Modeling Gravitational Recoil Using Numerical Relativity
MModeling Gravitational Recoil Using NumericalRelativity
Yosef Zlochower, Manuela Campanelli, Carlos O. Lousto
Center for Computational Relativity and Gravitation, School of MathematicalSciences, Rochester Institute of Technology, Rochester, New York 14623, USAE-mail:
Abstract.
We review the developments in modeling gravitational recoil from mergingblack-hole binaries and introduce a new set of 20 simulations to test our previouslyproposed empirical formula for the recoil. The configurations are chosen to representgeneric binaries with unequal masses and precessing spins. Results of these simulationsindicate that the recoil formula is accurate to within a few km/s in the similar mass-ratio regime for the out-of-plane recoil.PACS numbers: 04.25.Dm, 04.25.Nx, 04.30.Db, 04.70.Bw
Submitted to:
Class. Quantum Grav. a r X i v : . [ g r- q c ] N ov odeling Gravitational Recoil
1. Introduction
The field of Numerical Relativity (NR) has progressed at a remarkable pace since thebreakthroughs of 2005 [1, 2, 3] with the first successful fully non-linear dynamicalnumerical simulation of the inspiral, merger, and ringdown of an orbiting black-holebinary (BHB) system. In particular, the ‘moving-punctures’ approach, developedindependently by the NR groups at NASA/GSFC and at RIT, has now become themost widely used method in the field and was successfully applied to evolve genericBHBs. This approach regularizes a singular term in space-time metric and allows theblack holes (BHs) to move across the computational domain. Previous methods usedspecial coordinate conditions that kept the BHs fixed in space, which introduced severecoordinate distortions that caused orbiting-BHB simulations to crash. Recently, thegeneralized harmonic approach method, first developed by Pretorius [1], has also beensuccessfully applied to accurately evolve generic BHBs for tens of orbits with the use ofpseudospectral codes [4, 5].Since then, BHB physics has rapidly matured into a critical tool for gravitationalwave (GW) data analysis and astrophysics. Recent developments include: studies of theorbital dynamics of spinning BHBs [6, 7, 8, 9, 10, 11, 12], calculations of recoil velocitiesfrom the merger of unequal mass BHBs [13, 14, 15], and the surprising discovery thatvery large recoils can be acquired by the remnant of the merger of two spinning BHs[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 9, 29, 30, 31], empirical modelsrelating the final mass and spin of the remnant with the spins of the individual BHs[32, 33, 34, 35, 36, 37, 38, 39], comparisons of waveforms and orbital dynamics of BHBinspirals with post-Newtonian (PN) predictions [40, 41, 42, 43, 44, 45, 46, 47], andsimulations reaching mass ratios [48] q = 1 / The first in-depth modeling of the recoil from the merger of non-spinning asymmetricBHBs was done in Ref. [15], where it was shown that the maximum recoil is limited to ≈
175 km s − . Soon after, other groups showed that the maximum recoil for spinningbinaries is much larger. In Ref. [16] and [30] (which were released within a few days ofeach other), it was shown that the maximum recoil for a spinning binary with one BHspin aligned with the orbital angular momentum and other anti-aligned id 475 km s − .Within a day of the initial release of the preprint for Ref. [16], our group released apreprint [17] for more generic spin and mass configurations, showing that the recoil isconsiderably larger if the spins are anti-aligned and pointing in the orbital plane. In [17]we measured recoil velocities beyond the maximum predicted for the configurationsin [16, 30].An initial analysis of the results in [17] indicated that the maximum velocityexceeded 1300 km s − for spins lying in the orbital plane. This initial estimate wasbased on two ‘generic’ configurations. A subsequent analysis that incorporated theangular dependence of the projection of the spin on the orbital plane, showed that odeling Gravitational Recoil ∼ − [49]. Based on our conclusions in the preprintof [17], the group at Jena performed a set of two simulations in this ‘maximum-kick’configuration and measured recoils of around 2600 km s − . Our modeling showed thatthis recoil velocity actually depends sinusoidally on the angle that the spins make withthe infall direction. By evolving a set of configurations with spins at different initialangles in the orbital plane, we found that the recoil reaches a maximum of ∼ − [18, 63]. In Ref. [49], as part of our analysis, we proposed an empirical formula for therecoil based on post-Newtonian expressions for the radiated linear momentum. Theformula correctly predicts the sinusoidal dependence of the ‘maximum-kick’ recoil.In Ref. [26] the recoil for unequal-mass, spinning binaries, with total spin equal tozero and individual spins lying in the initial orbital plane, was measured. These S = 0configurations are preserved under numerical evolution and lead to minimal precessionof the orbital pane. Interestingly, they found that the recoil scales with the cube of themass ratio q , rather than the expected q seen in post-Newtonian expressions for therecoil. In a subsequent study [19], our group found that the recoil scales as q for themore astrophysically important precessing case. In [17] we introduced an empirical formula for the recoil, augmented in [39] which hasthe form. (cid:126)V recoil ( q, (cid:126)α ) = v m ˆ e + v ⊥ (cos ξ ˆ e + sin ξ ˆ e ) + v (cid:107) ˆ n (cid:107) ,v m = A η (1 − q )(1 + q ) [1 + B η ] ,v ⊥ = H η (1 + q ) (cid:20) (1 + B H η ) ( α (cid:107) − qα (cid:107) ) + H S (1 − q )(1 + q ) ( α (cid:107) + q α (cid:107) ) (cid:21) ,v (cid:107) = K η (1 + q ) (cid:20) (1 + B K η ) (cid:12)(cid:12) α ⊥ − qα ⊥ (cid:12)(cid:12) cos(Θ ∆ − Θ )+ K S (1 − q )(1 + q ) (cid:12)(cid:12) α ⊥ + q α ⊥ (cid:12)(cid:12) cos(Θ S − Θ ) (cid:21) , (1)where η = q/ (1 + q ) , the index ⊥ and (cid:107) refer to perpendicular and parallel to theorbital angular momentum respectively, ˆ e , ˆ e are orthogonal unit vectors in the orbitalplane, and ξ measures the angle between the unequal mass and spin contribution tothe recoil velocity in the orbital plane. The constants H S and K S can be determinedfrom new generic BHB simulations as the data become available. The angles, Θ ∆ andΘ S , are the angles between the in-plane component of (cid:126) ∆ = M ( (cid:126)S /m − (cid:126)S /m ) or (cid:126)S = (cid:126)S + (cid:126)S and the infall direction at merger. Phases Θ and Θ depend on theinitial separation of the holes for quasicircular orbits. A crucial observation is thatthe dominant contribution to the recoil is generated near the time of formation of thecommon horizon of the merging BHs (See, for instance Fig. 6 in [50]). The formula(1) above describing the recoil applies at this moment (or averaged coefficients aroundthis maximum generation of recoil), and has proven to represent the distribution of odeling Gravitational Recoil A = 1 . × km s − , B = − . H = (6 . ± . × km s − , K = (6 . ± . × km s − , and ξ ∼ ◦ .Note that v (cid:107) is maximized when q = 1 (i.e. equal masses), the in-plane spins aremaximal ( α = α = 1), and the two spins are anti-aligned.In this paper, we use a new set of simulations of precessing binaries to test therecoil formula and obtain a new estimate for K S .
2. Numerical Techniques
To compute the numerical initial data, we use the puncture approach [52] along withthe
TwoPunctures [53] thorn. In this approach the 3-metric on the initial slice hasthe form γ ab = ( ψ BL + u ) δ ab , where ψ BL is the Brill-Lindquist conformal factor, δ ab is the Euclidean metric, and u is (at least) C on the punctures. The Brill-Lindquistconformal factor is given by ψ BL = 1 + (cid:80) ni =1 m pi / (2 | (cid:126)r − (cid:126)r i | ) , where n is the total numberof ‘punctures’, m pi is the mass parameter of puncture i ( m pi is not the horizon massassociated with puncture i ), and (cid:126)r i is the coordinate location of puncture i . We evolvethese BHB data-sets using the LazEv [54] implementation of the moving punctureapproach [2, 3] with the conformal factor W = √ χ = exp( − φ ) suggested by [11] Forthe runs presented here we use centered, eighth-order finite differencing in space [55]and an RK4 time integrator (note that we do not upwind the advection terms).We use the Carpet [56] mesh refinement driver to provide a ‘moving boxes’ stylemesh refinement. In this approach refined grids of fixed size are arranged about thecoordinate centers of both holes. The Carpet code then moves these fine grids aboutthe computational domain by following the trajectories of the two BHs.We use AHFinderDirect [57] to locate apparent horizons. We measure themagnitude of the horizon spin using the Isolated Horizon algorithm detailed in [58].This algorithm is based on finding an approximate rotational Killing vector (i.e. anapproximate rotational symmetry) on the horizon ϕ a . Given this approximate Killingvector ϕ a , the spin magnitude is S [ ϕ ] = 18 π (cid:90) AH ( ϕ a R b K ab ) d V, (2)where K ab is the extrinsic curvature of the 3D-slice, d V is the natural volume elementintrinsic to the horizon, and R a is the outward pointing unit vector normal to the horizonon the 3D-slice. We measure the direction of the spin by finding the coordinate linejoining the poles of this Killing vector field using the technique introduced in [8]. Ouralgorithm for finding the poles of the Killing vector field has an accuracy of ∼ ◦ (see [8]for details). Note that once we have the horizon spin, we can calculate the horizon massvia the Christodoulou formula m H = (cid:112) m + S / (4 m ) , where m irr = (cid:112) A/ (16 π ) and A is the surface area of the horizon. We measure radiated energy, linear momentum, odeling Gravitational Recoil ψ , using the formulae provided in Refs. [59, 60].However, rather than using the full ψ , we decompose it into (cid:96) and m modes andsolve for the radiated linear momentum, dropping terms with (cid:96) ≥
5. The formulae inRefs. [59, 60] are valid at r = ∞ . We obtain accurate values for these quantities bysolving for them on spheres of finite radius (typically r/M = 50 , , · · · , l = 1 /r , and extrapolating to l = 0 [3, 61, 44, 62].Each quantity Q has the radial dependence Q = Q + lQ + O ( l ), where Q is theasymptotic value (the O ( l ) error arises from the O ( l ) error in r ψ ). We perform bothlinear and quadratic fits of Q versus l , and take Q from the quadratic fit as the finalvalue with the differences between the linear and extrapolated Q as a measure of theerror in the extrapolations. We found that extrapolating the waveform itself to r = ∞ introduced phase errors due to uncertainties in the areal radius of the observers, as wellas numerical noise. Thus when comparing Perturbative to numerical waveforms, we usethe waveform extracted at r = 100 M .In order to model the recoil as a function of the orientation and magnitudes of thespins, we use the techniques introduced in [19] to locate the approximate orbital planeat merger and 3D rotation such that infall directions are the same for each simulation.Briefly, this technique uses three points on the trajectories, given by fiducial choices ofthe BH separations, to define the orbital plane and preferred orientation.
3. Simulations
Config x /M x /M P/M m p m p S x /M S y /M Q33TH000 4.882446 -1.607923 0.101163 0.171173 0.723529 0 0.045983RQ33TH000 4.885558 -1.609045 0.101153 0.171170 0.723523 0 0.045982Q50TH000 4.360493 -2.163334 0.119252 0.230648 0.618813 0 0.082098
Table 1.
Initial data parameters for the quasi-circular configurations. The puncturesare located at (cid:126)r = ( x , ,
0) and (cid:126)r = ( x , , P = ± (0 , P, (cid:126)S = ( S x , S y , (cid:126)S = − q (cid:126)S , mass parameters m p . The configuration are denoted byQXXXTHYYY where XXX gives the mass ratio (0.33, 0.50) and YYY gives the anglein degrees between the initial spin direction and the y -axis. In all cases the initialorbital period is M ω = 0 .
05 and the spin of the smaller BH is α = 0 .
72. Initial dataparameters for the Q50TH000 and Q33TH000 configurations are given. The remainingconfigurations are obtained by rotating the spin directions, keeping all other parametersthe same. For the RQ33THxxx configurations, (cid:126)S is rotated by 90 ◦ with respect tothe corresponding Q33THxxx configuration. We evolve a set of configuration that initially have ∆ = 0, as well as a set ofconfiguration with one spin rotated by 90 ◦ for mass ratios q = 1 / q = 1 / odeling Gravitational Recoil
4. Results
In Table 2 we summarize the results of the simulations. The table shows the radiatedenergy and recoil (prior to any rotation). Note that these results can be used inadditional fits of the final remnant BH formulae for the mass, spin, and recoil velocityof the remnant, as was done in Ref. [39] using the then currently available results in theliterature.Config 100( δE/M ) V x V y V z Q50TH000 2 . ± .
018 76 . ± .
47 234 . ± .
12 167 . ± . . ± . − . ± .
03 10 . ± . − . ± . . ± .
016 189 . ± .
77 97 . ± . − . ± . . ± .
017 306 . ± .
23 319 . ± . − . ± . . ± .
018 78 . ± .
66 235 . ± . − . ± . . ± .
017 43 . ± .
32 108 . ± .
94 131 . ± . . ± .
017 295 . ± .
26 341 . ± .
78 741 . ± . . ± .
012 119 . ± .
36 212 . ± .
78 144 . . . ± .
012 67 . ± .
59 132 . ± . − . ± . . ± .
012 02 . ± .
87 132 . ± . − . ± . . ± .
012 210 . ± .
34 235 . ± . − . ± . . ± .
012 119 . ± .
36 212 . ± . − . ± . . ± .
012 73 . ± .
11 150 . ± .
73 59 . ± . . ± .
012 73 . ± .
79 127 . ± .
58 272 . ± . . ± .
013 210 . ± .
45 249 . ± .
87 489 . ± . . ± .
013 229 . ± .
07 132 . ± .
81 539 . ± . . ± .
012 45 . ± .
11 162 . ± . − . ± . . ± .
012 106 . ± .
16 82 . ± . − . ± . . ± .
012 182 . ± .
67 188 . ± . − . ± . ± .
012 121 . ± .
42 79 . ± .
45 462 . ± . Table 2.
The radiated energy and recoil velocities for each configuration. Note thatsome of the error estimates, which are based on the differences between a linear andquadratic extrapolation in l = 1 /r of the observer location, are very small. Thisindicates that the differences between the extrapolation can underestimate the trueerror. All quantities are given in the coordinate system used by the code (i.e. theuntransformed system). Table 3 gives the components of the radiated angular momentum in the original x, y, z frame (that of the initial data) using the Cartesian decomposition as in Ref. [60].In Table 4 we give the recoil velocities in a frame rotated such that the orbital odeling Gravitational Recoil δJ x δJ y δJ z Q50TH000 0 . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± . − . ± . − . ± . . ± . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± . − . ± . − . ± . . ± . − . ± . − . ± . . ± . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± . . ± . − . ± . . ± . Table 3.
The radiated angular momentum. Note that some of the error estimates,which are based on the differences between a linear and quadratic extrapolation in l = 1 /r of the observer location, are very small. This indicates that the differencesbetween the extrapolation can underestimate the true error. All quantities are givenin the coordinate system used by the code (i.e. the untransformed system). plane coincides with the xy plane and the infall direction in this plane is fixed. In orderto fit the data, we chose K = 5 . × based on previous work [17, 19, 63] and thenfit the Q
50 simulations for K S , Θ , and Θ . We found K s = − . ± .
8. While, K and K S are fixed, Θ and Θ depend on the configuration. To obtain these angles wefit the the recoil formula for the Q33Txxx and RQ33Txxx configurations separately.We then compared the predicted recoil for each configuration with the measured recoil.The results are summarized in Table 4. Note that the errors are typically less than6 km s − . The largest relative errors are ∼ . ∼ ∼ − . This can be traced to threemain sources the errors. The uncertainty in how the the recoils produced by unequalmasses and out-of plane spins contribute to the total in-plane recoil, the error in themeasurement in the out-of-plane spine due to errors in measuring the orientation of the odeling Gravitational Recoil V x V y V z V z (predict) errorQ50TH000 17.47 -164.42 -248.44 -245.851 2.591Q50TH045 -71.85 -100.95 196.47 190.25 -6.222Q50TH090 44.52 -180.82 553.49 551.759 -1.73Q50TH130 31.47 -242.82 846.74 848.283 1.540Q50TH180 18.34 -165.91 248.56 245.987 -2.575Q50TH210 26.93 -153.45 -81.52 -84.3437 -2.823Q50TH315 28.58 -242.02 -832.71 -834.713 -1.999Q33TH000 117.61 -157.99 -203.81 -208.612 -3.311Q33TH045 110.76 -132.11 102.02 105.811 1.937Q33TH090 122.60 -120.81 334.67 337.689 -0.893Q33TH130 144.38 -155.39 549.60 553.834 -2.540Q33TH180 117.60 -157.99 203.63 208.286 3.140Q33TH210 109.72 -138.43 -18.17 -18.7138 0.501Q33TH260 118.62 -122.29 -258.97 -268.242 -6.040Q33TH315 144.43 -160.93 -546.70 -551.418 2.112RQ33TH000 168.76 -118.44 -564.09 -569.559 -5.469RQ33TH090 116.46 -133.88 49.68 49.8794 0.203RQ33TH130 121.09 -92.33 421.93 413.897 -8.033RQ33TH210 155.20 -147.31 391.98 386.733 -5.246RQ33TH315 125.67 -89.20 -460.12 -464.877 -4.757 Table 4.
The recoil velocities after rotation and a comparison of the out-of-planecomponent of the recoil (in coordinates aligned with the orbital plane at merger)with the predictions of the empirical formula with coefficients K = 5 . × and K S = − .
5. Conclusion
We report here on a new set of generic simulations, with no symmetries and differentchoices of the binary’s mass ratio and spin direction and magnitudes. We computethe radiation emitted by this binaries, and in particular focus on the magnitude anddirection of the final recoil of the merged BH. While this set of runs can be used forfitting to additional subleading terms in the empirical formulae for the remnant BHrecoil, we choose to use them to test the previously fitted values, extended to includeeffects linear in the spin, as accurate determinations of these parameters will requiremany more simulations. In this paper we showed that the empirical recoil formulaprovides accurate predictions for the recoil velocity from BHB mergers for the new sets ofconfigurations. These configurations have fewer symmetries than previous comparisons odeling Gravitational Recoil V z measured recoilpredicted recoil Figure 1.
The predicted and measured out-of-plane recoil for each configuration inTable 4. The configurations are labeled 1-20 starting from Q50TH000 as ordered inTable 4. and can be considered of generic nature regarding spin orientations and intermediatemass ratios and spin magnitudes. This shows that our empirical formula (1) can beused as a first approximation for astrophysical studies of statistical nature, as we did inRef. [64] and also used for realistic recoil magnitudes and direction when modeling theobservational effects of recoiling BHs in a gaseous environment such as accretion disks[65].
Acknowledgments:
We gratefully acknowledge NSF for financial support from grantsPHY-0722315, PHY-0653303, PHY-0714388, PHY-0722703, DMS-0820923, PHY-0969855, PHY-0903782, PHY-0929114 and CDI-1028087; and NASA for financialsupport from grants NASA 07-ATFP07-0158 and HST-AR-11763. Computationalresources were provided by Ranger cluster at TACC (Teragrid allocation TG-PHY060027N) and by NewHorizons at RIT.
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