Modelling the Ringdown from Precessing Black Hole Binaries
MModelling the Ringdown from Precessing Black Hole Binaries
Eliot Finch ∗ and Christopher J. Moore † Institute for Gravitational Wave Astronomy & School of Physics andAstronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (Dated: February 17, 2021)Modelling the end point of binary black hole mergers is a cornerstone of modern gravitational-waveastronomy. Extracting multiple quasinormal mode frequencies from the ringdown signal allows theremnant black hole to be studied in unprecedented detail. Previous studies on numerical relativitysimulations of aligned-spin binaries have found that it is possible to start the ringdown analysis muchearlier than previously thought if overtones (and possibly mirror modes) are included. This increasesthe signal-to-noise ratio in the ringdown making identification of subdominant modes easier. In thispaper we study, for the first time, black hole binaries with misaligned spins and find a much greatervariation in the performance of ringdown fits than in the aligned-spin case. The inclusion of mirrormodes and higher harmonics, along with overtones, improves the reliability of ringdown fits with anearly start time; however, there remain cases with poor performing fits. While using overtones inconjunction with an early ringdown start time is an enticing possibility, it is necessary to proceedwith caution. We also consider for the first time the use of numerical relativity surrogate models inthis type of quasinormal mode study and address important questions of accuracy in the underlyingnumerical waveforms used for the fit.
I. INTRODUCTION
The gravitational wave (GW) observatories LIGO [1]and Virgo [2] have now observed dozens of GW events[3, 4], mostly from binary black hole (BBH) mergers.Particularly prominent in the GW signals of the higher-mass systems are the final few wave cycles, known asthe ringdown , emitted as the system settles into its finalstate: a Kerr black hole (BH). The ringdown signal con-tains a superposition of oscillatory modes, the frequencyspectrum of which is characteristic of the remnant BH.The characteristic oscillations of the remnant BH arecalled quasinormal modes (QNMs), so-called because,unlike normal modes, they decay over time. The QNMfrequencies are complex, ω = 2 πf − i/τ , with the realpart f giving the oscillation frequency and the reciprocalof the imaginary part τ giving the damping time. TheQNM frequencies can be calculated within the frameworkof linearised gravity, treating the gravitational field in thevicinity of the remnant as a small (linear) perturbation ofthe Kerr metric [5]. Therefore, the QNM description ofthe GW signal is only expected to be valid at sufficientlylate times, when the nonlinearities from the merger havelargely decayed away.The remnant Kerr BH has no hair; it is fully describedby only a final mass, M f , and a dimensionless final spinparameter, χ f = | χ f | . The same is true of the spectrumof QNM frequencies, ω (cid:96)mn ( M f , χ f ), which are also func-tions of only the mass and spin. Individual QNMs are in-dexed by the triplet ( (cid:96), m, n ) which are the polar ( (cid:96) ≥ − (cid:96) ≤ m ≤ (cid:96) ) and overtone ( n ≥
0) numbersrespectively. The QNM spectrum is further complicatedby the necessity of including mirror modes ω (cid:48) (cid:96)mn [5–8] ∗ efi[email protected] † [email protected] with negative real frequency f (cid:48) (cid:96)mn (as opposed to regularmodes ω (cid:96)mn with f (cid:96)mn >
0) for a complete descriptionof the ringdown. The spectrum of mirror modes containsthe same information as the regular modes (albeit withnontrivial relationships between them, see Eqs. 8) whichhas sometimes led to them being neglected. Whetherthey can, in fact, be neglected will depend on the relativeexcitation amplitudes of the regular and mirror modesand their differing decay times. In general, the ringdownwill contain a superposition of all these modes with differ-ent excitation amplitudes and phases (see Eq. 2). Usu-ally, the GW strain is dominated by the (cid:96) = | m | = 2modes. Furthermore, the overtones decay more quickly(i.e. τ decreases) with increasing n so that at late timesthe signal will be dominated by the fundamental n = 0modes. Therefore, the most prominent QNM in the ring-down is expected to be the ( (cid:96), m, n ) = (2 , ,
0) mode,and the observational challenge is usually to detect thepresence of other, subdominant modes.The study of QNMs has applications in both astro andfundamental physics. The highly constrained dependenceof the QNM spectrum on only the remnant mass and spinmeans that, conversely, if a QNM frequency is measured,then the mass and spin of the final BH merger can be in-ferred. For high-mass systems, where only the ringdownsignal is observable, this may be the only informationavailable about the nature of the source [6, 9]. For lower-mass systems, measuring QNM frequencies allows us toestimate the remnant properties independently of the restof the signal, and so consistency tests can be performed.For example, a test of the BH area theorem can be per-formed in this way [10, 11]. A similar consistency testusing full inspiral-merger-ringdown models and a sharpcut in the frequency (rather than time) domain was per-formed on GW150914 [12]. Each additional QNM thatcan be detected in the ringdown provides a separate esti-mate of the mass and spin of the remnant. Therefore, if a r X i v : . [ g r- q c ] F e b multiple QNM frequencies can be identified, a ringdown-only consistency test on the expected Kerr-like nature ofthe remnant BH can be performed [13, 14] (this is possi-ble only if the ( (cid:96), m, n ) of the modes are known). In thesetests, deviations from the expected results may point tonew physics beyond general relativity.QNMs also have practical uses in waveform mod-elling. They are used in full inspiral-merger-ringdownBBH waveforms produced in both the phenomenological[15–17] and effective-one-body approaches [18–20].A prerequisite for any ringdown analysis is a suitablechoice for the start time , t , of the ringdown. Start-ing too early risks a GW signal contaminated with non-linearities that cannot be described by a model basedsolely on QNMs and obtaining biased measurements asa result. On the other hand, starting too late leaves ashort signal that is already decaying and without enoughsignal-to-noise to make useful measurements. The diffi-culties of defining a suitable start time and some surpris-ing data analysis consequences of this were discussed in[21]. Previous studies on numerical relativity (NR) wave-forms have tended to use just a single [22] or relativelysmall number ( ≤
4) [8, 23–25] of QNMs, and have useda wide range of start times. Typically the start time isreferred to the maxima of some time-dependent quantity(e.g. the 22 mode of the strain, the modulus of the Weylscalar, or the total GW luminosity; these quantities peakat times that typically differ by a few tens of M ) and typ-ical choices were 10 − M after the peak (although, see[26] where a range of different start times are explored).Several studies caution against starting too early.More recently, [27] (see also [28]) looked to allow theringdown to start significantly earlier by including mul-tiple overtones. In particular, [27] found that by includ-ing up to seven overtones the ringdown analysis can bestarted as early as the peak in the 22 mode of the strain.This might be considered a surprising result; the signalpeak is expected to occur when the remnant BH (to theextent that it yet even makes sense to consider it as such)is most highly distorted and linear perturbation theoryis not expected to be valid. The failure of this intuitionwas investigated in [29] which suggests much of the non-linearity is trapped behind a forming common apparenthorizon and never makes it out to future null infinity inthe form of GWs. Even more surprising, in [7] this ap-proach was extended by the inclusion of mirror modesalong with overtones (thereby doubling the number ofQNMs) and it was found that it was possible to startthe ringdown analysis even earlier (up to 10 M before thepeak). Clearly it is not surprising that a model with somany free parameters is able to fit the GW signal well;the important point is that it is able to do so withoutobtaining biased values for the final mass and spin. Aringdown model with overtones was successfully appliedto GW150914 in [30] to extract the fundamental QNMand the first overtone from the noisy data, and more re-cently to events in GWTC-2 [31].To the best of our knowledge, previous studies mod- elling the ringdown from NR simulations with QNMshave only considered aligned-spin BBH systems. It iswell known that misalignment between the orbital angu-lar momentum and the spins of the component BHs causethe orbit to precess during the inspiral phase of the evo-lution leading to qualitatively different GW signals atearly times (e.g. see [32]). It is less clear what effect,if any, misaligned component spins would have on thelate time ringdown signal which is generally associatedwith the remnant BH. The primary aim of this paperis to address this question by systematically extendingthe analyses of [7, 27] to a large number of precessingBBH simulations from the SXS catalog [33, 34]. We findthat for BBH systems with misaligned spins, and thattherefore exhibit precession during their inspiral phase,a model consisting only of overtones (with or withoutmirror modes) cannot be reliably applied from the peakof the 22 strain. A more conservative ringdown starttime corresponding to the peak of the GW energy fluximproves reliability, but we still see significant variationin performance across different simulations. The intro-duction of a higher harmonic (QNMs with (cid:96) >
2) to theovertone model helps to reduce this variation, hinting atthe importance of mode mixing.Previous studies have also focused on using full NRsimulations to test ringdown models. In this paper webriefly investigate the use of surrogates, which providean opportunity to test models over a continuous parame-ter space. We find caution should be taken, particularlyfor surrogates of precessing systems, due to errors in thesurrogate waveforms.In section II we reproduce some important results from[7, 27], which are later compared with those for precess-ing systems in section III. With precessing systems, it isnecessary to perform a rotation to account for the factthat the spin of the remnant BH will not be aligned withinitial coordinate axes used to set up the simulation; theprocedure for doing this is also discussed in section III.In section IV we comment on the use of NR surrogatesto test ringdown models. Finally, concluding remarks arepresented in section V. Throughout, we use natural unitsin which G = c = 1. II. ALIGNED-SPIN SYSTEMS
Along with reproducing some important results forspin-aligned systems from [7, 27], this section serves asan introduction to QNM modelling and describes our nu-merical QNM fitting procedure.The full GW signal far from a source of total mass M can be expanded in terms of spin-weight s = − h ( t, r, θ, φ ) = Mr ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) h (cid:96)m ( t ) − Y (cid:96)m ( θ, φ ) . (1)The h (cid:96)m ( t ) coefficients are referred to as the sphericalharmonic modes of the GW signal. The (cid:96) = | m | = 2modes are typically largest, while the remaining “highermodes” are generally subdominant. The output of anNR simulation usually includes the first few modes (e.g. (cid:96) ≤
8) with the asymptotic radial dependence scaled out.The spherical harmonic modes are defined with respectto a particular frame at infinity. This frame is chosento be one in which the centre-of-mass of the system isat rest at some initial time; but this still leaves freedomto perform an overall rotation. By convention, the
NRframe ( θ, φ ) is uniquely fixed by requiring that initially the two component BHs are located along the x -axis andthe orbital angular momentum, L , points along the z -axis.At late times ( t ≥ t , where t is to be determined)the signal is modelled as a sum of QNMs. We note thatas QNMs are not complete , in the sense of being deriv-able from a underlying self-adjoint operator, this modelis necessarily an approximation. The most general QNMringdown model is a sum over ( (cid:96), m, n ) including both theregular ( ω (cid:96)mn ) and the mirror ( ω (cid:48) (cid:96)mn ) mode frequencies(see, e.g. [6]), h ( t, r, θ (cid:48) ,φ (cid:48) ) = M f r ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) ∞ (cid:88) n =0 (cid:104) C (cid:96)mn e − iω (cid:96)mn ( t − t ) − S (cid:96)mn ( θ (cid:48) , φ (cid:48) ) + C (cid:48) (cid:96)mn e − iω (cid:48) (cid:96)mn ( t − t ) − S (cid:48) (cid:96)mn ( θ (cid:48) , φ (cid:48) ) (cid:105) , (2)for t ≥ t . Here, − S (cid:96)mn ( θ (cid:48) , φ (cid:48) ) are the spheroidal har-monics of spin weight −
2, which are the most naturalangular basis for the radiation produced by a perturbedKerr BH. This model is constructed in the ringdownframe ( θ (cid:48) , φ (cid:48) ) in which the remnant BH is at rest with itsspin vector pointing along the positive z -direction (sucha frame is unique up to an unimportant φ (cid:48) rotation aboutthe z -axis). For aligned-spin BBH systems, which do notprecess during the inspiral, the ringdown and NR framesremain aligned with each other (at least up to an over-all sign; some systems with strongly negative componentspins can exhibit “spin flip” where the final spin pointsin the negative z -direction). For misaligned-spin systemsthe remnant spin can point in essentially any directionand the NR and ringdown frames are misaligned. Theringdown frame will also be moving with respect to theNR frames as a result of the recoil, or kick , from theanisotropic emission of GWs near merger. The kick di-rection can also serve to single out a preferred φ (cid:48) direc-tion in the ringdown frame. The effects of the kick areneglected here; it is assumed that the NR and ringdownframes are related by a rotation (see section III).Following Giesler et al. [27], the spherical harmonicmodes of the ringdown signal can be modelled by writingeach as a sum of N overtones: h N(cid:96)m ( t ) = N (cid:88) n =0 C (cid:96)mn e − iω (cid:96)mn ( t − t ) , for t ≥ t . (3) This overtone model is a restriction of the sum inEq. 2, where overlaps between different harmonic (cid:96) in-dices (mode mixing) [35] as well as mirror modes areneglected. As in [27], we model each spherical harmonicmode individually as a sum of QNMs. An alternativeapproach would be to model several spherical harmonicmodes simultaneously with a shared set of QNM ampli-tudes which might give improved fits, especially whenmode mixing is significant (see, e.g. [36]). In [27], theefficacy of this model for l = m = 2 was demonstratedby performing least squares fits to the h ( t ) mode for aselection of aligned-spin SXS simulations. The authorsfurther note that this was also verified for other values of( (cid:96), m ).The overtone model in Eq. 3 contains 2( N + 1) freeparameters in the complex amplitudes, C (cid:96)mn , plus thetwo parameters M f , χ f that determine the ω (cid:96)mn fre-quencies. All of these parameters depend on the prop-erties of the progenitor binary, but we do not studythese dependencies here. We now briefly describe the nu-merical procedure used to search over these parametersand obtain a least-squares fit with the NR strain data d = h (cid:96)m ( t ). Writing the difference between d and theovertone model in Eq. 3 in its discretely sampled form(at times t , t , . . . , t K − ) gives a linear matrix equa-tion (we temporarily drop the (cid:96) and m indices here forclarity), || d − h N(cid:96)m ( t ) || = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d ( t ) d ( t )... d ( t K − ) − e − iω ( t − t ) e − iω ( t − t ) · · · e − iω N ( t − t ) e − iω ( t − t ) e − iω ( t − t ) · · · e − iω N ( t − t ) ... ... . . . ... e − iω ( t K − − t ) e − iω ( t K − − t ) · · · e − iω N ( t K − − t ) C C ... C N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (4)As a first step, we consider varying only the complex amplitudes C n . Eq. 4 turns the minimisation problem −
20 0 20 40Start time, t − t h peak [ M ]10 − − − − M i s m a t c h , M N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 FIG. 1. The mismatch for the overtone model (Eq. 3) whenfitting to the NR simulation SXS:BBH:0305 as a function ofringdown start time t . When using only a single QNM (thefundamental ( (cid:96), m, n ) = (2 , , N = 7 overtones the GW signal can be modelledusing QNMs starting from as early as the peak strain. Thedashed grey curve shows the estimate of the error in the un-derlying NR simulation and is described in appendix C. into a linear algebra problem that can be efficiently solvedwith, for example, numpy.linalg.lstsq [37]. The sec-ond step, fitting over M f and χ f , is then performed usinga simple 2-dimensional grid search algorithm that callsthe linear algebra routine at each iteration.Once the least-squares fit to the data has been ob-tained, the quality, or goodness-of-fit , is quantified viathe mismatch and the error on the remnant parameters.The mismatch between signals h and h is defined as M = 1 − |(cid:104) h | h (cid:105)| (cid:112) (cid:104) h | h (cid:105) (cid:104) h | h (cid:105) , (5)where we use the following complex inner product [38] (cid:104) h | h (cid:105) = (cid:90) Tt h ( t ) h ( t ) d t, (6)(an overbar denotes complex conjugation). We integratefrom the ringdown start time, t , to an upper limit T chosen such that the whole ringdown is captured (we use T = t + 100 M ). However, as noted in [27], a smallmismatch is not sufficient by itself to justify the model.The overtone model contains more parameters as N isincreased, and it is necessary to check for over-fitting. Toaddress this, we check to see if the remnant BH propertiesare correctly recovered by the model. The combined erroron the remnant mass and spin is quantified by [27] (cid:15) = (cid:115)(cid:18) δM f M (cid:19) + ( δχ f ) , (7) .
60 0 .
65 0 .
70 0 .
75 0 .
80 0 . χ f . . . . . . . . . R e m n a n t m a ss , M f [ M ] t = t h peak N = 7 N = 0 N = 1 N = 2 N = 3 N = 4 N = 7 − − − − − L og m i s m a t c h ,l og M FIG. 2. The recovery of the remnant properties for theovertone model (Eq. 3) when fitting to the NR simulationSXS:BBH:0305 starting from the peak in the h strain.The heat map shows the mismatch for the fit with N = 7overtones, which shows a pronounced minimum close ( (cid:15) =3 . × − ) to the true remnant parameters (indicated by thehorizontal and vertical lines). The sequence of crosses showsthe locations of the minima for fits performed with the over-tone model using different values of N , all using the samestart time (the cross colours correspond to the colours usedin Fig. 1; crosses for N = 5 and 6 are omitted to avoid crowd-ing the plot, but they converge towards the true remnantparameters). where δM f = M bestfit − M f , and δχ f = χ bestfit − χ f . Thebest fit values are those which minimise the mismatch,while the true values are taken from the metadata forthe SXS simulation. A ringdown model can be said toperform well if it yields small values for both M and (cid:15) .Following [27], we now apply these ideas to the simula-tion SXS:BBH:0305 [39, 40]. This simulation has sourceparameters consistent with GW150914 and was originallychosen to demonstrate the success of the overtone model.Fig. 1 shows the mismatch values obtained with the over-tone model when using the true values of M f and χ f .With N = 7 (that is, eight QNMs = the fundamental+ seven overtones) the h ( t ) mode can be fitted all theway back to the time of its peak amplitude, t h peak , whilestill achieving the smallest possible mismatch. Using asmaller number of overtones requires a later choice forthe start time to achieve the smallest possible mismatch.In addition to giving a small ( ∼ − ) mismatch, the N = 7 overtone model, with t = t h peak , also achievesthis minimum mismatch with the correct values for theremnant properties; this is shown by the heat map inFig. 2 where the values of M f and χ f are now allowedto vary. We find, for the N = 7 model, a remnant error (cid:15) = 3 . × − . Importantly, this is larger than the NRerror on the remnant properties (which is estimated tobe (cid:15) NR = 2 . × − , see appendix C for details). Thisconfirms that this is really the true scale of the bias in − − − Remnant mass-spin error, (cid:15) N u m b e r o f s i m u l a t i o n s − − Mismatch, M t = t h peak t = t h peak − MN = 0 N = 3 N = 7 N = 7, withmirror modes FIG. 3. Left: histograms of the mass-spin remnant error (cid:15) from an overtone model fit to 85 aligned-spin SXS simulations forseveral different overtone numbers N . Right: histograms of the mismatch from a fit with the true remnant mass and spinparameters, with the same overtone models and SXS simulations as in the left histogram. The solid histograms show resultsfrom fits performed starting at the peak of the h mode with N overtones of the fundamental (cid:96) = m = 2 mode. The reddashed line shows results from a N = 7 model that also includes mirror modes (see section II A) and was fitted with a ringdownstarting 5 M before the peak in the strain. the inferred remnant parameters when using the overtonemodel, and not just the numerical noise floor in the NRsimulation. Again, using a smaller number of overtonesand starting the ringdown as early as t h peak gives inferiorresults with the minimum in the mismatch being biasedaway from the true parameters. The results in Figs. 1and 2 show that the overtone model performs well forSXS:BBH:0305 (i.e. yields small M and (cid:15) ) even whenstarting the ringdown as early as the peak in the strain.In order to see how robust the conclusions drawn fromSXS:BBH:0305 are in general, the calculations of (cid:15) and M were repeated for a wider selection of SXS simula-tions. Following [27], we consider only aligned-spin sim-ulations with initial spin magnitudes | χ , | = χ , < . q <
8. We also require that the z -component of χ f is greater than zero, which eliminatesthe “spin flip” systems. The simulations were chosen inthe ID range SXS:BBH:1412 to SXS:BBH:1513, as thesecover a range of initial spin magnitudes and mass ratios.After applying these cuts, this left 85 spin-aligned SXSsimulations in our test population. For each simulation,fits were performed using the overtone model with N = 0,3, and 7 and with a start time of t = t h peak . The resultsare shown in Fig. 3. We see distributions similar to thosein Fig. 3 of [27]. The inclusion of additional overtonessystematically shifts the entirety of both the (cid:15) and M histograms to smaller values. We note, as it will becomeimportant later, that the worst cases in these histogramsimprove, along with the median values. This demon-strates that, when using the overtone model on systemswith aligned spins, the ringdown reliably starts as earlyas the peak in the h ( t ) mode of the strain. A. Mirror Modes
For a given (cid:96) , m and n , the equations governing QNMfrequencies allow two solutions: one, ω (cid:96)mn = 2 πf (cid:96)mn − i/τ (cid:96)mn , with a positive real part; and another, ω (cid:48) (cid:96)mn =2 πf (cid:48) (cid:96)mn − i/τ (cid:48) (cid:96)mn , with negative real part [6, 7]. Thefrequencies of the mirror modes ω (cid:48) (cid:96)mn are related to theregular modes ω (cid:96)mn by f (cid:48) (cid:96)mn = − f (cid:96) − mn , τ (cid:48) (cid:96)mn = τ (cid:96) − mn ⇒ ω (cid:48) (cid:96)mn = − ω ∗ (cid:96) − mn . (8)A new ringdown model which explicitly includes themirror modes can be written as h N, mirror (cid:96)m ( t ) = N (cid:88) n =0 (cid:104) C (cid:96)mn e − iω (cid:96)mn ( t − t ) (9)+ C (cid:48) (cid:96)mn e − iω (cid:48) (cid:96)mn ( t − t ) (cid:105) for t ≥ t . This mirror mode model is an extension of the overtonemodel in Eq. 3; if C (cid:48) (cid:96)mn = 0 the mirror modes aren’texcited and we recover the previous overtone model. Thismodel has twice as many free parameters as the overtonemodel; 4( N + 1) in the complex amplitudes, plus the tworemnant parameters M f , χ f . The mirror mode modelis still a restriction of the full sum in Eq. 2 as overlapsbetween modes with different (cid:96) indices (i.e. mode mixing)are still not included. Substituting for ω (cid:48) (cid:96)mn using theconjugate symmetry property in Eqs. 8, we can rewritethe mirror mode model in the form h N, mirror (cid:96)m ( t ) = N (cid:88) n =0 (cid:104) C (cid:96)mn e − iω (cid:96)mn ( t − t ) (10)+ C (cid:48) (cid:96)mn e iω ∗ (cid:96) − mn ( t − t ) (cid:105) for t ≥ t . It is this form of the mirror mode model that was imple-mented.As was shown in [7], the inclusion of mirror modes canimprove the ringdown modelling of aligned-spin systems.In particular, the ringdown can be considered to starteven earlier in the waveform, whilst still recovering thecorrect remnant properties. We confirm this here by re-peating the above analysis for the same set of spin-alignedSXS simulation, but now using the mirror mode model inEq. 10 with N = 7 and an earlier choice for the ringdownstart time, t = t h peak − M . Although [7] demonstratedthe mirror mode model starting 10 M before the peakin the h strain, we adopt a more conservative choiceof 5 M . The results are shown in Fig. 3 plotted using adashed line. The addition of mirror modes gives a smallimprovement in the mismatch, but this is to be expectedwith the increased number of parameters. However, the (cid:15) histogram shows that the overall performance of themirror mode model is comparable to that of the N = 7overtone model, despite the use of an earlier start time. III. MISALIGNED-SPIN SYSTEMS
The analyses in section II, and in the previous studies[7, 8, 22–28], was limited to BBH systems with compo-nent spins that are aligned with the orbital angular mo-mentum, L . This is a potentially serious limitation asmisaligned spins are expected to be a generic feature ofastrophysical BBHs. Misaligned spins generally lead toprecession of the orbital plane during the inspiral phaseof the evolution and a richer phenomenology in the GWsignals [32]. Several of the GW detections already showsigns of precession, both individually [4, 41] and whenconsidered as a population [42]. Most notably for ourpresent purposes, the very high-mass system GW190521[43, 44] might show some signs of precession while alsohaving a high fraction of the observable signal-to-noiseratio in the ringdown. In this section we investigate theeffect of precession on the modelling of the ringdown byrepeating analyses like those in section II, but now onprecessing NR simulations.The general ringdown signal in Eq. 2 is written in theringdown frame ( θ (cid:48) , φ (cid:48) ) which is aligned with the remnantspin vector (i.e. χ f points along the positive z -directionin this frame). For BBH systems with misaligned spinsthat undergo precession, the ringdown frame differs fromthe frame typically used in NR simulations ( θ, φ ) whichis aligned with L at some arbitrary start time in theinspiral. These two frames are related by a rotation, R .The direction from which a GW source is viewed affectsthe observed signal (e.g. you see circularly/linearly po-larised GWs with a larger/smaller amplitude when view-ing parallel/perpendicular to L ). These differences inthe GW signals also manifest themselves at the level ofindividual modes as amplitude modulations. The framein which the expansion is performed affects the values ofthe spherical harmonic modes. The GW signal, originally given in the NR frame in Eq. 1, can be re-expanded inthe ringdown frame as follows: h (cid:48) ( t, r, θ (cid:48) , φ (cid:48) ) = Mr ∞ (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) h (cid:48) (cid:96)m ( t ) − Y (cid:96)m ( θ (cid:48) , φ (cid:48) ) . (11)When analysing the ringdown, it is most natural to usethe spherical harmonic modes in the ringdown frame, h (cid:48) (cid:96)m ( t ), as these are adapted to the remnant BH. In par-ticular, we will focus on modelling the (cid:96) = m = 2 spher-ical harmonic mode in the ringdown frame, h (cid:48) ( t ).For aligned-spin systems, the h ± are usually thedominant modes in the sum in Eq. 1. This is relatedto the fact that the GW signal amplitude is largest whenviewed along the direction of the orbital angular momen-tum: L or − L . For misaligned-spin systems undergo-ing precession, other modes become important. This inturn is related to the constantly changing direction ofthe orbital angular momentum, L ( t ). Changing into thenon-inertial, coprecessing frame in which L always pointsalong the z -direction has been found to account for mostprecessional effects and makes the precessing waveformremarkably similar to a non-precessing one. This trans-formation into the coprecessing frame has been success-fully used to help model the full inspiral-merger-ringdownwaveforms for precessing systems [45, 46] in the context ofphenomenological [15, 47, 48], effective-one-body [49, 50]and NR surrogate [51–53] modelling. There is an anal-ogy with the approach taken here for the modelling ofthe ringdown. In order to simplify the task, we choose to t ˙ E peak − t h peak [ M ]0 . . . . . . N o r m a li s e d c o un t Aligned spinsMisaligned spins
FIG. 4. Histogram of the differences between the two pos-sible start times considered in section III: the peak of the(rotated) strain mode h (cid:48) , and the peak of the GW energyflux. The normalised distribution of the differences betweenthese times is shown both for the 85 spin-aligned systems usedin section II and for the 252 precessing simulations consideredin section III. The peak of the flux almost always occurs laterthan the peak of the strain, making this a more conservativechoice for the ringdown start time. We note that there isa much greater variation amongst the population with mis-aligned spins. −
20 0 20 40Start time, t − t ˙ E peak [ M ]10 − − − M i s m a t c h , M N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 FIG. 5. The mismatch for the overtone model (Eq. 3) whenfitting to the NR simulation SXS:BBH:1856 as a function ofringdown start time t . When using N = 7 overtones, thelowest mismatch is achieved starting slightly ( ∼ M ) beforethe peak in the GW energy flux. However, the minimum mis-match is ∼
100 times larger than that obtained for the exam-ple spin-aligned system SXS:BBH:0305 in Fig. 1. The dashedgrey curve shows the estimate of the error in the underlyingNR simulation and is described in appendix C. work in a frame adapted to final spin angular momentumof the remnant, χ f . Although, in our case, the rotationrequired to get into this frame is not time dependent andour chosen frame is therefore inertial.The spin-weighted spherical harmonics, − Y (cid:96)m , trans-form in a particularly simple manner under rotations.Rotations have the effect of mixing together modes withdifferent m indices, but preserving the same (cid:96) . The mix-ing coefficients in these transformations are the Wigner D -matrices D (cid:96)µm ( R ). The transformation properties ofthe − Y (cid:96)m functions under rotations means that the ro-tated h (cid:48) (cid:96)m modes in the ringdown frame are related to the h (cid:96)m modes in the NR frame (included in the NR output,and as used in section II) by the sum (see, for example[45, 54, 55]) h (cid:48) (cid:96)m ( t ) = (cid:96) (cid:88) µ = − (cid:96) D (cid:96)µm ( R ) h (cid:96)µ ( t ) . (12)The rotation R can be obtained from the direction of theremnant BH spin vector (which is provided as metadatafor all SXS simulations). Specifically, R is any rotationthat maps the z -axis onto the final spin vector.We now apply the overtone model to the ringdownof an example precessing simulation SXS:BBH:1856 [53].This simulation initially (at the reference time) has amass ratio of q = 2 .
78 and dimensionless spins χ =(0 . , − . , − .
45) and χ = ( − . , − . , − . h ( t ).The final spin vector is χ f = ( − . , − . , .
42) andthe rotated mode h (cid:48) ( t ) was computed using Eq. 12. .
40 0 .
45 0 .
50 0 .
55 0 .
60 0 . χ f . . . . . . . . . R e m n a n t m a ss , M f [ M ] t = t ˙ E peak N = 7 N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 − . − . − . − . − . − . − . − . − . L og m i s m a t c h ,l og M FIG. 6. The recovery of the remnant properties for theovertone model (Eq. 3) when fitting to the NR simulationSXS:BBH:1856 starting from the peak in the flux. The heatmap shows the mismatch for the fit with N = 7, while thecrosses show the locations of the minima in the mismatchfor fits performed with different values of N . The mismatchshows a much broader and less deep minimum than that seenfor the spin-aligned system SXS:BBH:0305 in Fig. 2. Theminimum in the mismatch is also biased away from the trueremnant parameters with (cid:15) = 0 .
025 for the N = 7 fit. Thesequence of crosses for fits with different values of N also donot show the same convergent trend towards the true remnantparameters that was observed for SXS:BBH:0305 in Fig. 2. The overtone model in Eq. 3 was fitted to the rotated (cid:96) = m = 2 mode of the strain, h (cid:48) ( t ), in the same wayas was done for the aligned-spin systems in section II.There is some ambiguity in how to choose the ringdownstart time t in a way that gives as fair a comparison aspossible with the non-precessing case. We cannot use thepeak of the h ( t ) strain, as was done in section II, as thismode suffers from precession induced amplitude modu-lations. One option would be to use instead the peak ofthe rotated strain mode h (cid:48) ( t ). However, we find thatusing the peak of the GW energy flux, ˙ E , (which can becomputed from the modes in either frame, see Eq. 3.8 in[56]) gives more consistent results between simulations.For example, some precessing configurations show a peakin the (rotated) strain relatively early in the signal, lead-ing to poorer fits. The use of the peak in the flux is alsoa conservative choice in the sense that t ˙ E peak > t h peak inalmost all cases (see Fig. 4).Fig. 5 shows how the mismatch varies forSXS:BBH:1856 as a function of ringdown start time,for different values of N in the overtone model Eq. 3.With each additional overtone, the minimum mismatchis reached at an earlier time (the same behaviour as wasseen in Fig. 1). However, the values of the minimummismatch are a factor of ∼
100 larger than thoseobtained in the aligned-spin case.The N = 7 model achieves a minimum mismatch ∼ M before the time of peak GW energy flux. This is − − − Remnant mass-spin error, (cid:15) N u m b e r o f s i m u l a t i o n s − − Mismatch, M t = t ˙ E peak t = t ˙ E peak − MN = 0 N = 3 N = 7 N = 7, withmirror modes FIG. 7. Left: histograms of the mass-spin remnant error (cid:15) from an overtone model fit to the rotated h (cid:48) modes of 252misaligned-spin SXS simulations for several different overtone numbers N . Right: histograms of the mismatch from a fit withthe true remnant mass and spin parameters, with the same overtone models and SXS simulations as in the left histogram. Thesolid histograms show results from fits performed starting at the peak of the energy flux with N overtones of the fundamental (cid:96) = m = 2 mode. The red dashed line shows results from a N = 7 model that also includes mirror modes and was fitted with aringdown starting 5 M before the peak in the energy flux. These histograms should be compared with those in Fig. 3; we notethat the effect of precession is to (i) significantly broaden the histograms (i.e. the quality of the fit is much more varied) and(ii) to significantly degrade the quality of the fit for some systems. fairly typical behaviour among the misaligned-spin SXSsimulations considered. However, we note there is a muchgreater variety of possible behaviours for misaligned-spinsystems than for the aligned-spin population. In order toemphasise this greater variation, in appendix A we repeatthe analysis in Figs. 5 and 6 for three more misaligned-spin simulations and highlight some of the observed dif-ferences. The greater variation amongst the misaligned-spin population has already been hinted at in Fig. 4,where the spread of start times is greater than in thealigned-spin cases.The heat map of Fig. 6 shows the mismatch as afunction of the remnant BH properties, for the N = 7model. The coloured crosses indicate the mismatch min-imum for different values of N . Comparing with Fig. 2,we see the mismatch minimum is less pronounced thanthe aligned-spin case, which is probably contributing tothe larger value of (cid:15) (for N = 7 we find (cid:15) = 0 . (cid:15) NR = 8 . × − ). In addition, the convergent behaviourwith increasing N is not present. For N ≥
1, all mis-match minima appear randomly distributed around thetrue remnant properties. If we reproduce this figure witha earlier start time of t = t ˙ E peak − M (motivated bythe time of minimum mismatch for N = 7 in Fig. 5), theheat map remains unchanged, and the value of (cid:15) recov-ered for N = 7 is not significantly improved ( (cid:15) = 0 . (cid:15) for N ≤ χ θ , satisfies π/ <χ θ < π/
16. We again require initial spin magnitudes χ , < . q <
8. The 252 simu-lations were chosen in the ID range SXS:BBH:1643 toSXS:BBH:1899, as these cover a range of mass ratios andinitial spin configurations.The results are shown in Fig. 7. When compared tothe N = 0 model, the addition of three overtones reducesthe remnant error and mismatch. However, the inclusionof additional overtones does not change the (cid:15) histogram,and produces only a minor reduction in the mismatch.Comparing the N = 7 histogram for (cid:15) to that found inFig. 3, we see that, on average, (cid:15) increases by a factorof ∼
10 and, in the worst cases, by a factor of ∼ (cid:15) reflect the behaviour of Fig. 6, where models with N ≥ t ˙ E peak − M changed the recovered distribution on (cid:15) . This choicewas motivated by the location of the mismatch mini-mum typically seen for misaligned-spin simulations (e.g.see Fig. 5). The results are shown in appendix B. Itwas found the N = 7 model results did not significantlychange. However, the N = 3 and N = 0 models per-formed worse. Finally, we also note that all of the his-tograms are wider than those in Fig. 3. This may be dueto mirror modes and/or higher harmonics having a moreimportant role for precessing systems (see below). A. Mirror Modes
We repeat the population analysis with the N = 7mirror mode model, again shifting the ringdown start −
20 0 20 40Start time, t − t ˙ E peak [ M ]10 − − − M i s m a t c h , M N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 FIG. 8. The mismatch for the mirror mode model (Eq. 10)fitted to the NR simulation SXS:BBH:1856 as a function ofringdown start time t . Comparing with Fig. 5, the locationsof the mismatch minima are roughly unchanged in time, butthe inclusion of mirror modes reduces the mismatch to valuessimilar to those in Fig. 1. The dashed grey curve shows theestimate of the error in the underlying NR simulation and isdescribed in appendix C. time back by 5 M to make a clear comparison to Fig. 3.The histogram for (cid:15) doesn’t reach values as high as theovertone model (with worst-case values of (cid:15) ∼ .
04 com-pared to the overtone model’s ∼ . (cid:15) ). For example,Figs. 8 and 9 show how mirror mode fits perform forSXS:BBH:1856. We see significantly smaller mismatches,and a stronger mismatch peak around the true remnantproperties. However, on average this does not trans-late to smaller values of (cid:15) for the N = 7 model (as canbe seen from the red dashed histogram in Fig. 7). ForSXS:BBH:1856, the N = 7 model gives (cid:15) = 0 . t ≥ t h peak for the N = 7 overtone model). In addition, the mismatch min-imum stays centred on the true remnant properties until .
45 0 .
50 0 .
55 0 . χ f . . . . . . . . . R e m n a n t m a ss , M f [ M ] t = t ˙ E peak − MN = 7, withmirror modes N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 − . − . − . − . − . − . − . L og m i s m a t c h ,l og M FIG. 9. The recovery of the remnant properties for themirror mode model (Eq. 10) when fitting to the NR simu-lation SXS:BBH:1856, starting from 5 M before the peak ofthe flux. The heat map shows the mismatch for the fit with N = 7, while the crosses show the locations of the minimain the mismatch for fits performed with different values of N ( N = 0 lies outside the figure, and is not included for clarity).Comparing with Fig. 6, the inclusion of mirror modes sharp-ens the mismatch peak and achieves smaller mismatch values.However, when averaged across the population of precessingsimulations, the mirror mode model doesn’t give smaller val-ues for the remnant error (see dashed curve in Fig. 7). Here, (cid:15) = 0 .
014 for the N = 7 model. numerical noise takes over. Applying the N = 7 overtonemodel to the misaligned-spin simulation SXS:BBH:1856,we see that the location of the mismatch minimum movesaround the mass-spin plane as start time is varied. Evenat late times, it never settles on the true remnant prop-erties. The inclusion of mirror modes, as seen in Fig. 9,narrows the mismatch minimum. The movement of themismatch minimum around the mass-spin plane is re-duced as well, however it still doesn’t settle on the lo-cation of the true remnant properties. This behaviourmay explain some of the observed widening of the his-tograms, and perhaps hints something is missing fromthe ringdown model. B. Higher Harmonics
As demonstrated by Fig. 7 (and also Fig. 13 in ap-pendix A), the overtone and mirror mode models consid-ered so far achieve median values for the remnant error (cid:15) ∼ .
01, a factor of 10 or more higher than the aligned-spin fits of Fig. 3. In addition, the spread of (cid:15) valuesrecovered is significantly larger, leading to values of (cid:15) upto ∼ .
1. These models perform significantly worse insome cases for precessing systems than aligned-spin sys-tems.We now investigate whether the inclusion of higherharmonics (that is, QNMs with (cid:96) >
2) can improve the0 − − − Remnant mass-spin error, (cid:15) N u m b e r o f s i m u l a t i o n s − − − Mismatch, M t = t ˙ E peak , N = 7 L = 2 L = 3, withmirror modes L = 4, withmirror modes FIG. 10. Left: histograms of the mass-spin remnant error (cid:15) from harmonic model fits (Eq. 14) to same 252 misaligned-spinSXS simulations used in Fig.7. Shown (in dashed lines) are the N = 7, L = 3 and L = 4 models with mirror modes. Alsoshown in green is the overtone model with N = 7 and L = 2 (no mirror modes); this is the same as the green histogram inFig. 7 and is included here to aid comparison. Right: histograms of the mismatch from a fit with the true remnant mass andspin parameters, with the same models and SXS simulations as in the left histogram. The harmonic model, which includesmany free parameters, achieves small mismatches but without significant improvement in the remnant error. We note that theinclusion of L = 4 does not bring any additional improvements over L = 3. fits to h (cid:48) ( t ). These higher harmonics were neglected byboth the overtone (Eq. 3) and mirror mode (Eq. 10) mod-els. However, mode mixing does occur as a consequenceof the different angular basis functions used in the wave-form decompositions in Eqs. 1 and 2 and the fact thatthese basis functions are not mutually orthogonal [35].The amount of mode mixing between the spherical mode − Y (cid:96)m and the spheroidal mode − S (cid:96)mn is determinedby the remnant spin χ f and the QNM frequency. Thiscan be quantified by how much these functions fail to beorthogonal; i.e. by the integral µ (cid:96)m,(cid:96) (cid:48) m (cid:48) n (cid:48) = δ mm (cid:48) (cid:90) Ω − Y (cid:96)m (Ω) − ¯ S (cid:96) (cid:48) m (cid:48) n (cid:48) (Ω) dΩ , (13)where Ω denotes the angles θ, φ . A translational offsetbetween the NR and ringdown frames (e.g. due to a kick)can also lead to mixing between m -modes [58]; this ef-fect is neglected here. To include the contribution fromhigher harmonics, we define a new ringdown model forthe spherical harmonic modes which now allows for asum over different (cid:96) : h N, L, mirror (cid:96)m ( t ) = N (cid:88) n =0 L (cid:88) l =2 (cid:104) C lmn e − iω lmn ( t − t ) (14)+ C (cid:48) lmn e iω ∗ lmn ( t − t ) (cid:105) for t ≥ t . This harmonic model contains all of the allowed QNMsin Eq. 2, including the mirror modes and the overtones.This comes at the expense of a large number of free pa-rameters; there are 4( N + 1)( L − (cid:96) + 1) in the complexamplitudes, plus the two remnant parameters M f , χ f that determine the complex QNM frequencies.Multiple variations of this harmonic model were tri-alled (varying N , L , and the inclusion of mirror modes) on the same population of 252 misaligned-spin SXS simu-lations. Fig. 10 shows the chosen subset of results, whichincludes the N = 7, L = 3 and L = 4 models (both withmirror modes). As before, we fit to the rotated h (cid:48) ( t )spherical harmonic mode. To make a clear comparisonwith the previous models, we again use a ringdown starttime corresponding to the peak of the GW energy flux.The inclusion of higher harmonics (dashed histogramsin Fig. 10) drastically improves the mismatch. A smallmismatch is not surprising for a model with so many freeparameters, and in some of these cases we are likely push-ing beyond the limits of accuracy of the NR simulations.See appendix C for a discussion of the numerical errors.There is a modest reduction in (cid:15) for some systems, andin particular we see less systems with (cid:15) > .
01 (at leastfor L = 3). This hints at the importance of higher har-monics in some precessing systems. Despite this, we stillsee worst-case values of (cid:15) ∼ . IV. SURROGATES
NR simulations are computationally expensive, and al-though the number of simulations available in public cat-alogs is growing they are still limited in their parameterspace coverage. NR surrogate models [52, 53, 59, 60]would appear to be an attractive alternative. These mod-els use reduced-order and surrogate modelling techniquesto extend the results of a set of NR simulations smoothlyacross parameter space. The use of surrogates could, inprinciple, allow us to extend the results of the previoussection to include many more systems as well as allow-ing us to study how the excitations of the various QNMsvary during a smooth exploration of parameter space.However, care must be taken as the surrogate modellingnecessarily introduces an additional source of error into1 q − − − − R e m n a n t e rr o r , (cid:15) FIG. 11. Comparison of the remnant error (cid:15) from twosurrogate models several of SXS simulations. All are zero-spin. The labels on each cross correspond to the SXS ID. Thedashed line indicates where we are outside the training rangeof NRSur7dq4. the waveforms, on top of the errors originally in the NRwaveforms themselves.When attempting to fit QNM ringdown models withovertones to NRSur7dq4 [53] waveforms, it was foundthat incorrect values for M f and χ f were being re-covered (particularly at high mass ratios). This be-ing the case even for aligned-spin or non-spinning sys-tems. Although the NRSur7dq4 waveforms do not pro-vide the remnant properties, these can be obtained viaNRSur7dq4Remnant [53] (it was found the problem didnot lie with the values returned by NRSur7dq4Remnantbut rather with the waveform surrogate).To investigate the performance of NRSur7dq4 ring-down waveforms, a series of zero-spin waveforms withincreasing mass ratio q from 1 to 6 were used. The N = 7overtone model (Eq. 3) was fitted to the h ( t ) mode ofeach starting from the peak strain (as in section II) andthe remnant error (cid:15) (Eq. 7) was calculated for each. Theresults are shown in Fig. 11, along with the results forsimilar fits performed directly on 11 zero-spin SXS sim-ulations at discrete values of the mass ratio. The fitsto the NRSur7dq4 surrogate produce values for (cid:15) thatare 1-2 orders of magnitude higher than for the equiva-lent SXS simulations. Also shown are the results froma similar analysis with the more restrictive aligned-spinsurrogate NRHybSur3dq8 [60]; this was found to be inclose agreement with the SXS simulations.Residuals and mismatches can also be computed be-tween surrogate and NR waveforms (taking care toalign the waveforms in both time and phase). ForSXS:BBH:0168, the q = 3, zero-spin simulation used inFig. 11, we find ∼
2% residuals in the ringdown whencomparing to the NRSur7dq4 surrogate with the sameparameters. This leads to a mismatch between the sur-rogate and SXS:BBH:0168 of 3 . × − , when integratingover the ringdown. For comparison, we have a ∼ − mismatch between the ringdown model Eq. (3) and theSXS simulation. The relatively high mismatch betweenthe NRSur7dq4 and SXS waveforms translates to the rel-atively high values of (cid:15) seen in Fig. 11.It seems that the high-dimensional precessing surro-gate NRsur7dq4 is not yet sufficiently accurate in theringdown for the purposes of QNM overtone studiesthat, by virtue of their large number of free parame-ters, fit the ringdown with very small mismatches. Bycontrast, the lower-dimensional aligned-spin surrogateNRHybSur3dq8 does appear to be sufficiently accuratefor such studies. V. DISCUSSION
This paper has made a first systematic attempt at us-ing QNMs to model the ringdown of BHs formed fromBBHs with misaligned component spins in the inspiral.Previously, for aligned-spin systems, it has been foundthat the ringdown can be modelled with low mismatchand low remnant errors using a model that includes over-tones of the fundamental QNM [27]. For seven overtones,the ringdown can be reliably modelled from the peak ofthe h ( t ) strain for a range of SXS simulations. Ad-ditionally, the inclusion of mirror modes can allow theringdown to be modelled from even earlier times [7]. Inthis paper, which generalised these studies to precessingsystems, we find that while QNM models can reliablyachieve small mismatches, in the worst cases the rem-nant error are more than a factor of 10 higher. This isthe case even when choosing to start the ringdown at themore conservative (i.e. later) peak in GW energy flux.The inclusion of higher harmonics reduces the remnanterror in some cases, perhaps a sign that mode mixing inthe ringdown is generally more important in precessingsystems. However, in other cases, a bias remains in therecovered remnant properties. We conclude that it is notpossible to reliably model the ringdown from the peak inthe flux, or indeed from the peak in the strain.We end by sounding a brief note of caution to any whoattempt to construct a QNM model starting at or beforethe peak flux or strain. While such a model will work insome cases, it risks biased results in others. This risk issubtle because QNM models can give small mismatcheseven when they fail to adequately describe the remnant. ACKNOWLEDGMENTS
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20 0 20 4010 − − − M i s m a t c h , M N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 .
65 0 .
70 0 . . . . . R e m n a n t m a ss , M f [ M ] t = t ˙ E peak N = 7 N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 −
20 0 20 4010 − − − M i s m a t c h , M .
60 0 .
65 0 . . . . . R e m n a n t m a ss , M f [ M ] t = t ˙ E peak N = 7 −
20 0 20 40Start time, t − t ˙ E peak [ M ]10 − − − − − M i s m a t c h , M . . . . . χ f . . . . . . R e m n a n t m a ss , M f [ M ] t = t ˙ E peak N = 7 − . − . − . − . − . L og m i s m a t c h ,l og M − . − . − . − . L og m i s m a t c h ,l og M − . − . − . − . L og m i s m a t c h ,l og M FIG. 12. A selection of results for modelling the ringdown of precessing NR simulations from the SXS catalog [33, 34, 40] usingthe overtone model in Eq. 3. These plots show the results for the three systems described in the table that have been chosento illustrate the wider range of behaviours that occur for precessing systems, from good at the top to bad at the bottom. Theleft hand column of plots also shows the difficulty in identifying a general start time for the ringdown as mismatch is minimisedfor a range of different times and sometimes there isn’t even a clear first minimum.
SXS:BBH ID Figure row Remnant error (cid:15) ( (cid:15) NR ) Mass ratio q Component spins χ , χ Remnant spin χ f . × − (1 . × − ) 2.64 ( − . , , . − . , − . , .
06) ( − . , , . . × − (8 . × − ) 3.49 (0 . , . , . − . , . , .
47) (0 . , − . , . . × − (4 . × − ) 3.72 (0 . , . , − . − . , − . , − .
17) (0 . , . , . Appendix B: Overtone Model Fits to a Population of Precessing NR Systems Starting Before the Peak Flux
The analysis on the population of misaligned-spin simulations performed in section III (results plotted in Fig. 7) isrepeated here using an earlier start time for the ringdown: t = t ˙ E peak − M . This was done to check whether a poorchoice of start time was responsible for some of the poor fits obtained using the overtone model in Eq. 3. The newresults are plotted in Fig. 13. We find that the N = 7 model results do not significantly change with the new starttime. The N = 3 and N = 0 model results do change and generally give a worse fit with the earlier start time, asmight be expected. This analysis shows that the overtone model (with or without mirror modes) can not be reliablyapplied to precessing systems at early times. − − − Remnant mass-spin error, (cid:15) N u m b e r o f s i m u l a t i o n s − − Mismatch, M t = t ˙ E peak − Mt = t ˙ E peak − MN = 0 N = 3 N = 7 N = 7, withmirror modes FIG. 13. Left: histograms of the mass-spin remnant error (cid:15) from an overtone model fit to the rotated h (cid:48) mode of 252misaligned-spin SXS simulations for several different overtone numbers N . Right: histograms of the mismatch from a fit withthe true remnant mass and spin parameters, with the same overtone models and SXS simulations as in the left histogram. Theseresults are similar to those in Fig. 7 in the main text, but use a start time that is earlier by 10 M . The solid histograms showresults from fits performed starting 10 M before the peak of the energy flux with N overtones of the fundamental (cid:96) = m = 2mode. The red dashed line shows results from a N = 7 model that also includes mirror modes and was fitted with a ringdownstarting 15 M before the peak in the energy flux. Appendix C: Numerical Relativity Errors
It is important to remember the finite accuracy of the NR simulations used in ringdown studies. This is particularlytrue when using models with many QNMs which, by their very nature, use a large number of free parameters andregularly achieve very small ( ∼ − ) mismatches. If care is not taken, we risk fitting our models to the numericalnoise. In this appendix we describe the numerical checks performed on the 5 individual simulations used in this paper:SXS:BBH:0305, 1856, and the three simulations shown in Fig. 12. In each case the numerical errors were estimatedby comparing results obtained using data from the two highest resolutions (levels) available in the SXS catalog.First, we quantify the numerical error in the mismatch. This was done by calculating the mismatch between thetwo NR resolutions from a time t to a time T = t + 100 M , for a range of t . For each start time, we optimally alignthe two waveforms in time (taking the absolute value in the mismatch automatically optimises the mismatch overphase). The alignment in time can be done by matching the time of peak strain, for example, or by numerically rollingthe waveform to find the optimal time shift for each mismatch calculation. The results are shown by the grey dashedlines in the mismatch vs start time plots in Figs. 1, 5 (duplicated in Fig. 8) and the 3 panels of Fig. 12. Generally, wesee numerical error estimates at or below the model mismatches, particularly at late times, indicating that we are notfitting to the numerical noise. The main exception is Fig. 8 where the mirror mode model is applied to a precessingsystem. This is expected; precessing NR simulations, and those with high mass ratios are generally expected tohave larger numerical errors. Additionally, the mirror mode and harmonic models have the highest numbers of freeparameters making them more likely to reach the accuracy of the NR simulation.Second, we investigate the numerical error on the remnant mass and spin. We quantify the numerical error with (cid:15) NR , the Euclidean distance (Eq. 7) between the remnant properties reported in the two highest resolution levels ofthe NR simulation. The (cid:15) NR values are reported in the main text and in the table in appendix A. In all cases (cid:15) NR < (cid:15)< (cid:15)