Black Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris
BBlack Hole - Neutron Star Binary Mergers: TheImprint of Tidal Deformations and Debris
Bhavesh Khamesra a , Miguel Gracia-Linares b , Pablo Laguna b a Center for Relativistic Astrophysics and School of Physics, Georgia Institute ofTechnology, Atlanta, GA 30332, U.S.A. b Center for Gravitational Physics, Department of Physics, The University of Texas atAustin, Austin, TX 78712, U.S.A.E-mail: [email protected], [email protected],[email protected]
Abstract.
The increase in the sensitivity of gravitational wave interferometers willbring additional detections of binary black hole and double neutron star mergers. Itwill also very likely add many merger events of black hole - neutron star binaries.Distinguishing mixed binaries from binary black holes mergers for high mass ratioscould be challenging because in this situation the neutron star coalesces with theblack hole without experiencing significant disruption. To investigate the transitionof mixed binary mergers into those behaving more like binary black hole coalescences,we present results from merger simulations for different mass ratios. We show howthe degree of deformation and disruption of the neutron star impacts the inspiral andmerger dynamics, the properties of the final black hole, the accretion disk formed fromthe circularization of the tidal debris, the gravitational waves, and the strain spectrumand mismatches. The results also show the effectiveness of the initial data methodthat generalizes the Bowen-York initial data for black hole punctures to the case ofbinaries with neutron star companions.
1. Introduction:
The Gravitational-Wave Catalogue by LIGO and Virgo has been recently updated tobring the total number of detections to 50 [1], with 36 of the events confirmed binaryblack hole (BBH) mergers and two double neutron stars (NSs) mergers (GW170817 [2]and GW190425 [3]). Although not fully confirmed, the remaining two detections(GW190814 [4] and GW190426 152155 [1]) suggest that the gravitational waves (GWs)detected were produced from mergers of black hole - neutron star (BHNS) binaries. AsLIGO and Virgo reach design sensitivity, we will have more GW detections from BHNSbinaries. Characterising these events calls for numerical simulations that are not onlymore accurate but that include the relevant micro-physics.Numerical studies of BHNSs have considered different aspects of the merger. Somehave focused on the formation of the accretion disk from the tidal debris as well asthe relativistic jets emanating from the remnant black hole (BH). Specifically, the a r X i v : . [ a s t r o - ph . H E ] J a n lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris ≥ g/cm [28]. On the otherhand, for systems with high mass ratio and low BH spin, the NS barely suffers anydisruption before reaching ISCO and can be swallowed almost completely by the BHhardly leaving any trace of matter to generate detectable electromagnetic signatures. Inthe absence of any significant disruption, the BHNS systems behaves as a BBH, withalmost identical GW signatures [29].The work in this paper has two main objectives. One is to test the effectivenessof the initial data method introduced in Ref. [30]. The method generalizes the Bowen-York [31] approach for initial data with BHs modeled as punctures to the case of NSs.The second is to provide further insights on the transition of a BHNS into a BBH-likebehavior as the effects from the disruption of the NS change with the mass ratio of thebinary. Our results show that for low mass ratio cases, a considerable amount of energyand angular momenta, that otherwise would have been radiated in GWs, gets trappedin the accretion disk and redistributed as the BH accretes the material. The tidal debrisalso affects the ringing of the final BH when compared with the BBH case. For all thecases considered, the BHNS binary merges earlier than the corresponding BBH. Thisis due to the tidal deformation that the NS experiences. The deformation introduces acorrection to the potential that increases the orbital velocity and thus the emission ofGWs [28]. Our results have limitations since we model the NS as a polytrope and donot include magnetic fields or neutrino transport. At the same time, we demonstratethat the initial data method has promising feature, such as simplicity of implementationand generalization to realistic equations of state.The paper is organized as follows: Section 2 provides a summary of the initial datamethod developed in [30]. Section 3 details the parameters of the initial BHNS andBBH configurations. The section also includes the setup of the numerical simulationsand convergence tests. Results are presented in Section 4 organized by i) inspiral andmerger dynamics, ii) the final BH, iii) accretion disk, iv) GWs, and v) spectrum andmismatches. Conclusions are given in Section 5. We use geometrical units in which G = c = 1 and express all dimensions in terms of M , the total initial mass of the binarysystem. When necessary, we will also use physical units (SI units). Indices with Latinletters from the beginning of the alphabet denote space-time dimensions and from themiddle of the alphabet spacial dimensions. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris
2. Initial Data
We will briefly review the approach we introduced in Ref. [30] to construct initial data forbinaries with NS companions. Under the 3+1 decomposition of the Einstein equations,initial data consist of the spatial metric γ ij of the constant time initial hypersurface, theextrinsic curvature K ij in this hypersurface, and the projections ρ H ≡ n a n b T ab (1) S i ≡ − γ ij n b T jb (2)of the stress-energy tensor T ab , with n a the unit time-like normal to the hypersurface.For the present work we will only consider perfect fluids. Thus, T ab = ( ρ + p ) u a u b + pg ab , (3)with ρ the energy density, p the pressure, u a the four velocity of the fluid, and g ab = γ ab − n a n b the space-time metric. With this form for T ab , ρ H = ( ρ + p ) W − p (4) S i = ( ρ + p ) W u i , (5)where W = − n a u a is the Lorentz factor, which can be rewritten as W = 12 (cid:32) (cid:115) S i S i ( ρ + p ) (cid:33) . (6)The initial data { γ ij , K ij , ρ H , S i } must satisfy the constraints R + K − K ij K ij = 16 πρ H (7) ∇ j (cid:0) K ij − γ ij K (cid:1) = 8 πS i , (8)namely the Hamiltonian and Momentum constraints, respectively. Here, R is the Ricciscalar, and ∇ j is the covariant derivative associated with γ ij .We solve Eqs.(7) and (8) following the conformal-transverse-traceless (CTT)approach pioneered by Lichnerowicz [32], York and collaborators [33]. The central ideaof this approach is to apply the following transformations to isolate the four quantitiesobtained by solving the constraints: γ ij = ψ ˜ γ ij (9) K ij = A ij + 13 γ ij K (10) A ij = ψ − (cid:16) ˜ A ijT T + ˜ A ijL (cid:17) , (11) (cid:101) ∇ j ˜ A ijT T = 0 (12)˜ A ijL = 2 (cid:101) ∇ ( i V j ) −
23 ˜ γ ij (cid:101) ∇ k V k (13)˜ ρ H = ρ H ψ (14)˜ S i = S i ψ , (15) lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris ψ is the conformal factor. The last two transformations imply that ˜ ρ = ρ ψ ,˜ p = p ψ , ˜ u i = u i ψ and (cid:102) W = W .We adopt also the common choices of conformal flatness (˜ γ ij = η ij ), maximal slicing( K = 0), and ˜ A TT ij = 0. With these choices and the CTT transformations above, theHamiltonian and momentum constraints take the following form:˜∆ ψ + 18 ψ − ˜ A ij ˜ A ij = − πψ − ˜ ρ H (16) (cid:101) ∇ j ˜ A ij = 8 π ˜ S i . (17)Bowen and York [31] found point-source solutions to the source-free momentumconstraint (17) that can be used to represent BHs with linear momentum P i and spin J i . The solutions read:˜ A ij = 32 r (cid:2) P ( i l j ) − ( η ij − l i l j ) P k l k (cid:3) (18)˜ A ij = 6 r l ( i (cid:15) j ) kl J k l l (19)where l i = x i /r , a unit radial vector.In Ref. [30], we followed Bowen’s approach [34] to construct solutions to themomentum constraint that represent NSs. The solutions assume spherically symmetricsources and are given by˜ A ij = 3 Q r (cid:2) P ( i l j ) − ( η ij − l i l j ) P k l k (cid:3) + 3 Cr (cid:2) P ( i l j ) + ( η ij − l i l j ) P k l k (cid:3) (20)˜ A ij = 6 Nr l ( i (cid:15) j ) kl J k l l , (21)where Q = 4 π (cid:90) r σ ¯ r d ¯ r (22) C = 2 π (cid:90) r σ ¯ r d ¯ r (23) N = 8 π (cid:90) r χ ¯ r d ¯ r . (24)The source functions σ and χ are radial functions with compact support r ≤ R and aresuch that ˜ S i = P i σ (25)˜ S i = (cid:15) ijk J j x k χ . (26)Outside the sources, Q = N = 1 and C = 0; thus, the extrinsic curvatures (20) and (21)reduce to those of point sources, i.e. (18) and (19) respectively.Since ˜ S i = ( ˜ ρ + ˜ p ) W ˜ u i , we set σ = ( ˜ ρ + ˜ p ) / K (27) χ = ( ˜ ρ + ˜ p ) / N (28) lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris K and N obtained from K = 4 π (cid:90) R ( ˜ ρ + ˜ p ) r dr (29) N = 8 π (cid:90) R ( ˜ ρ + ˜ p ) r dr . (30)Given these solutions for the extrinsic curvature, we solve the Hamiltonian constraint(16), assuming that the conformal factor has the form ψ = 1 + m p / (2 r ) + u where m p isthe bare or puncture mass of the BH. To solve (16), we used a modified version of the TwoPunctures code [35] which handles the source ˜ ρ H .The method to construct BHNS initial data in Ref. [30] for a BH with irreduciblemass M h and NS with mass M ∗ follows similar steps to that for BBHs initial data withpunctures. That is, one selects the target values for M ∗ and the mass ratio q = M h /M ∗ .For BBH systems, one usually chooses instead of M ∗ the total mass M of the binary.Next, one carries out iterations solving the Hamiltonian constraint until the targetvalues for q and M ∗ are obtained. After each Hamiltonian constraint solve iteration,one computes M h from the irreducible mass of the BH. The challenge is in finding anappropriate definition for the mass M ∗ of the NS in the binary. Options are the ADMmass M A or rest mass M of the NS in isolation, which in isotropic coordinates read M A = 2 π (cid:90) R ρ ψ r dr (31) M = 4 π (cid:90) R ρ ψ r dr (32)respectively, with ρ the rest-mass density. The approach we suggested in Ref. [30] isto compute the mass after each Hamiltonian constraint solve iteration from M ( n ) ∗ = ξ ( n − M ( n )0 where ξ ( n − = M ( n − A / M ( n − , namely the ratio of the ADM and rest massof the star in isolation. Here M = (cid:90) ρ W √ γd x = (cid:90) ˜ ρ W ψ − d x (33)is the rest mass of the NS after each Hamiltonian constraint solve. For n = 1, M (1) ∗ = M (1) A ; thus, ξ (0) = 1. We have found that for the simulations we have considered, ξ ≈ .
93, with variations less than 1 % throughout the iteration procedure; thus, ourmethod is close to those in which the value of the rest mass of the NS is the target.
3. Initial Parameters, Numerical Setup, and Convergence Tests
We study mixed binaries with mass ratio q = 2 , P = κ ρ Γ0 equation of state. In all cases,we set M ∗ = 1 . M (cid:12) , κ = 93 . M (cid:12) , Γ = 2, and coordinate separation 9 M , where M = M h + M ∗ . The momenta P i for each compact object in the binary is obtained bysolving the 3 . lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q M /M (cid:12) M h /M (cid:12) Ω M ¯ m p ¯ M ¯ M A ¯ ρ c C Q2 2 1.456 2.7 0.0319 0.2733 0.1549 0.1436 0.1381 0.1529Q3 3 1.456 4.05 0.0318 0.4123 0.1553 0.1439 0.1391 0.1536Q5 5 1.457 6.75 0.0318 0.6907 0.1557 0.1443 0.1401 0.1543Table 1: Initial configuration parameters: q = M h /M ∗ binary mass ratio, M isthe rest mass of the NS, M h irreducible mass of the BH, Ω M orbital frequency,¯ m p = m p κ / bare or puncture mass of the BH, ¯ M = M /κ / rest mass ofthe NS in isolation, ¯ M A = M A /κ / ADM mass of the NS in isolation, ¯ ρ c = ρ c κ ,central density of the isolated NS, and C = M ∗ /R ∗ compactness of the isolated NS.For all cases, κ = 93 . M (cid:12) , Γ = 2, coordinate separation 9 M , and M ∗ = 1 . M (cid:12) .at separation where the numerical relativity simulation begins. Table 1 shows the initialparameters for the simulations at the end of the construction of the initial data. Ourconfigurations closely mimic models M . M . M .
145 in Ref. [7], models B and A A and D in Ref. [14].We use the MAYA code [36, 37, 38] for the simulations; the code is our local versionof the
Einstein Toolkit code [39]. It solves the BSSN [40, 41] form of the Einsteinevolution equations and follows the implementation in the
Whisky code [42, 43, 44] forthe hydrodynamical evolution equations. We use the Marquina solver [45] to handlethe Riemann problem during flux computation and the piece-wise parabolic method[46] for reconstruction of primitive variables. The BH apparent horizon is found usingthe
AHFinderDirect code [47]. We use two methods to track the NS. One methodtracks the maximum density within the star. The other tracks the star using the
VolumeIntegrals thorn in the
Einstein Toolkit [39]. The properties of the BH,mass, spins and multipole moments, are computed using the
QuasiLocalMeasures thorn[48] based on the dynamical horizons framework [49]. The GW strain is computed fromthe Weyl scalar Ψ [48, 50, 51]. To compute the radiated quantities, we follow themethod developed in [52]. The gauge choice for the evolutions is the moving puncturegauge [53, 54].We use the moving box mesh refinement approach as implemented by Carpet [55].The starting point in setting the grid structure and number of refinement levels is thenumber of points needed to resolve the BH and the NS. For the results in this work,we ensure that at the finest level both, the BH and the NS, are completely enclosedby a mesh with at least 100 points across. This translates to a grid-spacing of ∼ ∼ M , which is a suitable resolution for GW extraction.For the mass ratios in this study, the end result is 9 levels of refinement from the BHup to the coarsest and 8 for the NS.To test convergence, we carried out three simulations with q = 2 at initial coordinate lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris . Witha convergence rate of k and refinement actor θ , (Medium - Low) = θ k (High- Medium). The figure shows the left and right hand side of this equation for θ = 1 . k = 2 . M . The resolutions at the finest mesh covering the NS are M/
16 (low), M/
24 (medium), and M/
36 (high). These correspond to resolutions 372 (low), 248(medium) and 166 (high) meters, respectively. That is, there is a factor of θ = 1 . . With a convergence rate of k and refinement factor θ , (Medium - Low) = θ k (High - Medium). The figure shows theleft and right hand side of this expression for θ = 1 . k = 2 .
37. The disagreementin the ringdown phase is because at the lowest resolution the matter accretion is under-resolved; an effect also observed by [7]. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris
4. Results
During the late inspiral stage of a BHNS binary, the NS will face a constant battlebetween the tidal forces from the BH and its self gravity. Depending on the mass ofthe BH, this could lead to the complete disruption of the star before it gets swallowedby the BH. The tidal forces by the NS could also inflict deformations in a companion,such as in a double NS binary merger. However, although not generally accepted [56],there is strong evidence that BHs are immune to tidal deformations [57]. Thus, thereare potentially fundamental differences between a BBH and a BHNS.For compact object binaries, the luminosity of gravitational radiation and the rateof change of radiated angular momenta depend on the mass ratio as q / (1 + q ) [32]. Asa consequence, the higher the mass ratio, the longer it takes for the binary to merge. InFigure 2, we show the evolution of the coordinate separation of the binary for each of theBHNS systems described in Table 1 and its BBH counterpart. The delay as a functionof q for both, the BHNS binaries and BBHs, is evident in this figure. For a given q ,we also see that the coordinate separation of the BHNS binary decreases faster than itscorresponding BBH. For the q = 5 case, the BBH and BHNS follow each other up toa separation ∼ . M . For q = 3, the binaries diverge a little earlier at a separationof approximate 8 M . The earliest deviation occurs for the q = 2 case, approximately lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris M from the start of the simulation. That is, as the mass ratio increases,the BHNS binaries resemble longer a BBH. The difference in binary separation betweenBBH and BHNS systems grow stronger as the merger is approached.Regarding the time when compact objects merge, for BBH, it is marked by thesudden formation of a common apparent horizon. The apparent horizon appears a few M s before the gravitational radiation reaches peak luminosity. For BHNS binaries, wedo not have the formation of a common apparent horizon since the only horizon isthe one from the single BH in the binary. Therefore, when making comparisons nearcoalescence, we will focus on the time when the gravitational radiation reaches peakluminosity (corresponding to the peak of | Ψ | ).In Table 2, T mx denotes the time to peak luminosity and ∆ mx the final BHoffset at T mx . Notice that the BHNS binaries reach peak luminosity earlier than theircorresponding BBH. We will address the reasons for this difference when we discuss theGWs emitted by the binaries. An interesting aspect to point out is that this differencedoes not decrease monotonically with q . The same applies to the final BH offset ∆ mx at peak luminosity. For BBH systems, the offset increases with q , but this is not thecase for BHNS binaries.Also in Figure 2, denoted with solid squares is the coordinate separation when thebinary reaches the tidal radius as estimated by [32, Eq.17.19] R T M h (cid:39) . q − / C − . (34)Relative to the ISCO radius R I (cid:39) M h , the tidal radius is given by R T R I (cid:39) (cid:16) q . (cid:17) − / (cid:18) C . (cid:19) − . (35)For q ≥ . R T ≤ R I , and the NS is swallowed by the BH relatively intact. Forreference, the tidal radius in units of the total mass M is given by R T M (cid:39) q / q (cid:18) C . (cid:19) − . (36)Thus, R T /M (cid:39) . , . . q = 2 , M To get an overall sense of the inspiral and merger, Figures 3, 4 and 5 show snapshotsof the rest mass density in the orbital plane for all the cases under consideration. For q = 2, the tidal forces from the BH trigger mass shedding early on, at approximately440 M from the beginning of the simulation when the binary separation is approximately6 . M . This happens roughly 96 M before peak luminosity. Figure 3 shows fourevolution snapshots for this case. The BH is represented by a black circle with whiteboundary. The initial central density of star is 0 . M − (7 . × g/cm ). Top leftpanel shows a snapshot at time 96 M before the peak luminosity, when the NS begins tobe disrupted. Top right panel shows the stellar disruption at the time of merger. Noticethat the NS has been completely destroyed, deforming into a spiral arm around the BHwhich extends to 7 M beyond the hole. Bottom left panel show the circularization stage lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q E ADM /M J
ADM /M e/ − T mx /M ∆ mx /M BBH 2 0.9901 0.8293 6.3 648 0.027BHNS 2 0.9914 0.8293 6.8 537 0.746BBH 3 0.9918 0.7023 5.5 743 0.058BHNS 3 0.9921 0.7023 8.3 662 0.782BBH 5 0.9939 0.523 10.5 957 0.037BHNS 5 0.9941 0.523 9 848 1.03Table 2: Binary system, mass ratio q , ADM energy E ADM and angular momentum J ADM , eccentricity e , time to peak luminosity T mx , and final BH offset ∆ mx at T mx .of matter around the BH about 100 M after the merger. We found that about 90% of thestar’s material falls into the BH within the first 100 M of evolution while the remainingmaterial continues to expand outwards slowly morphing into an accretion disk. Thebottom right panel shows the final state of the accretion disk 500 M after the merger,reaching a core density of 10 − M − ( 10 g/cm ). The corresponding BBH q = 2 caseis completely different. As one can see from Table 2, the BBH takes approximately twomore orbits to merge.The q = 3 BHNS merger follows the q = 2 steps but not as dramatic in terms ofdisruption effects. In Figure 4, the top left panel shows the beginning of tidal disruptionand tail formation 78 M before the merger. The channel of mass transfer is muchnarrower due to weaker tidal interactions. This is followed by complete disruption ofstar at the merger shown in the top right panel more than 95% of which is consumed bythe BH within 30 M . The bottom left panel shows matter circularization 30 M afterthe merger. The bottom right panel depicts the formation of a very tenuous accretiondisk 500 M after peak emission with characteristic density 10 − M − .As mentioned before, the q = 5 for a BHNS behaves more like a BBH, with thestar remaining almost intact by the time it reaches R I since R T (cid:39) . M . The top leftpanel in Figure 5 shows a snapshot at 45 M prior to the merger. There are hints ofmaterial being stripped from outer layers of the star. The top right panel shows thesituation 20 M before the merger and the bottom left panel at the merger. The bottomright panel shows the result 20 M after the merger. At that point, 99% of the NS hasbeen swallowed by the hole. This leaves a remnant state with extremely low densities.Since there is very little change in this case of triggering electromagnetic signatures,the BHNS and BBH are almost indistinguishable from each other. This will be moreapparent when we compare GW emissions. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q = 2 BHNS. The BH isrepresented by a black circle with white boundary. Panel on the top left showsthe beginning of stellar disruption 96 M before the merger, i.e. peak luminosity.Top right panel shows the stellar disruption at the time of merger followed bycircularization of matter forming an spiral arm around BH 100 M after the merger(bottom left panel). The last panel shows the final state of the accretion disk500 M after the merger. The mass and spin of the final BH in a BHNS merger will depend on the extent to whichthe NS is devoured by the BH. Table 3 shows M h the irreducible mass of the initialBH (mass of the larger BH in BBH cases), M ∗ the initial mass of the NS (irreduciblemass of the smaller BH in BBH cases), M f the irreducible mass of the final BH, M c Christodoulou mass of final BH, M r mass left outside the final BH, a f dimensionlessspin of the final BH, E rad radiated energy, and v k kick of the final BH.First thing to notice is that the irreducible and Christodoulou masses of the finalBH in the BHNS and BBH are comparable. On the other hand, the energy radiated lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q = 3 BHNS merger. Top left panel shows the beginningof tidal disruption and tail formation 78 M before the merger, and the top rightpanel shows the consumption of disrupted star by the BH at the time of merger.The bottom left panel shows matter circularization 30 M after the merger. Thebottom right panel depicts the formation of a very tenuous accretion disk 500 M after peak emission.and the spins and kicks of the final BH differ significantly. BHNS mergers produce afinal hole with higher spin but with a lower kick. The differences in both the final spinand kick decrease as q increases since the binary becomes more BBH-like. The mainculprits of the differences are again the tidal deformations and disruption of the NS.To understand the differences in the mass, spin and kicks of the final BH, we plotin Figure 6 their evolution. The top left panel shows with solid lines the growth ofthe irreducible mass of the BH in BHNS mergers. Dashed horizontal lines denote thefinal mass of BH, M f , for the corresponding BBH merger. T = 0 is the time at peakluminosity. Notice that as expected, for q = 5, the growth is abrupt because the NS isswallowed almost intact, thus mimicking a BBH in which a common apparent horizon lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q = 5 binary merger. The top left panel showsthe rest mass density at 45 M prior to the merger. The top right panel shows thesituation 20 M before the merger and the bottom left panel at the merger. Thebottom right panel shows the result 20 M of merger.suddenly appears to signal the merger. For q = 2 the transitions takes much longer andthe final mass of the BH does not get closer to the mass of the BBH final BH. This isbecause of the material left behind. The rates at which the mass of the final BH changesare depicted in the top right panel of Figure 6. The rates clearly emphasize that thegrowth is sharp for q = 5 and smoother for q = 2.Regarding the spin of the final BH, the middle left panel in Figure 6 shows with solidlines the growth of the spin of the final BH for BHNS binaries given by S z /M , and, forreference, dashed horizontal lines denote the spin of the final BH for the correspondingBBH merger. Middle right panel shows the corresponding spin growth rate ˙ S z /M . Herealso one observes that for lower q the transition is smoother. Important to notice that S z /M is not the dimensionless spin of the final BH. The dimensionless spin is given a f = S z /M c with M c the Christodoulou mass of the final BH. The reason why a f for lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q M h /M M ∗ /M M f /M M c /M M r /M a f E rad /M v k (km/s)BBH 2 0.667 0.333 0.9073 0.9598 0.617 0.0287 149.4BHNS 2 0.667 0.333 0.8984 0.9658 0.0192 0.683 0.0075 33.09BBH 3 0.750 0.250 0.9319 0.9712 0.5405 0.0209 169.5BHNS 3 0.750 0.250 0.9308 0.974 0.0113 0.563 0.0099 30.42BBH 5 0.833 0.167 0.9598 0.9823 0.4166 0.0118 135.8BHNS 5 0.833 0.167 0.9603 0.9834 0.0024 0.4203 0.0102 85.89Table 3: M h irreducible mass of the initial BH (mass of the larger BH in BBHcases), M ∗ initial mass of the NS (irreducible mass of the smaller BH in BBHcases), M f irreducible mass of the final BH, M c Christodoulou mass of final BH, M r mass left outside the final BH, a f dimensionless spin of the final BH, E rad radiated energy, and v k its kick.BHNS are higher is because, as we will see later, the emission of gravitational radiationcarrying out angular momentum is lower; thus, at merger, the final BH is left with highangular momentum.For the kicks of the final BH the situation reverses. The gravitational recoil islower for BHNS mergers. This is because most of the accumulation of the gravitationalrecoil in compact object binaries takes place in the last few orbits, but this is preciselythe stage when BBH and BHNS differ the most. As the NS undergoes disruption, andthus lose its compactness, the BHNS binary radiates less and with it the opportunityto carry out linear momentum. This is clear from the bottom panels in Fig. 6 wherethe left panel shows the accumulation of linear momentum emitted by GWs for both,the BHNS (solid lines) and BBH systems (dashed lines). It is interesting to notice thatwhile the magnitude of the kicks for BBHs are Q < Q < Q
3, consistent with theresults in Ref. [58], the kicks for BHNS systems are Q < Q < Q We have seen from Figure 3 that the disruption of the NS would leave behind materialin the vicinity of the BH. To get a better understanding of how the remnant materialoutside the BH depends on q , we plot in Figure 7 the rest mass M r in a shell1 . M ≤ r ≤ M which to a good approximation accounts for the mass in the accretiondisk.For q = 2, 90% of the NS’s mass falls into the BH within first 100 M of the merger.After a time 400 M , the accretion slows down leaving behind 5% of the mass of the NS.The q = 3 BHNS binary follows a similar trend. During the first 100 M of evolution, theBH has already consumed 95% of the stellar material, and we are left with an accretiondisk with slightly lower density than in the is q = 2 case. As anticipated, the case q = 5 lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris T = 0 is the time atpeak luminosity. Top right panel shows the corresponding mass growth rate ˙ M h .Middle left panel shows with solid lines the growth of the spin of the BH for BHNSbinaries. For reference, dashed horizontal lines denote the spin of the final BH forthe corresponding BBH merger. Middle right panel shows the corresponding spingrowth rate ˙ S z . Bottom left panel shows the accumulation of linear momentumemitted by GWs for both, the BHNS (solid lines) and BBH systems (dashedlines). Bottom right panel depicts the corresponding rate of linear momentumaccumulation. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris M r in a shell 1 . M ≤ r ≤ M which to a goodapproximation account for the mass in the accretion disk.is significantly different since 99% of star is devoured by hole in 20 M of the merger,leaving outside barely any material. Our results are consistent with similar cases inRefs. [7, 8, 14]. Figure 8 shows the real part of the (2,2) mode of the Weyl scalar Ψ for the BHNSbinaries (solid line) together with their corresponding waveform for the BBH (dashedline). The insets show the waveforms early on between 150 ≤ T /M ≤ q grows. Inthis figure, it is also evident that Ψ for BHNS binaries reaches its maximum amplitudeearlier. Since peak luminosity also signals that the binary merges around that time, thisalso implies that BHNS binaries merge earlier than their corresponding BBH system.Before addressing the reasons for the prompt merger of BHNS binaries, we will discussthe differences in the peak luminosity.Figure 9 depicts the amplitude of Ψ . The left panel shows the waveform amplitudesof both the BBH and BHNS mergers around peak luminosity. For the BBH amplitudes,from highest to lowest are q = 2 , lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris . From top tobottom q = 2 , ≤ T /M ≤ lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris . The left panel shows the waveformamplitudes of both the BBH and BHNS mergers around peak luminosity. Theright panel shows the waveform amplitudes for BHNS binaries during the timewindow of the insets in Figures 8.as q / (1 + q ) . Interestingly, the situation reverses for BHNS systems. The amplitudesare not only lower than those of the BBHs, but now instead from highest to lowest are q = 5 , q = 2 and 3, the decrease in amplitude is due to the disruptionthe NS experiences that makes it loose compactness and thus decrease the quadrupolemoment of the binary. The q = 5 BHNS case is comparable to the BBH case because,once again, this is the case in which the star merges with the hole without significantdisruption. The “bump” observed in this case is an artifact of the way the sphericaldecomposition is done. It assumes that the coordinate system is centered at the originof the computational domain. From Table 2, we see that at merger time the center ofmass of the binary for q = 5 is already displaced 1 M from the origin. As a consequencethe (2,2) mode has contributions from higher modes. The other two q cases also undergodisplacements, but they are not as large, and the higher modes for low q ’s are not asdominant as in q = 5.The right panel in Figure 9 shows the waveform amplitudes for BHNS binariesduring the time window of the insets in Figure 8. Since this is during the early stageof the simulation, disruption effects do not play a significant role yet, and the situationresembles the BBH case in which from highest to lowest are q = 2 , ∼ ∼ ,
250 Hz. The oscillations do not affect the stability of the star. During the inspiral,the NS sheds no more than ∼ − of its mass before the onset of tidal disruption.Regarding the prompt merger of BHNS systems, there are only two channels to lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris | J ADMz − J GWz | with J ADMz is the ADM angular momentum in the initialdata and J GWz the angular momentum carried out by GWs. Dashed lines denotethe angular momentum of the final BH. For both, dashed and solid lines, bluedenotes the BBH case and red the BHNS. From top to bottom the cases q = 2, 3and 5, respectively. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris z -component. In Figure 10,we plot with solid lines | J ADMz | − | J GWz | from top to bottom the cases q = 2, 3 and 5,respectively. Here, J ADMz is the ADM angular momentum in the initial data, and J GWz is the angular momentum carried out by GWs. In Figure 10, dashed lines denote theangular momentum of the final BH. For both, dashed and solid lines, blue denotes theBBH case and red the BHNS. Since for BBHs there is only one channel, GWs, to removeangular momentum from the binary, | J ADMz | − | J GWz | after merger and ring-down closelymatches the value of the angular momentum of the final BH. The slight difference isbecause of the junk radiation in the initial data.The situation is different for BHNS binaries. The first thing to notice in Figure 10is that there is a gap between the value that | J ADMz | − | J GWz | reaches after merger andring-down and the value of the angular momentum of the final BH. This gap is closedif in addition one includes the angular momentum carried out by the tidal debris. Thegap is larger the lower the q because the tidal disruptions is stronger. The other featurein Figure 10 is that the decrease of | J ADMz | − | J GWz | is faster for BHNS binaries. Thedifferences start appearing after approximately 300 M , 400 M and 500 M of evolutionfor q = 2, 3 and 5, respectively. At those times, as clear from equation 36, the binary isfar from the tidal disruption separation. Therefore, the most likely culprit is the tidaldeformations on the NS. This effect was pointed out in Ref. [28] using post-Newtonianarguments. Specifically, it was noted that the deformation in the NS introduces acorrection term in the potential whose magnitude increases steeply with the decreaseof the orbital separation. The effect is an acceleration of the inspiral and thus on theemission of GWs, leading to a prompt merger. Next is to discuss the onset of the quasi-normal ringing of the final BH. Given themass and the spin of the final BH in Table 3, we compute from the standard fits inthe literature [59] the quasi-normal frequency and decay time for the (2,1), (2,2) and(3,3) modes. The values are given in Table 4. Figure 11 shows the amplitude (leftpanels) and phase (right panels) of the (2,1), (2,2) and (3,3) modes of Ψ after peakluminosity when the final BH is expected to undergo quasi-normal ringing. In theselog-linear for the amplitude and linear-linear for the phase plots, quasi-normal ringing(i.e. exponentially damped sinusoidal) would show up as linear dependence with timefor both the amplitude and the phase. For reference, the solid lines are the quasi-normalringing computed from Table 4. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q ω , f τ , f ω , f τ , f ω , f τ , f BHNS 2 0.471 11.73 0.546 11.8 0.866 11.49BHNS 3 0.446 11.42 0.499 11.42 0.794 11.06BHNS 5 0.421 11.26 0.454 11.27 0.726 10.86Table 4: Quasi-normal frequencies and damping times computed from the massand the spin of the final BH in Table 3 using the standard fits in the literature [59].System q ω , τ , ω , τ , ω , τ , BHNS 2 0.155 55.62 0.543 9.898 0.603 9.86BHNS 3 -0.332 13.1 0.492 11.32 0.483 11.52BHNS 5 0.436 10.83 0.449 11.08 0.71 10.08Table 5: Quasi-normal frequencies and damping times computed from fitting thedata in Figure 11.For the (2,1) mode, we see that the only case showing quasi-normal behaviour is the q = 5, the one with the more BBH-like characteristics. For the other two cases, thereare two factors that prevent a clean quasi-normal ringing. One is that the geometry ofthe tidal debris does not favor excitation of the final BH in this mode. The other is that,during the time spanned in the figure for the decay of Ψ ( ∼ M ), the final BH is stillgrowing as one can see from Figure 6. The (2,2) mode is the one with more noticeablequasi-normal characteristics, in particular in the phase. The exponential decay of theamplitude is cleaner for the q = 3 case, and the q = 5 case shows the bumps associatedwith the contributions from higher modes due to the center of mass displacement. The q = 2 case shows exponential decay after 50 M , which according to the left panel inFigure 6 is when the BH has almost stopped accreting the debris from the disruptedNS. Interestingly, in all cases, the phase (3,3) modes shows an approximate linear growth(i.e. constant frequency of oscillation), but only in the case q = 5 the growth matchesthat of quasi-normal ringing. Similarly, exponential decay in the amplitude is not asclear with the exception of the q = 5. The oscillations in the q = 3 we conjectureare associated with the accretion of tail of debris observed in the bottom left panel inFigure 4. Ultimately, comparisons between BBH and BHNS systems would be incomplete if notlooked through the eyepiece of data analysis tools. The focus of a follow up paper willpay particular attention to observational signatures from the compactness of the NS. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris after peakluminosity for BHNS systems. The solid lines are the quasi-normal ringingcomputed from Table 4.For the present work, we start by showing in Figure 12 the strain of both the BHNS andBBH systems. It is evident how the BHNS and corresponding the BBH system agreeearly on for low frequencies, with the q = 5 following each other through merger.Next we show in Table 6 mismatches, 1 −(cid:104) h , h (cid:105) , relative to LIGO and the EinsteinTelescope, with the matches given by (cid:104) h , h (cid:105) ≡ Re (cid:90) ∞ ˜ h ∗ ( f )˜ h ( f ) S h ( f ) df (37)maximized by the time and phase at coalescence. In this expression, h are strainsincluding modes up to l = 8. The mismatches in Table 6 include three different lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris q Detector i = 0 i = π/ i = π/
32 LIGO 0.0845 0.0756 0.0742 ET 0.0767 0.0686 0.06663 LIGO 0.0611 0.0731 0.09843 ET 0.0464 0.0576 0.08145 LIGO 0.0046 0.0401 0.11945 ET 0.0030 0.0349 0.1072Table 6: Mismatches between BBH and BHNS waveforms for three differentinclination angle i and for the Einstein Telescope (ET) and LIGO.inclinations: i = 0 , π/ π/
3. As expected, for i = 0 (face-on), the mismatchis highest for q = 2 because the (2,2) mode dominates in this case; the mismatchdecreases with increasing mass ratios consistent with our previous observations. Acrossthe detectors, mismatches are smaller for the Einstein Telescope compared to LIGO,given the higher sensitivity of the former. For i = π/
6, the trend remains similar thoughthe mismatch values decrease for q = 2 while increase for other two cases because of thecontributions from higher modes. This situation becomes more visible for i = π/
3, wherethe contributions of higher modes increase significantly. The mismatch now increaseswith mass ratio with 12% mismatch between BBH and NSBH waveforms for q = 5 and10% mismatch for q = 3. This shows the importance of higher modes in the study ofmixed binaries at high mass ratios. lack Hole - Neutron Star Binary Mergers: The Imprint of Tidal Deformations and Debris
5. Conclusions
For high mass ratio systems, distinguishing BHNS binaries from BBH binaries willincur challenges because the NS is swallowed by the hole without experiencing significantdisruption. To investigate the transition of the merger behaviour of a BHNS into a BBH-like system, we have carried out three BHNS merger simulations and their correspondingBBH mergers for mass ratios q = 2 , q = 2 representsthe case of total NS disruption before merger, and the q = 5 case is an example of aBBH-like merger. The focus was on the effects that the disruption of the NS imprints onthe inspiral and merger dynamics, the properties of the final BH, the accretion disk, theGWs, and the strain spectrum and mismatches. A secondary objective of the study wasto demonstrate the effectiveness of the method we developed in Ref. [30] to constructinitial data with a generalization of the Bowen-York data for BH punctures to the caseof NSs.The most noticeable feature observed in the simulations of the merger dynamicsof the BHNS binaries was that they merge earlier than their corresponding BBHs. Wefound that the dominant factor hardening the mixed binary is the enhanced angularmomentum emission carried out by the GWs due to the tidal deformations in the NS.On the other hand, the tidal disruption of the NS suppresses the gravitational recoil ofthe final BH in BHNS mergers when compared with BBHs. Regarding the final BH, itsmass is comparable between the BHNS and BBH systems. This, however, is not thecase regarding the final spin. For instance, in the case of q = 2, the tidal debris as isaccreted by the hole increases the spin by approximately 10 %. The same tidal debrishas an influence in the quasi-normal ringing of the final BH. For low q ’s only the (2,2)mode exhibits a clean damped exponential sinusoidal behavior. In terms of mismatches,the most favorable configuration to distinguish between BHNS and BBH systems withlarge q ’s would be that for large inclinations where high modes are more influential. Acknowledgements
This work is supported by NSF grants PHY-1908042, PHY-1806580, PHY-1550461. MGL acknowledges the support from the Mexican NationalCouncil of Science and Technology (CONACyT) CVU 391996. We would also liketo acknowledge XSEDE (TG-PHY120016) and the Partnership for an AdvancedComputing Environment (PACE) at the Georgia Institute of Technology, Atlanta,Georgia, USA for providing necessary computer resources for this study. We thank ChrisEvans, Deborah Ferguson, Zachariah Etienne and Deirdre Shoemaker for discussions andsharing their resources for this work. Finally, the authors would also like to express theirgratitude to all the front-line workers for their efforts and dedication to keep us all safeduring these challenging times. [1] Abbott R et al. (LIGO Scientific, Virgo) 2020 (
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