Simulating merging binary black holes with nearly extremal spins Geoffrey Lovelace,1 Mark A. Scheel,2 and B´ela Szil´agyi2 1Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853, USA 2Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USA (Dated: January 12, 2011) Astrophysically realistic black holes may have spins that are nearly extremal (i.e., close to 1 in dimensionless units). Numerical simulations of binary black holes are important tools both for calibrating analytical templates for gravitational-wave detection and for exploring the nonlinear dynamics of curved spacetime. However, all previous simulations of binary-black-hole inspiral, merger, and ringdown have been limited by an apparently insurmountable barrier: the merging holes’ spins could not exceed 0.93, which is still a long way from the maximum possible value in termsofthephysicaleffectsofthespin. Inthispaper,wesurpassthislimitforthefirsttime,opening the way to explore numerically the behavior of merging, nearly extremal black holes. Specifically, 1 using an improved initial-data method suitable for binary black holes with nearly extremal spins, 1 0 we simulate the inspiral (through 12.5 orbits), merger and ringdown of two equal-mass black holes 2 with equalspins of magnitude0.95 antialigned with theorbital angular momentum. n PACSnumbers: 04.25.dg,04.30.-w a J 1 I. INTRODUCTION Rotational energy / rotational energy if extremal 1 1.1 1.1 1 1 ] Although there is considerable uncertainty, it is pos- 0.9 SKS initial data 0.9 c q sible that astrophysical black holes exist with nearly 0.8 Not accessible with (this paper): 0.8 r- etoxt1re,mthael stphienosre(tii.cea.,l uinppdeirmleinmsiitonfloerssausntaittsiosnpairnysbcllaocske emal 0.7 Binoitwiaeln d-aYtaork (B.Y.) fbiersyto BndB BH. Ym.e lrigmerit 0.7 g hole). Binary black hole (BBH) mergers in vacuum typ- extr 0.6 0.6 [ ot, 0.5 0.5 v3 iχscuar∼lrloyu0nl.e7dae−dd0tb.oy8mr[e1am–t3nt]e,arnattlhthehooruleegmshnwaifinttth’hsedsipmmineerntgysiipnoignclaehlsloyslecssopuainrldes E / Erotr 00..43 00..34 7 be higher than χ ∼ 0.9 [1, 3]. Black holes can reach 0.2 0.2 7 higher spins via prolonged accretion [4, 5]: thin accre- 0.1 0.1 7 2 tiondisks(withmagnetohydrodynamiceffectsneglected) 0 0 0. leadtospinsaslargeasχ∼0.998[6],whilethick-diskac- -0.-10.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.1 cretion with magnetohydrodynamic effects included can 2 1 Spin / (Mass) 0 leadtospinsaslargeasχ∼0.95[7,8]. Evenwithoutac- 1 cretion, at very high mass ratios with spins aligned with FIG. 1. The rotational energy of a Kerr black hole as a : theorbitalangularmomentum,binaryblackholemergers v function of the hole’s dimensionless spin parameter χ := i can also lead to holes with nearly extremal spins [9–11]. Spin/(Mass)2. The thick red line indicates the Bowen-York X There is observational evidence suggesting the existence limit: standard Bowen-York puncture initial data—used in r of blackholes with nearlyextremalspins inquasars[12], almost all numerical binary-black-holecalculations todate— a andsomeeffortsto inferthe spinofthe blackholeinmi- cannotyieldrotationalenergiesmorethan60%ofthewayto croquasar GRS 1915+105from its x-ray spectra suggest extremality. By using instead initial data based on two su- a spin larger than 0.98, though other analyses suggest perposedKerr-Schildholes(“SKSinitialdata”),inthispaper the spin may be much lower [13–15]. we surpass the Bowen-York limit (green circle), opening the way for numerical studies of merging, nearly extremal black MergingBBHs—possiblywithnearlyextremalspins— holes. are among the most promising sources of gravitational waves for current and future detectors. Numerical simu- lations of BBHs are important tools both for predicting tremality in terms of physical effects: a black hole with the gravitational waves that detectors will observe and spin 0.93 has less than 60% of the rotational energy of for exploring the behavior of nonlinear, highly dynam- an extremal hole with the same mass (Fig. 1). ical spacetimes. Following Pretorius’ breakthrough in 2005 [16], several groups have successfully simulated the Previoussimulationshavebeenunable toreachhigher inspiral, merger, and ringdown of two coalescing black spinsbecauseofthewaytheyconstructtheirinitialdata. holes in a variety of initial configurations; however, all Just as initial data for Maxwell’s equations must satisfy priorBBHsimulationshavebeen limitedto spinsof0.93 constraints (the electric and magnetic fields must have or less, which is quite far from extremal. The parameter vanishingdivergenceinvacuum),initialdatafortheEin- χ is a poor measure of how close a black hole is to ex- stein equations must satisfy constraint equations. Most 2 Mi/M 0.5000 MADM/M 0.9933 d0/M 15.366 magnitude 0.95 antialigned with the orbital angular mo- χzi -0.9498 JAzDM/M2 0.6845 Ω0M 0.014508 mentum. Some properties of the initial data used in this a˙0 -0.0007139 paper are listed in Table I. Following Ref. [35] and the references therein, we TABLE I. Properties of initial data evolved in this paper. construct constraint-satisfying initial data by solving ThequantityM denotesthesumoftheholes’Christodoulou the extended conformal thin sandwich equations with massesatt=0. Holei(wherei=A or B)hasChristodoulou mass Mi and dimensionless spin χzi along the z axis (i.e., in quasiequilibrium boundary conditions [36–41] using a thedirectionoftheorbitalangularmomentum). Alsolistedis spectral elliptic solver [42]. The initial spatial metric is the Arnowitt-Deser-Misner (ADM) mass MADM and angular proportional to a weighted superposition of the metrics z momentum JADM (e.g., Eqs. (25)–(26) of Ref. [35]). The of two boosted, spinning Kerr-Schild black holes. initial angular velocity Ω0, radial velocity a˙0,and coordinate We measure the quasilocal spin SAKV of each hole separation d0 were tuned to reducetheorbital eccentricity. in the initial data using the approximate-Killing-vector method summarized in Appendix A of Ref. [35], which is very similar to the prescription previously published BBH simulations begin with puncture initial data [17], by Cook and Whiting [43]. The dimensionless spin of which assumes that the initial spatial metric is confor- each hole χ is then related to SAKV by the formula mally flat (i.e. proportional to the metric of flat space). χ := SAKV/Mc2hr, where Mchr := pMi2rr+S2/4Mi2rr is Withthisassumption,3ofthe4constraintequationscan the Christodoulou mass, Mirr := pA/16π is the irre- be solved analytically using the solutions of Bowen and ducible mass, and A is the area of the horizon. (For a York [18, 19]; however, conformally flat initial data can- single Kerr black hole, Mchr reduces to the usual Kerr notdescribesingleorbinaryblackholesthatarebothin mass parameter.) equilibriumandpossesslinear[20]orangular[21,22]mo- To reduce eccentricity, we follow the iterative method mentum. Bowen-York puncture data can yield solutions of Ref. [44], which is an improvement of the earlier ofbinaryblackholeswithspinsaslargeasχ=0.984ini- method of Ref. [45]. For each iteration, we construct tially, but when such initial data are evolved, the holes an initial data set and evolve it for approximately 3 or- quickly relax to spins of about χ = 0.93 or less [23–25]. bits. Then, the initial angular and radial motion of the Several groups have evolved BBH puncture data with holes are adjusted to minimize oscillations in the orbital spins near but below this Bowen-York limit [9, 26, 27], frequency. Using this method, we reduce the orbital ec- with the simulationby Dain, Lousto,and Zlochower[28] centricity to approximately 10−3. coming the closest with spins of 0.967 at time t = 0 quickly falling to 0.924. ToreachspinsbeyondtheBowen-Yorklimit,onemust III. EVOLUTION begin with initial data that is conformally curved. Re- cently, Liu and collaborators [29] have constructed and We evolve our initial data using the Spectral Einstein evolved conformally curved initial data based on that of CodeSpEC[46]. BuildingonthemethodsofRef.[47]and Brandt and Seidel [30, 31] for a single black hole with the references therein, we have made several technical spinsashighasχ=0.99. Hannamandcollaborators[32] improvementstoourcodewhichbothenableustoevolve have constructed and evolved conformally curved BBH our χ = 0.95 initial data through merger and make our initial data [33, 34] for head-on mergers of black holes code more robust in general. Here we briefly summarize with spins as large as χ = 0.9. In Ref. [35], conformally someofthemostimportantimprovements;fulldetailsof curvedBBHdatawithspinsofχ=0.93wereconstructed these techniques will be described in a future paper. andevolvedthroughthefirst1.9orbitsofaninspiral,but We use a computational domain with the singularities no attempt was made to simulate the complete inspiral, insidethehorizonsexcised,andweuseatime-dependent merger, and ringdown. coordinate mapping to keep the excision boundaries in- Inthispaper,wedemonstratethatconformallycurved side the individual apparenthorizons as the horizons or- initial data is suitable for simulations with nearly ex- bit and slowly approach each other [48]. Our coordinate tremal spins by using it to compute the first inspiral, mapping also ensures that the excision surfaces’ shapes merger,andringdownoftwoblackholeswithspinslarger conform to those of the horizons which enclose them. than the Bowen-York limit. By surpassing this limit, One important ingredient of our improved binary-black- ourresultsopenthewayfornumericalexplorationofthe holeevolutionsisthatthecoordinatemappingisadjusted gravitationalwaveformsandnonlineardynamicsofblack adaptively throughoutthe evolution,whichis helpful be- holes that are nearly extremal. cause the horizons’ dynamics change from slow to fast during the simulation. Because we apply no boundary condition on the exci- II. INITIAL DATA sionsurfaces,thesesurfacesmustbepure-outflowbound- aries(i.e.,musthavenoincomingcharacteristicfields)in We evolve a low-eccentricity initial data set: a BBH order for the evolution to be well posed. A second im- where the holes have equal masses and equal spins of provement to our code is that we now can adjust the 3 Spin of individual horizon Spin of common horizon 0.4 -0.9485 0.4 5 0.2 -0.949 -0.93 -0.9495 N3.3 y 0 0 N4.3 N5.3 0.3 -0.95 zχ 0 1000 2000 3000 0.375 zχ -5 -0.2 -0.94 N3 B N4 A 0.37 0.2 -0.4 N5 D C -5 x0 5 3800 3900 0.4 0.365 35004000 -0.95 M0.2 0.1 0 1000 t / M2000 3000 3500 t / 4M000 h / 0 r -0.2 FIG.2. Coloronline. Leftpanel: Thez componentχz ofthe -0.40 1000 2000 3000 4000 dimensionlessquasilocalspinofoneindividualhorizonvstime t / M t. (Theindividualholes’ spinsareequalwithin numerical er- ror.) Right panel: spin of the common horizon vs. time and FIG. 3. Color online. The orbital trajectory of the centers the final spins predicted by the fitting formulae in Ref. [49] of the individual horizons and the individual and common (“A”),Ref.[1](“B”),Ref.[50](“C”),andRef.[9](“D”),and horizons at the end of the inspiral (top left) and the real (for “B”,“C”, and “D”) the error bars corresponding to the part of the ℓ = 2,m = 2 mode of the emitted gravitational fittingformulae’slisteduncertainties. (Notethatthehorizon- waveform h extracted at radius r = 405M (bottom). The tal positions of points “A”–“D” on the figure are arbitrary.) holes travel through about 12.5 orbits before merging. All Our results are shown for several resolutions (labeled Nx or data is from resolution N5.3. Nx.y, where x ∈ 3,4,5 and y ∈ 1,2,3 label the resolution used before and after merger, respectively). the next 11.8 orbits, until just before merger, when the magnitude of the spin of each hole drops sharply. This velocity of each excision surface to keep the character- result demonstrates that it is now possible to simulate istic fields outgoing there. For the χ = 0.95 simulation BBHs where the holes retain spins beyond the Bowen- considered here, this characteristic speed control is nec- Yorklimit throughoutthe inspiral;it alsoopens the way essary only during the last orbit before merger; earlier, for future explorations of the strong-field dynamics of it is sufficient to control the size of the excision surface merging,nearlyextremalholes—dynamics thatcan only using the method of Ref. [47]. be explored using numerical simulations. AthirdelementwhichwehaverecentlyaddedtoSpEC During the ringdown, the spin χ of the common hori- is spectral adaptive mesh refinement. During the evo- zonquicklyrelaxestoitsfinalvalueofχ=0.3757±0.0002 lution, we monitor the truncation error of each evolved (where the uncertainty is estimated as the difference be- field, the resolution requirements of the apparent hori- tween the highest and second-highest resolutions). This zons, and the local magnitude of constraint violation; is approximately consistent with but slightly largerthan to maintain a desired accuracy, we then add or remove the predictions obtained by extrapolating fitting formu- spectral basis functions as needed. In the simulation lae from simulations with lower initial spins of Ref. [49] presented in this paper, we use spectral adaptive mesh (χfit ≈ 0.371), Ref. [1] (χfit = 0.372±0.057), Ref. [50] refinement only during the final quarter orbit before (χfit =0.369±0.012),andRef. [9](χfit =0.366±0.002). merger. Throughouttheentiresimulation,wealsoadap- This result is a first step toward a better understanding tivelyadjusttheresolutionoftheapparenthorizonfinder oftherelationbetweenthepropertiesoftheremnanthole as the horizon becomes more distorted. andthoseofthemergingholeswhenthelatterarenearly extremal. Numerical simulations that directly measure thisrelation(insteadofextrapolatingfromlower-spinre- IV. RESULTS sults) will yield greater understanding of the properties of black holes produced from merging extremal holes. In Fig. 2, we plot the dimensionless quasilocal spin Figure 3 shows the individual horizontrajectoriesand χ measured on one individual horizon and also on the the real part of the (ℓ,m) = (2,2) spherical harmonic common horizon. From t = 0 to t = 50M, there is a mode ofthe emitted gravitationalwaveform. We extract sharp,numericallyresolveddropinthemagnitudeofthe wavesonaseriesofconcentricsphericalshells;the wave- dimensionless spin χ from 0.9498 to 0.9492. During the form shown was extracted on the outermost spherical remainderoftheinspiral,thespindrifts,withtheamount shell(atradiusr =405M). Accurategravitationalwave- of drift decreasing as resolution increases; at the highest forms obtained from this and future simulations with resolution (N5), the spin remains χ = 0.949 throughout spinsbeyondthe Bowen-Yorklimitwillbe usefulforcal- 4 ibrating analytic template banks for gravitational-wave ACKNOWLEDGMENTS searches. The Christodoulou mass of the final black hole is Mfinal/Mrelax = 0.9683 ± 0.0001, where Mrelax = We arepleasedto thank Nick Taylorfora gaugemod- 1.0003M is the sum ofthe massesofthe individualholes ification that allows us to use the nonsmooth maps of after the initial relaxation and where the uncertainties Ref. [47] throughout our evolutions and Larry Kidder, of Mfinal and Mrelax are estimated as the difference be- Robert Owen, Harald Pfeiffer, Saul Teukolsky, and Kip tween the highest and second-highest resolution. Under Thorneforhelpfuldiscussions. This workwassupported theassumptionthateachholehasaconstantmassMrelax in part by grants from the Sherman Fairchild Founda- throughout the inspiral (which holds within O(cid:0)10−5(cid:1) in tion to Caltech and Cornell and from the Brinson Foun- our simulation after the holes have relaxed), the quan- dation to Caltech; by NSF Grants No. PHY-0601459 tity1−Mfinal/Mrelax representsthefractionoftheinitial and No. PHY-1005655 at Caltech; by NASA Grant No. mass that would have been radiated from t = −∞ to NNX09AF97G at Caltech; by NSF Grants No. PHY- t=+∞,hadoursimulationcontainedtheentireinspiral 0969111andNo. PHY-1005426atCornell;andbyNASA instead of just the final 12.5 orbits. Grant No. NNX09AF96G at Cornell. The numerical Our results demonstrate for the first time that it is computationspresentedinthispaperwereperformedpri- possibletosimulatemergingblackholeswithspinslarger marily on the Caltech compute cluster zwicky, which than the Bowen-York limit of χ=0.93, the highest spin was cofunded by the Sherman Fairchild Foundation. previously obtainable. Because astrophysicalblack holes SomecomputationswerealsoperformedontheGPC su- may be nearly extremal, these simulations have astro- percomputer at the SciNet HPC Consortium; SciNet is physical as well as physical relevance. In particular, this funded by: the Canada Foundation for Innovation un- work opens the way to use numerical simulations to ex- der the auspices of Compute Canada; the Government plore the strong-field dynamics of merging, nearly ex- of Ontario; Ontario Research Fund - Research Excel- tremalblackholes,to gaina better understanding ofthe lence; and the University of Toronto. 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