short simulations are well resolved, a similar distribution is recovered when using earlier epochs of longer signals (i.e., $M\simeq 100M_{\odot }$).

Figure 10

High mass: best fit approximately recovers the final state: At $M\simeq 1000M_{\odot }$, this scatter plot shows the true $(M_{\mathrm{sim}},a_{\mathrm{sim}})$ and best-fit $(M_{\mathrm{fit}},a_{\mathrm{fit}})$ final black-hole masses (bottom panel) and spins (top panel), derived from our precessing simulations (blue or dark) and nonprecessing simulations (red or light). The best-fit final black-hole properties are derived from the IMRPhenomB best-fit parameters using Eqs. (11, 12). In both figures, a dotted black line is provided to guide the eye onto $a_{\mathrm{f},\mathrm{sim}}=a_{\mathrm{f},\mathrm{fint}}$. Because the IMRPhenomB model claims to reproduce nonprecessing binaries, the distribution of the red (light) points can be used to estimate systematic uncertainties in this method. For example, in the top panel, the systematic offset between the red (light) points and the line $a_{\mathrm{f},\mathrm{sim}}=a_{\mathrm{f},\mathrm{fit}}$ suggests significant systematic differences between our simulations’ ringdown frequencies and the model of IMRPhenomB. In the bottom panel, the significant scatter in both red (light) and blue (dark) points reflects our inability to measure total masses to better than a few percent by a fitting procedure; see also the bottom panel in Fig. 2. This figure includes a separate point for each individual simulation, including some simulations with physically identical parameters but performed with different resolutions. To better resolve the final black-hole quasinormal modes, we have explicitly excluded all simulations performed with $h=M/77$.

Figure 11

Comparing resolutions: Top panel: A plot of $P_{\max ,\mathrm{corot}}$ versus mass for the Sq(4, 0.6, 270, 9) simulation, performed at resolutions $h=M/100$ (red or outermost), $M/120$ (blue or second), $M/140$ (green or innermost), $M/160$ (purple or third from outside), and $M/180$ (black). Though significant, the differences between resolutions are nonetheless smaller than the typical differences between simulations; compare to Fig. 1. Bottom panel: The path of the preferred direction derived from simulations with different resolution, viewed in projection in a frame aligned with the precession cone (i.e., with $\widehat{W}\simeq \widehat{J}$; see Ref. 14).

Figure 12

Signal duration and the low-mass error: Top panel: The best-fitting match $P_{\max ,\mathrm{sim}}$ between IMRPhenomB and the (2, 2) mode of an unequal-mass aligned-spin binary [Tq(1.75, 0.2, 0, 10)]. The dotted curve shows $P_{\max }$ derived using the whole signal, while the solid curve shows $P_{\max }$ derived after first truncating the signal, to mimic the results of a shorter simulation starting at $d=6.2$. Comparing with Fig. 1 and the text, this illustration shows that signal duration dominates our error at low mass and has little impact on our results at high mass. Bottom panel: At $M=100M_{\odot }$ (blue and black arrows) and $M=250M_{\odot }$ (purple arrows), we plot the best-fitting IMRPhenomB parameters which reproduce Tq(1.75, 0.2, 0, 10), truncated to different lengths. For comparison, the red point indicates the corresponding physical properties expected from the initial data. Arrows connect longer to shorter signals, indicating the effect of truncating the signal: a significant error in $\eta$ and a moderate bias in $\chi$. As illustrated by the discrepancy between the best-fit points and physical parameters, differences between the IMRPhenomB model and our simulations also contribute significantly to systematic error in parameter recovery, particularly in mass ratio.