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Towards a Realistic Neutron Star Binary Inspiral Mark Miller(1,2) and Wai-Mo Suen(2,3) (1) 238-332 Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109 (2)McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, Missouri 63130 and (3)Physics Department, Chinese University of Hong Kong, Hong Kong An approach to general relativity based on conformal flatness and quasiequilibrium (CFQE) assumptions has played an important role in the study of the inspiral dynamics and in providing initialdatafor fully generalrelativistic numericalsimulations of coalescing compact binaries. How- ever,theregime of validity of theapproach has neverbeen established. To thisend,we develop an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. With this analysis, we show that the CFQE assumption is significantly violated even at relatively large orbital separations in thecase of corotational neutron star binaries. We also demonstrate that the innermost stable circular orbit (ISCO) determined in 3 0 theCFQE approach for corotating neutron star binaries may have noastrophysical significance. 0 2 PACSnumbers: 04.25.Dm,04.30.Db,04.40.Dg,02.60.Cb n a a. Introduction The analysis of general relativistic four constraintequations plus one of the dynamical Ein- J binary neutron star processes is an important yet chal- stein equations under the CF and QE assumptions (the 7 lenging endeavor. The importance of studying binary QEassumptionaffectstheconstraintequationsonlyindi- 2 neutron star coalescence is rooted in observational as- rectly). The inspiral is assumed to proceedalong the se- tronomy; the process is considered to be a candidate quence ina quasistationarymanner ona timescalemuch 1 v for the sources of detectable gravitational waves as well longer than the orbital timescale. The slow secular evo- 2 as X- and γ-rays. Recently, there has been a large ef- lution is driven by gravitational radiation, which can be 1 fort expended towards numerically simulating the inspi- calculated using e.g., the standard quadrupole formula 1 raland coalescenceofcompact objects in orderto deter- (see, e.g., [11]). In the corotating case, when the orbit 1 mine their signaturegravitationalwaveformsinprepara- shrinks to a certain radius (which will depend on the 0 tion for the detection of gravitational waves by the new equationofstate(EOS)oftheneutronstars),theCFQE 3 0 generation of gravitationalwave observatories,including configurationbecomessecularlyunstable,withtheADM / LIGO,LISA,VIRGO,TAMA,andGEO.Thecomplexity mass of the system attaining a minimum among all con- c of the nonlinear Einstein field equations, the lack of ex- figurations in the sequence. The orbit at this radius is q - actsymmetryofthesystem,andthecouplingofdifferent referredtoastheinnermoststablecircularorbit(ISCO). r timescales in the process combine to make the study of One expects the binary to become dynamically unstable g the neutron star inspiral coalescence a major challenge. at an orbital radius that is slightly less than this ISCO : v Arecentapproachusedtoinvestigatethe inspiralcoa- point. This ISCO point is therefore thought to be the i X lescence problem that has drawn much attention [1, 2, point at which the system enters the “plunge phase”, in r 3, 4, 5, 6, 7, 8, 9, 10] is based on the separation of which the subsequent merging occurs within an orbital a twotimescales,namely,the orbitalmotiontimescaleand timescale. the gravitational radiation timescale. This approach, Besides providing a description of the inspiral coales- which we refer to as the conformally flat quasiequilib- cence process, the CFQE approach is important for an- rium (CFQE) approach, seeks to construct general rela- other reason: it is expected to provide a starting point tivistic configurationscorrespondingto two compact ob- for numerical investigations in general relativity. The jects in circular orbit by assuming that the spacetime is setting of initial data is a sensitive issue in the numer- essentially stationary with an approximate Killing vec- ical integration of the Einstein equations. On the one tor(quasiequilibrium,QE).Theslowsecularevolutionis hand, astrophysically relevant initial data must be used driven by the gravitational radiation which has a much in order to make contact with gravitational wave obser- longer timescale than the orbital motion timescale, so vations. On the other hand, we are forced to use ini- thatthespacetimeisthoughttobedescribablebyacon- tial data in the late state of the inspiral, due to limited formally flat (CF) spatial metric. computational resources. As the CFQE approach gives The CFQE approach leads to a two-phase description configurationsthatsatisfytheconstraintequationsofthe of the inspiral and coalescence process. In the “slowly Einstein theory, one is tempted to start numerical inte- inspiraling phase”, the evolution is approximated by a grationwith a CFQE configurationnear the ISCO point time sequence (which we refer to as a CFQE-sequence) asdeterminedintheCFQEapproach,see,e.g.,[12]. The ofCFQEconfigurationswiththesamerestmassandspin danger in using the CFQE approach to determine the state (e.g. corotating or nonrotating). Each CFQE con- initial configuration for numerical integration is the un- figurationinthetimesequenceisobtainedbysolvingthe knownastrophysicalrelevanceoftheinitialdata,regard- 2 less of the fact that each CFQE configuration satisfies tropicEOS,P =KρΓ,withΓ=2andK =0.0445c2/ρ , n the constraints of general relativity and is therefore, in where ρ is nuclear density (2.3 x 1014 g/cm3). For this n principle, a legitimate initial data set for numerical inte- EOS,the maximum stable static neutron star configura- grations. In order for the results of the numerical evolution has an ADM mass of 1.79M⊙ and a baryonic mass tionstoberelevanttoobservations,e.g.,thegravitational of 1.97M⊙ (M⊙ is 1 solar mass). In this paper, we use waves emitted in neutron star coalescences, we have to neutronstarswhosebaryonicmassM0 is1.49M⊙,which make sure that the initial data actually correspondsto a is approximately 75% that of the maximum stable con- configurationina realisticinspiral. The setting ofphysi- figuration. TheADMmassofasinglestaticneutronstar cal initial data is a standard difficulty in numerical rela- for this configuration is 1.4M⊙. tivityanditisparticularlyacuteinthiscase;iftheinitial The corotational CFQE configurations with a fixed data is picked to be a CFQE configuration in a regime baryonic mass can be uniquely specified by the separa- where the CFQE-sequence is not expected to be valid, tion between the two neutron stars. We measure the theinitialdatamustberegardedashighlycontrivedand spatial separation on a constant time hypersurface by the resulting calculation will have no astrophysical sig- the spatial geodesic distance ℓ between the points of 1,2 nificance. maximum baryonic mass density in each of the neutron In this paper we analyze the limitations of the CFQE stars. This particular choice of measure of distance de- approach, which suffers from two basic problems. The pends on the choice of time slicing. In order to compare firstproblemis thatthe CFandQEassumptionsviolate with the CFQE-sequence approximation in an invariant theEinsteinequationswhich,forthissystem,forbidboth manner,weusethesameslicingcondition(maximalslic- an exact Killing vector and a conformally flat metric at ing) as that used in CFQE-sequence approach. Four all times (the metric can only be chosen to be confor- CFQE configurations denoted respectively NS-1 to NS-4 mally flat at one point in time but not along the entire areused as initial data in our generalrelativistic simula- time sequence). The second problem is that the CFQE tions in this paper: ℓ1,2/M0 = 23.44,25.94,29.78,35.72. approach itself does not provide a way to estimate the The corresponding orbital angular velocity parameters errorinvolvedin its assumptions. A determination of its are ΩM0 =0.01547,0.01296,0.01022,0.00746. accuracy has to be carried out with a completely differ- Inthefollowingwefirsttakecaretoconstructinvariant ent analysis. Because of this, the regime of validity of measures that can be used to compare, and give a sense the CFQE approach has never been determined. Here, of the difference between, two different spacetimes, one we calculate the rate of increase of the accuracy of the of which is obtained by solving the Einstein equations CF and QE approximations with increasing initial neu- andthe other representedby the CFQE-sequence,which tron star separation. For the corotating binaries studied does not satisfy the Einstein equations. (with a Γ = 2 polytropic equation of state), we demon- c. The QEapproximation. Onebasicassumptionof stratethataninitialseparationofatleast6neutronstar the CFQE approach is the existence of a timelike, heli- radii is needed in order for the violation of the QE as- cal Killing vector field. In the case of a corotating bi- sumptiontoremainbelow10%(insomereasonablemea- nary, the 4-velocity ua of the fluid is proportional to sure defined below) after a short evolution (a fraction of this Killing vector. This implies the vanishing of the an orbit). (The violation of the CF assumption is an or- type-(0,2) 4-tensor Q u + u +u a +u a , ab a b b a a b b a ≡ ∇ ∇ derofmagnitudesmalleratthisseparation.) Itissafeto where aa ub ua is the 4-acceleration of the fluid b ≡ ∇ conclude that, starting at a separation of 6 neutron star ( denotes the covariant derivative operator compat- a ∇ radii,theevolution(e.g.,theorbitalphase)ofthesystem ible with the 4-metric g ). The quantity Q ua van- ab ab is completely different from that of the CFQE-sequence ishes identically. We can thus monitor the space-space well before the CFQE ISCO separation(less than 3 neu- components of Q during full general relativistic simu- ab tron star radii) is reached. lations as a way of monitoring how well the 4-velocity In what follows, we present the results of our analysis ua stays proportional to a Killing vector field. Define of the CFQE-sequence approximation for binary, coro- Qij as the projection of Qab onto the constant t spa- tating neutron stars. The details of all calculations and tial slice: Qij = PiaPjbQab = Qab(cid:0)∂∂xi(cid:1)a(cid:0)∂∂xj(cid:1)b, where the subsequent analysis of the numerical results, along P = g + n n is the projection operator onto the ab ab a b with results on long timescale (e.g., multiple orbital pe- constant t spatial slices with unit normal na. The 3- riodtimescale)numericalevolutionswillbereportedelse- coordinate independent norm Q of this matrix Q is ij ij where [13]. thesquarerootofthelargestei|genv|alueofQ Qj ,where ij k b. A General Relativistic Analysis of the CFQE we have raised and lowered 3-indices with the 3-metric. Approximations for Corotating Neutron Star Binaries. To gain a sense of the size of Q , we normalize ij | | To analyze the validity of the CFQE-sequence approxi- Q by the norms of its two principle parts: Q ij | | ≡ mation, we compare it to solutions of the full Einstein Q /max (cid:12)Q (cid:12),(cid:12)Q (cid:12) , where Q u + u , | ij| {(cid:12) 1ij(cid:12) (cid:12) 2ij(cid:12)} 1ab ≡ ∇a b ∇b a equations whose initial Cauchy slice corresponds to spe- and Q u a +u a . A value of Q=0 signifies that 2ab ≡ a b b a cific CFQE configurations. We construct these corotat- the 4-velocity of the fluid is proportional to a timelike ingCFQEconfigurationsfollowingthealgorithmdetailed Killing vector as in the CFQE-sequence,while a value of in [1]. The neutronstars we use aredescribed by a poly- Q = 1 would signify that the assumption is maximally 3 0.25 reciprocal power series fit 0.25 reciprocal power series fit to error 0.2 >max 0.2 CFQE (a) Q IS < C O <Q>0.15 0.15 sep 0.1 NS-1, ∆x = 0.09580 M0, 5153 gridpoints 0.1 aration25 30 35 40 45 NS-2, ∆x = 0.1007 M, 5153 gridpoints 0.05 0 NS-3, ∆x = 0.1100 M, 5153 gridpoints 0.05 0 0.04 reciprocal power series fit NS-4, ∆x = 0.1050 M, 6433 gridpoints C reciprocal power series fit to error 0 max FQE 00 0.005 t Ω / (2π0.)01 0.015 <H> 0.03 ISCO (b) 0.02 sep FIG. 1: The quantity hQi is plotted as a function of time aratio n forfullyconsistentgeneralrelativisticnumericalcalculations, 0.01 25 30 35 40 45 using CFQE configuration NS-1to NS-4as initial data. l / M 1,2 0 FIG. 2: Fig. 2a shows the maximum value of hQi obtained violated. in our fully consistent general relativistic simulations using Since Q is meaningful only inside the fluid bodies, a CFQE configurations with different geodesic separation ℓ1,2 natural global measure of the magnitude of Q is its the as initial data. Fig. 2b shows the corresponding maximum baryonic mass weighted integral, denoted by Q : valueof hHi h i R d3x Q √γρW Q = | | , (1) h i R d3x√γρW willbe. This effectcanbe seendirectly inFig.2a,where the maximum value of Q obtained in the general rela- wheretheintegralsaretakentobeovertheentirespatial h i tivistic simulations is plotted as a function of the initial slice, with support only where ρ = 0. Here, we have denoted γ to be the determinant of6 the 3-metric and W geodesicseparationℓ1,2. Acarefulanalysisofthenumer- ical errors (shown as error bars in the figure) based on a to be the Lorentzfactor of the fluid so that R d3x√γρW Richardson extrapolation technique (for both truncation is the total (conserved) rest mass of the system. errorsandboundary errors)can be found in [13]. In this In Fig. 1, we plot Q as a function of time in the h i way,wecandeterminethe requiredseparationoftheini- general relativistic numerical simulations, using CFQE tial CFQE configuration such that the QE assumption configurationsNS-1to NS-4asinitialdata. For eachini- is valid to any required level in our fully consistent gen- tial CFQE configuration, the subsequent solution to the eralrelativisticsimulations. InFig.2a,wefitareciprocal Einstein equations has Q reaching a maximum value h i powerlaw, a1 + a2 + a3 + a4 ,tothemaximum afteraveryshorttime (between0.5%and1.5%ofanor- (ℓ1,2) (ℓ1,2)2 (ℓ1,2)3 (ℓ1,2)4 bit, 2π/Ω being the orbitalperiodcorrespondingto each value attained in our general relativistic calculations (as CFQE configuration). These results are in contrast to well as to the lower bound of the estimated error). We theCFQE-sequenceapproximation,where Q isexactly canseethat,e.g.,onewouldhavetouseaCFQEconfigu- zero. The rapid growth of Q to such a lahrgeivalue sig- rationinitialdatasetwithageodesicseparationbetween nalsthat,inthepresentcasheoifcorotatingneutronstars, the neutron stars of about ℓ1,2 = 47M0 in order for the after just a short time the CFQE-sequence is very dif- subsequentsolutionto havethe fluid4-velocityfollowing ferent from the spacetime described by the full Einstein a Killing vector to the 10% level. equations even when the latter is started with the same d. The CF Approximation. While the 3-metric can initial data. Within a small fraction of an orbit, the 4- bechosentobeconformallyflatonanyonetimeslice,the velocity of the fluid already is in no way proportional to assumptionthatthe 3-metricremainconformallyflatfor a Killing vector. alltimeviolatesthedynamicalEinsteinequations. Again Also, we see from Fig. 1 that the maximum value ob- we begin with an invariant construction of a measure of tained by Q decreases from approximately Q = 0.27 the violation. Because conformal flatness is a property h i h i to Q = 0.17 as the geodesic separation of the initial of a constant time slice, and because we are using the h i CFQE configuration increases from ℓ /M = 23.44 to same slicing conditions in our simulation as that used in 1/2 0 ℓ /M = 35.72. In principle one can obtain an arbi- theCFQE-sequenceapproximation,weseeka3-invariant 1/2 0 trarily accurate CFQE configuration by starting the in- (with respect to spatial coordinate transformations) to spiral evolution with a large enough separation. While monitor the CF assumption. The Bach 3-tensor Bijk = startingatalargerseparationdoesnotimplytheCFQE- 2 (cid:0)(3)R 1γ (3)R(cid:1) can be shown to vanish if and D[i j]k− 4 j]k sequence will be more astrophysically relevant at late only if the 3-metric γ is conformally flat. The Cotton- ij times, the fully relativistic simulation starting with it Yorktensor,H ,isrelatedtothe3-BachtensorbyH = ij ij 4 0.05 NS-1, ∆x = 0.09580 M, 5153 gridpoints attainedinour generalrelativistic numericalsimulations 0 NS-2, ∆x = 0.1007 M, 5153 gridpoints using the four CFQE configurations NS-1 through NS-4 0 0.04 NS-3, ∆x = 0.1100 M, 5153 gridpoints asinitialdata(see[13]foradetailedanalysisofthetrun- 0 NS-4, ∆x = 0.1050 M, 6433 gridpoints cation and boundary errors in our calculation, which we 0 indicate as error bars in the figure). As in Fig. 2a, we 0.03 fit a reciprocal power law to both the maximum value <H> of H and the lower bound of the error. We see that h i 0.02 a CFQE configuration with geodesic separation ℓ1,2 of approximately 47M or greater must be used as initial 0 data in order for the full solution to the Einstein field 0.01 equations to have a value of H to be 0.01 or less. h i e. Conclusions In this letter we present a method 00 0.005 0.01 0.015 0.02 foranalyzingtheregimeofvalidityoftheCFQE-sequence t Ω / (2π) approximation. We apply this method to the CFQE- sequence approximation of corotating binary neutron FIG. 3: The quantity hHi is plotted as a function of time stars. We show that for geodesic separations less than forfullyconsistentgeneralrelativisticnumericalcalculations, roughly 45M , which is about twice the geodesic sep- using CFQE configuration NS-1to NS-4as initial data. 0 aration of the ISCO configuration, the QE approximation is severely violated and the CFQE sequence cannot ǫmn B , where ǫ is the natural volume element 3- be taken as a reasonable approximation of the binary j mni ijk form. We define the scalarH asthe matrix normofH , evolution for these separations. Numerical simulations ij normalized by the size of the covariant derivative of the starting with CFQE configurations as initial data with 3-Ricci tensor: H = Hmn /p i(3)Rjk i(3)Rjk, where geodesic separations smaller than 45M0 cannot, there- H denotes the matr|ix nor|m oDf the H D. As before, we fore, be considered as approximating a realistic neutron ij ij |defin|e the baryonicmassdensity weightofH,denotedas star binary inspiral. H , by h i R d3x H √γρW H = | | , (2) h i R d3x√γρW I. ACKNOWLEDGEMENT wheretheintegralsaretakentobeovertheentirespatial It is a pleasure to thank Abhay Ashtekar, Lee Lind- slice, but have support only where ρ=0. 6 blom, Kip Thorne, and Clifford Will for useful discus- In Fig. 3, we plot H as a function of time for the h i sions and comments. Our application code, which solves general relativistic simulations using initial data CFQE the coupled Einstein-relativistic hydrodynamics equa- configurationsNS-1throughNS-4. Theprofileofthevio- tionsusestheCactusComputationalToolkit[14]forpar- lationoftheconformalflatnessassumption H issimilar h i allelization and high performance I/O. tothatoftheKillingvectorassumption Q : aquickrise h i to a maximum value at a timescale on the same order of Financial support for this research has been provided but slightly longer than that of Q . This is contrasted by the NSF KDI Astrophysics Simulation Collaboratory h i totheCFQE-sequenceapproximation,whichhas H ex- (ASC) project (Phy 99-79985) and the Jet Propulsion h i actly equal to 0. Laboratory(account100581-A.C.02). Computationalre- In Fig. 2b, we plot the maximum value of the quan- source support has been provided by the NSF NRAC tity H as a function of initial geodesic separation ℓ grants MCA02N022 and MCA93S025. 1,2 h i [1] T. W. Baumgarte, G. B. Cook, M. A. Scheel, S. L. Rev. D60, 087301 (1999). Shapiro, and S. A. Teukolsky, Physical Review D 57, [6] M. Shibata, Phys.Rev.D 58, 024012 (1998). 7299 (1998). 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