Mergers of irrotational neutron star binaries in ...coupled, nonlinear, elliptic ﬁeld equations,...

PHYSICAL REVIEW D 69, 124036 ~2004!

Mergers of irrotational neutron star binaries in conformally flat gravity

Joshua A. Faber, Philippe Grandcle´ment,* and Frederic A. RasioDepartment of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA

~Received 21 December 2003; revised manuscript received 12 April 2004; published 30 June 2004!

We present the first results from our new general relativistic, Lagrangian hydrodynamics code, which treatsgravity in the conformally flat~CF! limit. The evolution of fluid configurations is described using smoothedparticle hydrodynamics~SPH!, and the elliptic field equations of the CF formalism are solved using spectralmethods in spherical coordinates. The code was tested on models for which the CF limit is exact, finding goodagreement with the classical Oppenheimer-Volkov solution for a relativistic static spherical star as well as theexact semianalytic solution for a collapsing spherical dust cloud. By computing the evolution of quasiequilib-rium neutron star binary configurations in the absence of gravitational radiation back reaction, we have con-firmed that these configurations can remain dynamically stable all the way to the development of a cusp. Withan approximate treatment of radiation reaction, we have calculated the complete merger of an irrotationalbinary configuration from the innermost point on an equilibrium sequence through merger and remnant for-mation and ringdown, finding good agreement with previous relativistic calculations. In particular, we find thatmass loss is highly suppressed by relativistic effects, but that, for a reasonably stiff neutron star equation ofstate, the remnant is initially stable against gravitational collapse because of its strong differential rotation. Thegravity wave signal derived from our numerical calculation has an energy spectrum which matches extremelywell with estimates based solely on quasiequilibrium results, deviating from the Newtonian power-law form atfrequencies below 1 kHz, i.e., within the reach of advanced interferometric detectors.

DOI: 10.1103/PhysRevD.69.124036 PACS number~s!: 04.30.Db, 47.11.1j, 95.85.Sz, 97.60.Jd

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I. INTRODUCTION

Gravitational wave~GW! astronomy stands at a crucimoment in its history, with the LIGO~Laser InterferometerGravitational Wave Observatory! Scientific Collaboration re-porting results from their first scientific runs@1–3#, GEO600having completed two scientific runs@4,5#, TAMA takingdata @6,7#, and VIRGO reporting its first lock acquisitio@8–10#. As such, it is now more important than ever to haaccurate theoretical predictions of the main candidatenals, both to aid in their detection and to facilitate the intpretation of any future detections.

It has long been recognized that coalescing relativibinary systems containing compact objects, either neustars~NS! or black holes~BH!, are likely to be importantsources of detectable GWs. Recent population synthesisculations indicate that an Advanced LIGO detector shouldable to see at least tens of coalescences per year of NSNS-BH, and BH-BH binaries@11#. Empirical rate estimatesbased on three of the four observed binary pulsar systexpected to merge within a Hubble time, PSR B1913116,PSR B1534112, and PSR J0737-3039@12#, are in generalagreement, with the latter making the dominant contributto the probabilistic rate because of its short coalescenceof 85 Myr ~see @13,14# and references therein; the foursystem is located in a globular cluster, rather than the gatic plane!. NS-NS binaries are the only known systems wcoalescence times shorter than a Hubble time to have bconclusively observed, and it is the orbital decay

*Current address: Laboratoire de Mathe´matiques et de PhysiquTheorique, Universite´ de Tours, Parc de Grandmont, 37200 TouFrance.

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PSR1913116 that currently provides our best indirect evdence for the existence of GWs@15–17#.

At large separations, the dynamics of compact objectnaries can be well approximated by high-order poNewtonian~PN! ~see@18# and references therein! and otherformalisms@19,20# which treat the compact objects as poinmasses, including the effects of spin-orbit and spin-spingular momentum couplings for systems containing a B@21–23#. In general, these methods are more appropriatedescribing a BH than a NS, since the finite-size effectsnored by point-mass formulas are of greater magnitudesystems containing NS.

For smaller binary separations,r &10RNS, whereRNS isthe radius of the NS, these finite-size corrections playincreasingly significant role in the evolution of NS-NS binries. As long as the coalescence time scalet5r / r remainslonger than the dynamical time scale, the evolution is qsiadiabatic, and the binary will sweep through a sequencconfigurations representing energy minima for given binseparations. The infall rate will be given by

dr

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dt DGW

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, ~1!

where (dE/dt)GW is the energy loss rate to gravitational rdiation, and (dE/dr)eq is the slope of the equilibrium energcurve. Equilibrium energy sequences were first construcin Newtonian and then PN gravity~see@24# and referencestherein!. More recently, general relativistic~GR! sequenceshave been calculated for binary NS systems in quasicircorbits @25–32#. An important result from these studies is ththe slope of the equilibrium energy curve is flatter whrelativistic effects are included, compared to the Newton

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FABER, GRANDCLEMENT, AND RASIO PHYSICAL REVIEW D69, 124036 ~2004!

value. This becomes especially pronounced at small septions, near the point where equilibrium sequences encouan energy minimum~often associated with the ‘‘innermosstable circular orbit,’’ or ISCO! or terminate when a cusdevelops on the inner edge of the NS. Based on this obvation, it was noted that a flatter slope in the equilibriuenergy curve results in not only a faster infall rate balso a decrease in the energy spectrum,dE/d f'(dE/dr)/(d f /dr). In fact, based on the relativistic equlibrium sequences of Taniguchi and Gourgoulhon~@27#,hereafter TG!, Faberet al. ~@33#, hereafter FGRT! concludedthat the dependence of finite-size corrections on the NS cpactness~i.e., MNS/RNS) would leave an imprint in the GWenergy spectrum at frequenciesf '500–1000 Hz, within theband accessible to Advanced LIGO. Hughes@34# proposed amethod using these results which would allow for a deternation of the NS compactness to within a few percent baon 10–50 observations with Advanced LIGO if it employsnarrow-band detector in addition to the current broadbsetup, with the required number of observations dependupon both the true compactness and the particular setuthe detector.

Shortly before the ISCO or the end of the equilibriusequence~when it terminates at the formation of a cusp! thebinary will begin a transition toward a rapid plunge inwareventually leading to merger. Once the binary passespoint, the dynamical evolution becomes too complicateddescribe using semianalytical methods, requiring instea3D hydrodynamic treatment. Such calculations were pformed first for Newtonian gravitation, using both Euleriagrid-based codes and Lagrangian smoothed particle hydrnamics ~SPH; for a review, see, e.g.,@35# and referencestherein!. It was always recognized that any results wouldat best qualitatively accurate, since the extreme compactof the NS would induce a host of GR effects. Noting thsome attempts were made to calculate the evolution of binNS systems in lowest-order PN gravity@35–39#, using a for-malism developed by@40#. Unfortunately, using realistic NSequations of state~EOS! violated the basic assumption of thPN approximation that the magnitude of the 1PN termssmall relative to Newtonian-order effects. As a result, all Pcalculations were forced to make unphysical approximatioeither by evolving NS with a fraction of their proper physicmass@38,39#, or by reducing the magnitude of all 1PN term@@35–37#, hereafter denoted FR1-3, or collectively FR#.

While the ultimate goal of studying binary NS coalecences should be a full GR treatment, only one groupbeen able to calculate the full evolution of a binary systfrom an equilibrium binary configuration through merger athe formation of a remnant@41–43#. While these calculationsrepresent a breakthrough in our understanding of the hydynamics of coalescing NS binaries, they still leave a grdeal of room for further research into a variety of questioCurrently, full GR calculations are extremely difficult, anare vulnerable to several numerical instabilities. In orderguarantee the stability of calculation through coalesceand the formation of either a stable merger remnant or abinaries were started from the termination points of equirium sequences, from quasicircular configurations with z

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infall velocity. The calculations of Shibata and Uryu@41,42#used initial conditions generated by Uryu, Eriguchi, and claborators@31,32#, and the more recent work by ShibatTaniguchi, and Uryu@43# used initial conditions generateby TG. This restricts studying the behavior of the infall vlocity and GW signal when the stars begin their plunge, ding which time detected signals may yield important infomation, as mentioned above. The lack of information abthe dynamics of binaries before the plunge also hinders ting the validity of the initial matter and field configurationused in full GR calculations. While the assumptions definthe quasiequilibrium configuration are certainly reasonabit is difficult to gauge how well such configurations agrwith those evolved dynamically from larger separations. Aunknown in detail is the effect that errors in the initial cofiguration will have on the system as they propagate in nlinear fashion during the calculation.

A further technical difficulty with full GR calculationsresults from computational limits, since numerical grids acurrently constrained to have their boundaries lie within aproximately half of a GW wavelength, within the near zonThis can induce possibly significant errors into the GW etraction process, which would ideally be performed by studing the behavior of the metric in the wave zone.

A possible middle road, at least at present, is providedthe conformally flat~CF! approximation to GR, first de-scribed by Isenberg@44# and developed in greater detail bWilson and collaborators@45# ~note that their original ver-sion contained a mathematical error, pointed out by Flana@46#, accounting for spurious results with regard to ‘‘crusing’’ effects on NS prior to merger!. This formalism includesmuch of the nonlinearity inherent in GR, but yields a setcoupled, nonlinear, elliptic field equations, which canevolved stably. The first dynamical 3D calculations to mause of the CF framework were performed by Shibata, Baugarte, and Shapiro@47#, who created a PN variant by discarding some of the nonlinear source terms in the field equatiwhile retaining the vast majority of the nonlinearity in thsystem. They found, among other things, that the maximdensity of the NS is smaller for binary configurations thanisolation, and that NS in binaries have a higher maximmass as well, strongly indicating that collapse to a BH prto merger is essentially impossible. The first calculationsuse the full formalism were performed by Oechslinet al.@48#, using a Lagrangian SPH code with a multigrid fiesolver. Using corotating initial configurations, which athought to be unphysical~since viscous effects are thoughtbe much too weak to tidally synchronize the NS@49,50#!,they found that mass loss during the merger is suppresserelativistic gravitational schemes, as had been previousuggested based on calculations in PN gravity~FR3!.

While the CF formalism appears to be safer than full Gwith regard to evolving the system stably, the grid-basapproaches used so far suffer from many of the same plems faced by full GR calculations. Since the CF field equtions are nonlinear and lack compact support, approximboundary conditions for the fields must be applied atboundary of the grid, which can lead to errors in the solutiAdditionally, very large grids are required to satisfy th

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MERGERS OF IRROTATIONAL NEUTRON STAR . . . PHYSICAL REVIEW D 69, 124036 ~2004!

tradeoff between large grid sizes, required to improveaccuracy of the boundary conditions, and small grid csizes, necessary to improve the resolution of the NS. Mugrid techniques can be very valuable for such cases, butrefinement techniques can also act as a source of numeerrors. Noting this, we have taken an entirely new approto the dynamical evolution of binary NS systems with SPwhich requires no use of rectangular 3D grids to solve formetric fields. Instead, we use the LORENE numerical libries, developed by the Meudon group, which are freely avable online1 ~see@51#, hereafter BGM, and@52# for a thor-ough description of the numerical techniques usedLORENE!. These routines, which use spectral methodsiterative techniques to solve systems of coupled nonlinmultidimensional PDEs, have been used widely to study qsiequilibrium sequences of binary NS systems in Newton@53,54# and CF gravity~TG @26,55#!, as well as a number oother fluid configurations. The CF quasiequilibrium solutioof Gourgoulhonet al. @26# and TG have been used as tinitial configurations for the most recent full GR calculatioof Shibata, Taniguchi, and Uryu,@43#, and will be used inthis work as well.

Our work here represents the first time that sphericalordinates and spectral methods have been used to studdynamical evolution of binary NS systems in any gravitional formalism, including Newtonian gravity. It has lonbeen known that these techniques are ideal for describoth binary systems with large separations as well asmerger remnants formed during the coalescence, sincespherical coordinates correspond much more naturally tometric fields than rectangular coordinates can. Traditionarectangular grids have been used anyway, both becauseare more widespread throughout computational fluid dynaics, but also because they are viewed as more robust. Itoften been assumed that spherical coordinate field solmay fail to calculate fields properly during the merger, whthe matter can be described neither as two spheroids noone, but rather some combination of the two, with mass lstreams and other phenomena confusing the picture.show here, however, that spectral methods can be usedcessfully in this regime, taking advantage of the mudomain techniques of LORENE. This allows us to take avantage of the many advantages inherent to specmethods: improved speed, vastly improved compumemory efficiency, and a coordinate system which allowsa natural treatment of the exact boundary conditions.

The code we have developed to perform 3D, relativisnumerical hydrodynamic evolutions is well-suited for tstudy of a number of physical systems. While we focus hon merging binary NS, our code is capable of evolvingsentially any binary or single-body relativistic fluid configration. These include collapsing stellar cores and supermsive stars, and rapidly rotating fluid configurations. Moducurrently exist to handle a number of physically-motivatEOS, including models with phase transitions and boso

1http://www.lorene.obspm.fr

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condensates, and we plan to extend our studies to incappropriate treatments of these conditions in the future.

Our paper is structured as follows. In Sec. II, we summrize the theoretical basis for our new relativistic Lagrangcode, describing in turn the numerical methods usedimplement both the dynamical equations of the CF formism in SPH and the details of our spectral methods fisolver. In Sec. III, we turn to the computational aspectsthe code relevant to coalescing binary systems. We detaichoices we have made with regard to SPH discretization,techniques used to convert between particle-based and strally decomposed descriptions of various quantities, andcoordinate transformations implemented to describe binsystems, merging systems, and the resulting remnantsSec. IV, we report the results of several test calculatioincluding two well-known exact CF solutions as well as tevolution of quasiequilibrium configurations in the absenof dissipative effects, producing circular orbits. In Sec. V, wshow our results from the calculation of a full NS binacoalescence started from the innermost quasiequilibrconfiguration, and followed through the merger and the fmation of a merger remnant. We compare our results to pvious work in the field, including full GR calculations usinthe same initial condition. Finally, in Sec. VI, we summariour results and lay out some of the many classesproblems in relativistic hydrodynamics where our code mprove useful.

II. NUMERICAL METHODS

The CF approximation assumes that the spatial part ofGR metric is equal to the flat-space form, multiplied byconformal factor which varies with space and time. SettG5c51, as we will do throughout this paper, the CF mettakes the form

ds252~N22NiNi !dt222Nidtdxi1A2f i j dxidxj , ~2!

whereN is the lapse function,Ni is the shift vector, andAwill be referred to here as the conformal factor~see Table Ifor a comparison of our notation to those of@26,45,47,48#,which are all based on the same exact assumptions!. Theflat-space three-metric is denoted byf i j . We follow the stan-dard notation for relativistic tensors, denoting spatial thrvectors with Latin indices, and relativistic four-vectors wiGreek indices. While the CF approximation cannot repduce the exact GR solution for an arbitrary matter configration, it is exact for spherically symmetric systems, ayields field solutions which agree with those calculated usfull GR to within a few percent for many systems of intere@56#.

As is typical for any relativistic calculation utilizing apolytropic EOS, there is a single physical scale which dfines the units for the problem. We choose to define this scfor single-body test calculations~Secs. IV A and IV B! bydividing all mass, length, and time-based quantities bygravitational~ADM ! massM0 of the object. For all binarycalculations, we scale our results by the ‘‘chirp mass’’ of tsystem,M ch[m3/5Mt

2/5, whereMt is the total gravitationalmass of the system at large separation, andm[M1M2 /Mt isthe reduced~gravitational! mass. This quantity is expected tbe the most directly measurable physical parameter dedufrom any GW observation~see, e.g.,@57#!. For a binary sys-

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TABLE I. A comparison of our notation for various relativistic quantities to previous works using theformalism: @26,45,47,48#. For those cases where no unique terminology was defined, we give the simequivalent algebraic form. We also list the equation in this paper where the quantity is defined.

Quantity Here Gourgoulhon Oechslin Shibata Wilson See Eq

Lapse N N a a a ~2!

Shift Ni Ni 2b i 2b i 2b i ~2!

Conf. Fact. A A c2 c2 f2 ~2!

Rest dens. r* GnA3r r* r* Df6 ~5!

Lorentz Fact. gn Gn au0 au0 W ~6!

Velocity v i NUi1Ni v i v i Vi ~7!

Spec. Momentum uiwi ui ui

Si /(Df6) ~10!

Enthalpy h h w 11Ge h ~11!

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tem consisting of twoM051.4 M ( NS, the chirp mass isMch51.22 M ( , which yields characteristic distance antime scalesR51.80 km andT56.0031026 sec, respec-tively. To compare our results with the equivalent relativismodel ~run M1414! of @43#, who use the total gravitationamass of the binary at large separation as their unit, onedivide our quoted masses and radii by a factor of 21.2

52.30. To convert our time evolution results into the timunits used in@43#, which are defined in terms of the initiabinary orbital period, divide our time units by a factor of 44~their unit Pt5052.66 ms for two NS each withM051.4 M ().

A. CF smoothed particle hydrodynamics

In what follows, we will assume that the stress-enetensor of the fluid is that of a perfect fluid, with

Tmn5~r1re1P!umun1Pgmn , ~3!

where r is the rest-mass density,e is the specific internaenergy,P the fluid pressure, andum the four-velocity of thefluid. For our initial data, we assume a polytropic EOSP5krG, with constant values fork and the adiabatic indexG,and throughout the calculation we assume that the evoluis adiabatic, such thatP5(G21)re. The maximum infallvelocity of the two stars relative to each other during odynamical calculations was found to bev in50.06c, whereasthe sound speed at the center of the NS is initiallycs50.85c. Since the sound speed for our EOS dependsdensity such thatcs}r1/2, we expect that the only supersonfluid flows will occur in the very tenuous outer regions of tNS. During the merger, we do expect that low-density mafrom the inner edge of each NS will cross over to the bincompanion at high relative velocity (dv'0.3c), but the ve-locity field is dominated by the circular motion, rather thanconverging flow. All motion within the cores of the Nshould remain strongly subsonic throughout the merger pcess.

Using our CF metric and stress-energy tensor, Eqs.~2!and ~3!, the Lagrangian continuity equation is given by

dr*dt

1r* ¹ivi50, ~4!

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r* [Nu0A3r5gnA3r, ~5!

the Lorentz factor of the fluid is defined as

gn[Nu0, ~6!

and the physical velocity is given by

v i[ui /u05Ni1ui

A2u0. ~7!

Note that u0 is the timelike element of the covarian4-velocity; all other numerical superscripts refer to expnents. Following the SPH prescription, this conservatform of the continuity equation allows us to define a setparticles, each of which has a fixed massma and a welldefined velocity given bydxi /dt5v i . For each particle ‘‘a,’’we also define a ‘‘smoothing length,’’ha , which representsthe physical size of the particle. SPH particles do not hadelta-function density profiles; rather, each particle repsents a spherically symmetric density distribution centerethe particle’s positionxWa with compact support. This densitdistribution, determined by our chosen form of the smooing kernel function, is second-order differentiable, and droto zero at a radius equal to two smoothing lengths fromparticle’s position. For each particle, we define a setneighboring particles by the condition that all particlwhose centers fall within a given particle’s compactly suported density distribution are its neighbors. To determthe proper smoothing length for each particle, we defineideal number of neighbors for each particle,NN , and we usea relaxation technique to adjustha after every time step~asdescribed in detail in, e.g., FR1 and@58#!. We note in passingthat ‘‘neighborhood’’ is not a reflexive property; we handthis through the use of a ‘‘gather-scatter’’ algorithm~see@59#for details!.

The primary advantage of the SPH method over trational grid-based codes is that fluid advection is handlednatural way, such that one can define the edge of a fldistribution without recourse to artificial ‘‘atmospheres’’ oother tricks necessary to prevent matter from bleeding ithe vacuum. Particles simply follow their trajectories in t

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fluid flow. As such, SPH is extremely computationally efcient, since all numerical resources are focused automaticon those regions containing matter. Because of this, SPHallows for high spatial resolution. The primary disadvantaof SPH compared to shock-capturing grid-based codes ithe lower resolution of shock fronts, which, as we discubelow, is unimportant here.

All hydrodynamic quantities can be defined using stadard SPH summation techniques over each particle’s nebor list ~see, e.g.,@58,60,61# for a thorough discussion!, withthe rest mass densityr* taking the place of the standarNewtonian density. Thus the rest-mass density for part‘‘a’’ can be defined via SPH summation over a set of neigboring particles denoted by ‘‘b’’ as

~r* !a5(b

mbWab , ~8!

whereWab is theW4 smoothing kernel function for a pair oparticles first introduced by Monaghan and Lattanzio@62#,used in FR,@58# and many other implementations. The mmentum equation is given by

dui

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ui[hui , ~10!

and the specific enthalpy is defined as

h[~11e1P/r!511Ge. ~11!

In this expression and throughout this paper, covariantrivatives are associated with the flat-space metric. In thesence of nonadiabatic artificial viscosity terms, the eneequation merely implies that the value ofk in the EOS re-mains constant. In our calculation, we user* and ui as thebasic hydrodynamic variables~in addition to our uniformvalue of k). To find u0, which enters into the momentumequation, we take the normalization condition for t4-velocity,umum521, and find

gn25~Nu0!2511

uiui

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which can be solved implicitly in terms of the density athe field values.

To solve the field equations of GR, we need to fix tslicing condition for timelike hypersurfaces. Following thstandard approach to the CF formalism, we find that thetrinsic curvature tensor is given in terms of the shift vector

Ki j [1

2NA2 S ¹ iNj1¹ jNi22

3f i j ¹kN

kD , ~13!

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when we assume the maximal slicing condition (TrK50).To lower the indices of the extrinsic curvature tensor aother spatially-defined quantities, we use the conformal flspace metric~i.e.,Ki j 5A4d ikd j l K

kl in Cartesian coordinateswhered i j is the Cartesian flat-space metric!. Combining themaximal slicing assumption with the Hamiltonian constrayields a pair of elliptic equations forn[ ln(N) and b[ ln(AN) ~GGTMB!, in the form

¹k¹kn54pGA2~E1S!1A2Ki j K

i j 2¹in¹ ib, ~14!

¹k¹kb54pGA2S1

3

4A2Ki j K

i j 21

2~¹in¹ in1¹ib¹ ib!,

~15!

where the matter energy density and the trace of the stenergy tensor are given by

E[gn2~rh!2P, ~16!

S[gn

221

gn2 ~E1P!13P. ~17!

Note that Eqs.~14! and ~15! are algebraically equivalent tothe field equations found in other papers on the CF formism, although most groups have typically solved the corsponding Poisson-type equations forc[AA and Nc. Fi-nally, the momentum constraint gives us the equation forshift vector,

¹ j¹jNi1

1

3¹ i¹jN

j5216pGN

hgn~E1P!ui

12NA2Ki j ¹j~3b24n!. ~18!

Since the CF formalism is time-symmetric, we can defiseveral conserved quantities. The total baryonic mass,

Mb5E r* d3x, ~19!

is automatically conserved in our SPH scheme, since wethe rest mass density to define particle masses. Theangular momentum of the system can be defined as

Ji5« i jkE r* xj ukd3x. ~20!

Finally, the ADM mass is given by

M05E rADMd3x; rADM[A5/2S E11

16pGKi j K

i j D .

~21!

It is important for numerical reasons to note that the tterms that make up the ADM mass densityrADM have dif-ferent behaviors: the contribution from the matter enedensity,

r1[A5/2E, ~22!

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B. The spectral methods field solver

The spectral methods techniques we use to solve thefield equations, Eqs.~14!, ~15! and ~18!, discussed in greadetail in @52#, provide a number of important advantages npresent in traditional grid-based approaches. First and fmost is their speed and computational efficiency. Finite dferencing schemes typically require 3D grid sizes of*106

elements@43,48#. In contrast, GGTMB show that spectrmethod field solvers can be used to construct field solutiyielding ADM masses and angular momenta convergenwithin 1024 and satisfying the virial theorem to the samlevel using only 3 grids of size 17313312. These grids,extremely small compared to those used in Cartesian mgrid solvers, result in a great increase in speed. Additionathe use of spherical coordinates allows for a more nattreatment of boundary conditions. In the approach we utaken from GGTMB and summarized here, space isscribed using spherical coordinates, split into a set of ne‘‘computational domains.’’ The outermost domain cancompactified by rewriting the field equations in terms of 1r ,allowing us to impose the exact boundary conditions atfinity, rather than ‘‘fall-off’’ boundary conditions which approximate the behavior of the fields at the edges of a recgular grid. As such, we avoid the classic tradeoff betweegrid with large grid spacing, which yields accurate boundconditions but poor spatial resolution of the matter sourand a grid with small spacing, and the opposite concern

Combining the LORENE methods with particle-basSPH requires some small but significant changes fromprevious approach described in detail in Sec. IV AGGTMB. As in that work, we construct a set of three coputational domains around each star to evaluate theequations. The innermost domain has a spheroidal topolwith a boundary roughly corresponding to the SPH particonfiguration’s surface, as described in Sec. III A. The ottwo domains cover successively larger regions in radii, wthe outermost domain extending out to spatial infinity,shown in Fig. 1. The field equations are solved in eachmain and the global solution is obtained by matchingfunction and its first radial derivative at each boundary. Apropriate boundary conditions at radial infinity are also iposed. All fields can be described in one of two complemtary representations, either in terms of their coefficientsthe spectral decomposition, or by means of their valuesset of ‘‘collocation points.’’ The coordinate representatiofor these points are defined such that the origin of the sysdescribing each NS is located at the position of the stmaximum density, The collocation points are spaced equin both sinu andf, as required by the angular component

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te--

sto

ti-y,ale,-

ed

-

n-ay,

ef-ldy,

er

hs-

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the spectral decomposition. The radii of the points are demined from the collocation points of the Chebyshev polynmial expansion used for the radial coordinate, as describeBGM and GGTMB. For all calculations described in thpaper, these domains consisted of a 17313312 grid, interms of radial, latitudinal, and longitudinal directions, rspectively, an acceptable trade-off between speed and aracy, based on the description above.

A key feature of the LORENE libraries is their handlinof binary systems in a straightforward and natural way. Tinvolves ‘‘splitting’’ the source terms of the field equationinto two distinct components, each of which is centeredone of the stars in the binary. Since the field equations incase are nonlinear the split cannot be performed uniquthe fields present around one star cannot be determinedits source terms only. Rather, this method seeks to minimthe contribution of one star to the fields around the otherpractice, both field variables and hydrodynamic source tecan be broken down, as shown in Eqs.~79!–~87! ofGGTMB, such that

n5n^1&1n^2&5n^1&1n^2→1&5n^1→2&1n^2& , etc. ~24!

where quantities labeled1& and ^2→1& are defined at thecollocation points of star 1, and2& and ^1→2& at those ofstar 2. The autopotentials of each star,n^1& and n^2& , areprimarily generated by matter from the star itself, while t‘‘comp-potentials,’’n^2→1& and n^1→2& are primarily gener-ated by the other star. It is this conversion of fields betwethe two sets of coordinates which represents the greaamount of numerical effort during a calculation. In practicwe attempt to minimize the magnitude of the compotentials since they are centered around the other starnot as well described by spherical coordinates. The detadescription of how these quantities are defined and calated can be found in Sec. IV C of GGTMB.

III. NUMERICAL TECHNIQUES FOR COALESCINGBINARIES

Integrating the LORENE library routines into an SPHbased Lagrangian code introduces a number of rather sunumerical issues. The simplest of these is deciding the sh

FIG. 1. Radial domains used to evaluate the field equationthe CF method. The boundaries of the inner two domains are shas lattices, with all collocation points within these domains shoas well. The outermost domain, which extends to spatial infinitynot shown.

6-6

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nswinfo

b

oivftiophi

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of the innermost computational domain’s surface. The clothe boundary of the innermost domain lies to the surfacethe fluid, the more Gibbs effect errors are minimized, so loas the surface is sufficiently smooth and convex everywhGibbs effects, which are common to all spectral decomption techniques, result from the attempt to describe a nonferentiable density distribution as a weighted sum of smofunctions; they are relevant here when we attempt a spedecomposition of a region of space where the density drto zero within the boundary~see BGM for more details!.Unfortunately, the smoothness and convexity conditionsnot necessarily apply to the region in space where the Sdensity is nonzero, since a single SPH particle being sfrom the star can greatly affect the nonzero density boundeven though it may represent an insignificant amountmass. As a result, we have implemented an algorithmattempts to take the middle ground, defining a boundarythe innermost domain which encloses as much of the maspossible, in such a way that smoothness and convexityguaranteed. The second domain is bounded by a spheradius twice that of the outermost point of the first domaand a third domain extends to spatial infinity.

Once the configuration of the computational domainsdetermined, there are several choices which need to be mwith regard to the most accurate and efficient way to callate various terms in the evolution equations. Some hydronamic quantities, such as the rest mass densityr* , need tobe defined for each of the SPH particles. Here, we usN;100 000 SPH particles for each run, which was found tosufficient for achieving numerical convergence of the Gsignal to the ;1 –2 % level in our studies with postNewtonian SPH@35#. Other quantities, such as those appeing in the source terms for the field equations, need todefined at every point among the 17313312 spheroidalgrids of collocation points for each of the three domaiCompared to solutions computed using larger grid sizes,find that these agree with significantly larger grids to with;0.1% for the value of the shift vector, and even betterthe values of the lapse function and conformal factor.

In general, however, many quantities do not need todefined both ways, so long as a full set of thermodynamvariables is known in both representations. Details abwhich quantities are used in each representation are gbelow. Briefly, it is most efficient to perform the majority oour algebraic operations on quantities defined at collocapoints, reading in and exporting back as small a set oframeters as possible to the full set of SPH particles. Wreading quantities from SPH particles to collocation pointsrelatively quick, requiring an SPH summation at every clocation point position, the reverse process is much minvolved. To calculate the value of a quantity known in tspectral representation at SPH particle positions requiresforming a sum over all spectral coefficients with the weigappropriate to each particle position, consuming a greatof time.

A. Binary systems

To construct the initial SPH particle configuration forbinary NS evolution, we use the quasiequilibrium irrotation

12403

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models of TG, which are publicly available.2 These modelsdescribe the complete 3-dimensional structure of bothfield values and hydrodynamic quantities at every pointspace, in terms of a spectral decomposition. Specifically,take the results of their irrotational run for stars of equmass and equal compactnessGM/Rc250.14 with aG52polytropic EOS. Each star has a baryonic mass in isolagiven byGMB /Rc250.1461.

To convert these models, which are stored in the coecient basis, into spatially defined particle-based quantitwe first lay down a grid of SPH particles in a hexagonclose-packed~HCP! lattice with constant lattice spacing anparticle smoothing length. This grid is then treated as ifreflected around thez50 plane, to take advantage of thvertical symmetry inherent in the problem. Each particletreated as if it were really two particles for all SPH summtions, one located above thez50 plane, the other an equadistance below, both with half the true mass of the ‘‘reaparticle. Since the vertical symmetry is enforced onparticle-based level, we solve all our field equations onlyvertical angles 0,u,p/2, and reflect the solutions for apoints below the plane. The mass of each particle is initiaset to be proportional to the density at the particle’s positaccording to the quasiequilibrium model.

Next, using an iterative process, we calculate the Sexpression for the density of each particle, using Eq.~8!, andadjust each particle’s mass so that the SPH density matthe proper value from the initial model, stopping once tmaximum difference for any particle is less than 0.25%the star’s central density. Particle velocities are assignematch the quasiequilibrium model’s velocity field, as areother thermodynamic variables. Finally, we advance thelocities by half a time-step, using the same methods asstandard iteration loop~see below!, since we use a secondorder accurate leapfrog algorithm~described in detail in@58#!.

During each iteration, the first step is always the calcution of each particle’s neighbor list, and the associated Sforms for the density and other hydrodynamic terms. Onthis is done, we perform a Euclidean transformation oncoordinates into a new frame~denoted by primed quantities!,whose origin is defined to be the system center-of-mass, wthe center-of-mass of each star lying on thex axis, makingsure to transform the positions, velocities, and acceleratfor each particle. In terms of the inertial frame coordinatesthe NS centers-of-mass,xW1 and xW2, the transformation isgiven by

f8[tan21y22y1

x22x1, ~25!

xCoM[x11x2

2; yCoM[

y11y2

2, ~26!

x8~x,y![~x2xCoM!cosf81~y2yCoM!sinf8,~27!

y8~x,y![~y2yCoM!cosf82~x2xCoM!sinf8,~28!

2http://www.lorene.obspm.fr/data/

6-7

is for

er,

g

ding


FIG. 2. A pictorial demonstration of the coordinate transformations described in Secs. III A and III B; the particle configurationdemonstration purposes only, and was not taken from our calculations. In the upper left panel, we show a NS-NS binary in thex2y ~inertial!frame, as well as thex82y8 frame, defined so that the centers-of-mass of the stars~crosses! lie equidistant from the origin on thex8 axis@seeEqs.~25!–~29!#. In the upper right, we see the same system in thex82y8 frame. The anglesF1 andF2 are determined from the respectivmoment of inertia tensors using Eq.~30!. The best fit ellipsoidal configurations, determined from Eqs.~31!–~38! are shown around each staaligning with theXq82Yq8 axes, determined from Eqs.~59! and~60!. In the bottom left, we show isocontours for the radial functionsf 1 andf 2, defined by Eq.~58!, as well as the boundary of the overlap region~heavy solid line!. Finally, in the bottom right, we show the rescalinof the surface function for very close configurations, showing only star 2 for clarity. Here, the binary separation isr /Mch56.0, implying themaximum extent of the surface of star 2 is tox8/Mch5r /4Mch51.5. We linearly rescale the surface function, as well as the corresponvalues off 2, for all points withx8.0.

oo

sr-io,

tiotirin

ro,dsith

g-ity

otr isideasend-the

ri-en-the

xW1,28 5S 6R

2,0,0D ; R[A~x22x1!21~y22y1!2. ~29!

This transformation is shown in the upper left panelFig. 2. The next computational task is defining the shapethe innermost computational domains, calculated for setrays equally spaced inu andf, as measured from the centeof-mass of each star. We denote these surface functr (uq ,fq), where the ‘‘q’’ subscript, taking the value 1 or 2refers to angles measured in the primed frame of Eqs.~25!–~29! outward from the center-of-mass of starq. These sur-faces are used to determine the position of the collocapoints, which in turn are used as the basis for the enspectral decomposition. While it is easy to find the po

12403

ff

of

ns

net

along any ray where the SPH density drops to zer SPH(uq ,fq), we have found that such a set of points leato unacceptable results from the field solver, especially wregard to the convexity of the matter distribution. If we inore the density contributions of all particles whose densfalls below some fixed value, sayrmin50.0001, the resultingsurface functionr SPH(uq ,fq) is typically much more regu-lar, but still is not an ideal choice, since we still cannguarantee convexity. It should be noted that the field solveentirely capable of handling matter sources which lie outsthe innermost domain, although it does work best in the cwhere the surface of the matter matches the domain bouaries closely. For this reason, we restrict the shapes ofinnermost domains to triaxial ellipsoidal configurations, oented along the principal axes of the moment of inertia tsor for each star. The growing misalignment between

6-8

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stars and the~rotated! coordinate system is a well-knoweffect, reflecting the tidal lag that develops in close binaras matter tries to reconfigure itself in response to the rapchanging gravitational field@68#. Thus, for each star, we caculate the angleFq such that

Fq51

2tan21S 2I xy

I xx2I yyD , ~30!

whereI i j [(ama(xa82xq8) i(xa82xq8) j , and define our surfacefunction r (uq ,fq) such that

r ~uq ,fq!5F x2

a21

y2

b21

z2

c2G21/2

, ~31!

where

x~uq ,fq![sinuq~cosFqcosfq1sinFqsinfq!

5sinuqcos~fq2Fq!, ~32!

y~uq ,fq![sinuq~cosFqsinfq2sinFqcosfq!

5sinuqsin~fq2Fq!, ~33!

z~uq ,fq![cosuq , ~34!

anda, b, andc are the axis lengths of the ellipse.We have found it best to fix the axis ratios of the ellip

by computing the maximum extent of the surface definedr (uq ,fq) in each principal direction, such that

a05maxu r SPH~uq ,fq!• x~uq ,fq!u, ~35!

b05maxu r SPH~uq ,fq!• y~uq ,fq!u, ~36!

c05maxu r SPH~uq ,fq!• z~uq ,fq!u. ~37!

Since this prescription can lead to some particles withra.rmin falling outside of the surface, we multiply the distance in all directions by the smallest factorF0 required toencompass all SPH particles whose density is greaterrmin , typically leading to an increase in linear size of nmore than 2%, such thata5a0F0 , b5b0F0 , c5c0F0, and

F05maxF r ~uq ,fq!Ax2

a02

1y2

b02

1z2

c02G . ~38!

A typical surface fit is shown in the upper right panel of F2. The outer two domains are defined in terms of these ncoordinate systems as well, to allow us to match field valand their derivatives at the boundaries.

Using the collocation points derived from these surfacour next task is to calculate the value of the field equatsource terms at these points. Since calculatingE and S foreach SPH particle from Eqs.~16! and ~17! would require agreat deal of ultimately needless algebraic work, we docalculate the SPH expressions for these quantities at colltion points. Instead, at every collocation point, we calcul

12403

sly

y

an

.ws

s,n

ta-e

the SPH expression for the rest mass density, the ‘‘rest psure’’ P* [kr

*G , and the density-weighted average veloc

ui . Using these values, as well as the field values fromprevious iteration, we calculategn from Eq.~12!, and thenEand S at every collocation point, and proceed to solve tfield equations, using the iterative techniques describedGGTMB. Typically, we require'50 iteration loops toachieve a solution for which no field value varies by mothan 1 part in 109 from one iteration to the next.

Once we have solved the field equations, we use the stral expansion of the fields to calculate as much as we cathe terms in the force equation, before reading off the valat every particle position, which takes a significant amoof time. In practice, we use the spectral decompositioncalculate the prefactor for the pressure force term,NA3, thevector sum of the terms involving derivatives of the confomal factor and lapse function, the nine first derivatives ofshift vector, and the radiation reaction terms. While it woube faster to calculate the force term involving the derivatof the shift vector completely in the spectral basis, we hafound it to be inadvisable. This term alone is linear in tvelocity, and it is inconsistent to use an averaged velocitythis term on the RHS of the force equation to calculaterate of change of each particle’s individual velocity on tLHS. Thus, denoting terms calculated within the spectralsis and exported to particle positions by ‘‘sb,’’ and thocalculated using SPH techniques only by ‘‘SPH,’’ Eq.~9! istruly evaluated as

dui

dt5@2NA3#sbF¹i P

r*G

SPH

1FNh~gn221!

Agn¹iA2hgn¹iNG

sb

2@¹iNj #sb@ u j #SPH. ~39!

After calculating the forces on each particle and advancthe velocities by a full time step, we are still left with thtask of recomputing Eq.~7!, which relatesv i and ui . Sincethe sources for the field equations are velocity dependand we have just advancedu, we rerun the field solver withthe new values ofui , and compute Eq.~7! in the form

v i5@Ni #sb1F 1

A2hu0Gsb

@ ui #SPH. ~40!

Finally, we record the GW strains, ADM mass and systangular momentum, after rotating all positions, velocitieand accelerations back to the inertial frame by means ofinverse Euclidean transformation to the one at the beginnof the iteration.

There are several different ways to calculate the ADmass numerically, all of which should be equivalent to E~21!, yielding an important check on the code. First, we cculate the system’s ADM mass using by taking a surfaintegral at spatial infinity@see Eq.~65! of GGTMB#,

M0521

2p R ] iA1/2dSi . ~41!

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ys

th

m

ivbr

roheedeedp-s-ilthl,

orna

ins

ani

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e

-.

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ivede-r-

weto

i-

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This quantity can be compared to the particle-based expsion, with two important caveats. Since the extrinsic curture contribution to the ADM mass, Eq.~23!, does not havecompact support, there is no way to convert the integral ia sum over particles that have a different spatial extent. Sond, noting our concerns about exporting large numbersterms from the spectral representation to particle positiowe perform some of the algebra involved in determiningADM mass in the spectral representation. In the end,particle-based expression for the ADM mass becomes

M05S E r2d3xDsb

1(a

maF r1

r*G

sb

, ~42!

and we definer[r1 /r* as the ratio of ADM mass densitto rest mass density, calculated using LORENE techniqueall SPH particle positions.

In a similar fashion, there are two ways to calculatesystem angular momentum. We check the behavior ofJ inthe spectral basis@see Eq.~67! of GGTMB#,

Ji51

16pe i jk R ~xjKkl2xkK jl !dSl , ~43!

as well as the SPH expression for the angular momentu

Ji5e i jk(a

maxj uk . ~44!

Since the CF formalism is time-symmetric, the dissipateffects of gravitational radiation back reaction have toadded in by hand, just as they are for PN calculations. Pvious PN calculations of binary NS systems~FR and@39#!have typically employed the exact 2.5PN formulas intduced by@40# to describe lowest-order GW losses from tsystem. Unfortunately, those equations are not applicablCF calculations, since they are written in terms of fieldsfined in the PN approximation that differ from those definin the CF approximation. For this work, we follow the aproach of@45#, using the slow-motion approximation to etimate the radiation reaction potential of the system. Whthis method contains some obvious flaws, most obviouslyfixed spatial dependence of the radiation reaction potentiadoes yield a back reaction force which is quantitatively crect in overall magnitude. These approximations shouldaffect our calculations to a large degree. While the infvelocity of the binary prior to plunge is driven by the GWback reaction, a different regime occurs after dynamicalstability sets in. During this period, the evolution is almocompletely hydrodynamic in nature@58#. While the chosenGW back reaction treatment may affect the final massangular momentum of the resulting merger remnant, it wplay only a secondary role in the detailed evolution of tfluid configuration, since GW back reaction becomes limportant during the coalescence.

From Eq.~51! of @45#, the radiation reaction force in thslow-motion approximation is given by adding a term,

ai :reac5N2hu0¹ix, ~45!

12403

s--

oc-ofs,ee

at

e

,

eee-

-

to-

eeit-otll

-t

dll

s

to the RHS of Eq.~9!. We define the radiation reaction potentialx in a similar but slightly different way than their Eq~56!, such that

x51

5xkxlQkl

[5] , ~46!

similar to the approach taken in@48#. As they do, we definethe quadrupole moment as

Qkl5STFF E rADMxkxld3xG , ~47!

noting that our ‘‘rADM’’ corresponds to many other authorSc , or some multiple thereof. The expression ‘‘STF’’ refeto the symmetric, trace-free component of the tensor, whis linked to the gravitational radiation production in thquadrupole limit. As noted previously in our calculationsthe system’s ADM mass, we can evaluate the contributfrom r1 using standard SPH summation techniques, butsecond term can only be found using LORENE integrattechniques. Thus,

Qkl[Q11Q25STFF S (a

maraxakxa

l D 1S E d3xr2xkxl Dsb

G ,~48!

whereQ1 andQ2 reflect the contributions fromr1 andr2,respectively. Using SPH, we can take the first time derivatof the former half, so long as we ignore the Lagrangianrivative dr1 /dt, which should be essentially negligible duing our calculations. We find

~Q1!kl5STFF E r1~xkv l1xlvk!d3xG . ~49!

To calculate the rate of change of the extrinsic curvature,assume that the time variation in the tensor is due solelythe orbital motion~rather than an overall change in magntude of the tensor components in a corotating frame!, whichyields

~Q2!xx52~Q2!yy'2v~Q2!xy , ~50!

~Q2!xy'2vF ~Q2!xx2~Q2!yy

2 G , ~51!

where the factor 2v reflects the fact that the quadrupotensor makes two cycles during every orbital period. To cculate the fifth time derivative of the quadrupole tensor,use the same technique with both components of the tenfinding

Qkl[5]'16v4Qkl , ~52!

where in all cases the system’s instantaneous angular veity is calculated as the ratio of the angular momentum tomoment of inertia,

6-10

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s

amle

acaaccdoNumneaa

ru

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the

nel

are


v[

(a

ma„xa~vy!a2ya~vx!a…

(a

ma~xa21ya

2!

, ~53!

which holds exactly in the quadrupole limit for synchronizbinaries.

Calculating the GW signal and energy spectrum is a mstraightforward task which can be done after the calculais finished. We calculate the GW strain in two independpolarizations for an observer located at a distanced from thesystem perpendicular to the orbital plane from the loweorder quadrupole expressions,

dh15Qxx[2]2Qyy

[2] , ~54!

dh352Qxy[2] , ~55!

where we calculate the second time derivatives of the qurupole tensor by numerically differentiating the results froEq. ~49!. In terms of the Fourier transform of the quadrupomoment,

Qkl[2]~ f GW![E e2p i f tQkl

[2]~ t !dt, ~56!

where f GW[2 f orb is the GW frequency, the GW energyspectrum is computed as@63#

dE

d fGW5p f GW

2 S 8

15@~Qxx

[2]2Qyy[2] !21~Qxx

[2]2Qzz[2] !21~Qyy

[2]

2Qzz[2] !2#1

8

3@~Qxy

[2] !21~Qxz[2] !21~Qyz

[2] !2# D . ~57!

B. Merging systems

As the stars in the binary system spiral inward, they rea point where the density distributions begin to overlapmatter from the inner edge of each star falls onto the surfof the other. Our field solver can handle this situation, sinit does not assume that the matter sources are spatiallytinct, but the surfaces required to envelop the particles freach star would become poorer and poorer fits to the twoNoting this, we alter the approach described above in a nber of ways when the particles first cross through the inLagrangian point at the center of the system, in such a wthat the surfaces we define for each star always remsmooth and still do an acceptable job of describing the tdensity distribution in a meaningful way.

Once the binary separation shrinks sufficiently, matstreams from the inner edge of each NS toward the otherflowing along the surface of the companion. These counstreaming, low-density flows lead to the formation of a vtex sheet~FR3!. Mass transfer typically occurs before thtriaxial ellipsoidal surfaces used to define the two stars ovlap, since particles crossing from one star to another geally fall beneath the density cut used to define each surfTo account properly for the star to which each particle

12403

ent

t-

d-

hse

eis-mS.

-ryine

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bound, we declare particle ‘‘a’’ a member of the first starxa8,0 and a member of the second star ifxa8.0, at least forthe purposes of defining each star’s center of mass andlag angle, as described in Eqs.~25!–~38!.

This approach entails further alterations once the ellipsdal computational domains from each star begin to overUsing they8-axis as the dividing line between the two stais fine for determining the center-of-mass, tidal lag angand ellipsoidal surface for each star, but it is inappropriatedraw a fixed line atx850 when calculating field equationsource terms. These would induce a sudden density dfrom finite density at small negative values ofx8 to zerodensity at positivex8 values~for the first star, vice versa fothe second!, leading to large Gibbs effects. It is equally inappropriate to count one particle as a member of both stsince we would end up double-counting its density contribtion to the field equations. Instead, we introduce a weifunction f a[ f (xWa8), for each particle in the overlapping region in such a way thatf 50 at the surface of star 1,f 51 onthe surface of star 2, and 0, f ,1 in a spatially differentiableway within the overlap region. To do so, we definef 1 and f 2,the fractional squared distance outward a point lies fromcenter of each star to the surface. For the first star,

f q~x8![~Xq8!2

a21

~Yq8!2

b21

~z8!2

c2, ~58!

where

Xq8[xqcosFq1y8sinFq , ~59!

Yq8[2xqsinFq1y8cosFq , ~60!

are the rotated coordinates used to define the~tidally lagging!ellipsoidal surface of the star, andxq(x8)[x82xq8 is the dis-tance in thex8-direction from the center-of-mass of starq. Apicture of these quantities is shown in the bottom left paof Fig. 2. In terms off q , we define our overlap functionf afor each particle such that

f a5 f ~xWa8!5~12 f 1!3

~12 f 1!31~12 f 2!3. ~61!

For the first star, source terms in the field equationsevaluated as

~r* !15(a

maf aWa , ~62!

~P* !15~r* !1

~r* !11~r* !2k@~r* !11~r* !2#G,

~63!

~ ui !15

(a

maf a~ ui !aWa

(a

maf aWa

, ~64!

6-11

.


TABLE II. A summary of the runs presented in this paper.

Run Description See Sec. no

OV1 Equilibrium OV model,GM0 /Rc250.126 IV AOV2 Nonequilibrium OV model,R051.1Req IV AOV3 Same as OV2, w/relaxation drag IV ADC5 Collapsing dust cloud,M051, R055 IV BDC10 Collapsing dust cloud,M051, R0510 IV BDC50 Collapsing dust cloud,M051, R0550 IV BQC1 Quasicircular binary orbit w/o rad. reac.,

r 0 /Mch519.91IV C

QC2 Quasicircular binary orbit w/o rad. reac.,r 0 /Mch520.42

IV C

QC3 Quasicircular binary orbit w/o rad. reac.,r 0 /Mch522.98

IV C

RR1 Full binary evolution w/rad. reac.,r 0 /Mch

519.91V B

RR2 Full binary evolution w/rad. reac.,r 0 /Mch

522.98V A

es

su

rldeinlaetr

redf

e

-um

-

nolv

ointnglethe

thent oftherefore

racyndforo-er-of

cal-ib-a-s,in

ricrdi-beo-

V

where Wa is the smoothing kernel function evaluated btween the collocation point and overlapping particleComplementary expressions hold for the second star bystituting (12 f a) for f a .

This approach allows us to calculate the fields propeusing spectral methods well into the merger, but a final mofication is necessary to bring us to the point where the dsity profile of the matter can better be described by viewit as a single object. When the inner edge of one star overthe center of the other star, the density profile typically bcomes bimodal, leading to spurious results from the specexpansion. To guard against this happening, we use atively simple approach. If the surface of one ellipse extenmore than halfway fromx850 toward the center-of-mass othe other star atxq856R/2, we linearly rescale all surfacpoints that lie across they8-axis. Thus, definingxq(uq ,fq)5r (uq ,fq)sinuqcosfq in accordance with our previous notation, if the surface of either star extends to a maximvalue xq:max[max(uxqu).3R/4 @or in other words, ifmax„ur (uq ,fq)sinuqcosfq2xq8u….R/4], we adjust all pointson the other side of they8-axis, yielding

xq:new~uq ,fq!5R

21

R

4 S xq:old~uq ,fq!2R/2

xq:max~uq ,fq!2R/2Dfor all points with xq~uq ,fq!.R/2.

~65!

To evaluate the weight function, we adapt thex8-dependencecorrespondingly, such that for particles withxq.R/2

f a:new~xWq!5 f a:old~k•xWq!, ~66!

where the rescaling factork[R/21(xq2R/2)@(xq:max2R/2)/(R/4)#. The bottom right panel of Fig. 2 demonstrates this last coordinate transformation.

Eventually, the system will reach a point where it canlonger be properly described as a binary, and our field so

12403

-.b-

yi-n-gps-alla-s

er

fails to converge. Before this happens, we reach a pwhere we can describe the system as a rapidly rotating sistar, using all the methods described above for computingevolution, but now assuming that all particles comprisesame star. We have found that our results are independethe exact moment at which we make this conversion. Infollowing discussion, we will show the results for runs whewe perform the conversion at the earliest possible timewhich the field solver will converge to a solution when thmatter configuration is treated as a single star.

IV. TEST CALCULATIONS

We have performed several tests to check the accuand numerical stability of our code, for both single-star abinary systems. We have studied the behavior of the codespherically symmetric problems for which the exact field slution can be calculated semi-analytically: the OppenheimVolkov solution for a static spherical star and the collapsea pressureless dust cloud initially at rest. We have alsoculated the dynamical evolution of the binary quasiequilrium models calculated in TG, without the inclusion of rdiation reaction forces. A summary of all our calculationincluding those used for testing the code, can be foundTable II.

A. The Oppenheimer-Volkov „OV… test

Since the CF formalism is exact for spherically symmetsystems, which can always be described in isotropic coonates, it is fair to expect that any working code shouldable to reproduce the well-known Oppenheimer-Volkov slution exactly, noting that the traditional form of the Osolution needs to be rewritten into CF coordinates. Withr8[r(11e) as the total energy density~including both the restand internal energy densities!, the OV equations are typicallywritten

dm

dr54p r 2r8, ~67!

6-12

eqnedd

riooric

us

hees,ary

d

ultss.

edr as

tou-meger.ial

tost,ut

the

-ion

on

asueshasults

ardhe


dP

dr52

~r81P!~m14pPr3!

r 222mr, ~68!

dF

dr5

~m14pPr3!

r 222mr, ~69!

for an interior metric in the form

ds252e2Fdt21S 122m

rD 21

dr21 r 2dV2, ~70!

and exterior metric in the Schwarzchild form

ds25S 122M0

rD dt21S 12

2M0

rD 21

dr21 r 2dV2,

~71!

where the star’s total gravitational~ADM ! mass M0

[m( r s), andr s is the Schwarzchild coordinate radius of thstellar surface. A quick comparison with the CF metric, E~2!, shows thatN5eF inside the star, but the conversiobetweenA andF requires more care, since we have to dtermine the relationship between the two coordinate rar ( r ). The simplest way to determine the change of coornates is to solve for the values ofr s and r s , the radii of thestellar surface, using the asymptotic behavior of the extesolution ~following the same logic used in exercise 31.7@64#!. Comparing the radial and angular parts of the metwe find

A2dr25S 122M0

rD 21

dr2, ~72!

A2r 25 r 2. ~73!

Dividing and taking a square root, we find

dr

r5

dr

Ar ~ r 22M0!. ~74!

Integrating yields

ln r 1k52 ln~Ar 1Ar 22M0!, ~75!

and we find

kr5~Ar 1Ar 22M0!252r 22M012Ar ~ r 22M0!.~76!

The asymptotic behavior at infinity indicates that we mhavek54, so our final expression forr ( r ) takes the form

r 51

2~ r 2M01Ar ~ r 22M0!!. ~77!

For a star with Schwarzchild surface radiusr s , we find thatthe CF surface radius is given by

12403

.

-iii-

rf,

t

r s51

2~ r s2M01Ar s~ r s22M0!!, ~78!

the surface value for the conformal factor is

As5r

r5

2r s

r s2M01Ar ~ r 22M0!, ~79!

and the lapse function at the surface is given by

Ns5A122M0

r s

. ~80!

To solve for the interior metric, we add an equation to tOV set to take into account the different radial schemdefining a scale free conformal radius satisfying a boundcondition limr→0r 05 r whose radial behavior is given by

dr0

dr5

r 0

Ar ~ r 22m!. ~81!

This yields an expression forr 0( r ) which is defined up to anarbitrary multiplicative constantk, such thatr 05k• f ( r ),where f is determined from the mass distribution. To finr ( r ), we merely set k5r s /(r 0)s , which implies r

5 r r s /(r 0)s and determineA from Eq. ~73!.To test the code, we ran three calculations whose res

could be compared with well-known semi-analytic solutionFirst, we constructed an equilibrium model for an isolatNS, whose density profile was given by an OV solution foG52 EOS, with unit ADM mass, and a conformal radiur s56.874. The solution has a total baryonic massMb

51.066, and an areal radiusr s57.913. We started run OV1by taking this solution as an initial condition, and used ittest the overall stability of the code for equilibrium configrations. Next, we constructed a similar model with the samass and EOS, but scaled to an initial radius 10% larRun OV2 is a dynamical calculation started from this initcondition using our standard evolution code, enabling usstudy the oscillations around our equilibrium solution. Lawe took the nonequilibrium configuration from run OV2, badded a ‘‘relaxation’’ drag term of the form2ui /t relax to theRHS of the force equation, Eq.~9!, with t relax /M057.9.This provides an overdamped force for run OV3 sincedynamical time scaletD /M0[(Gr)20.5518. All three runswere followed until t/M05120, corresponding to 6.7 dynamical times for the NS, and in all cases radiation reactwas turned off.

In Fig. 3, we show the radial profiles of the lapse functiand conformal factor att/M05100 for runs A and C, alongwith the correct OV solution. It is no surprise that run A hessentially remained the same, since the initial field valwere essentially exact, but it is reassuring that run Cconverged as well toward the same solution. Indeed, resfor configurations at later times continue to converge towthe exact solution. In Fig. 4, we show the evolution of t

6-13

thitn

isld

nccachlisn

rya

om, atel-CFf t

orea

al

ach,arenshes anc-on

llyout

na-

tialr

neml,enth

ed

ch

0%ns

3butilib-


maximum central density for the three runs, as well aspredicted value from the OV solution. The maximum densfrom run A stays near this value throughout the evolutiowith small deviations which result from the unavoidable dcretization effects present in SPH; in general, SPH will yievery accurate global integrals over a mass distribution, sinumerical noise smoothes out, but demonstrates signifinoise in quantities defined for individual particles, whivary iteration to iteration as each particle’s neighboradapts to current conditions. The maximum density for ruoscillates around the proper value with a period ofT/M05112, showing no signs of systematic drift. This is veclose to the proper value for the limiting case of infinitesimradial variations,T/M05104, which we find by interpolatingfrom the values given in Table A18 of@65#, after scaling theirresults to our units.

B. Spherical dust cloud collapse

To further test the dynamical aspects of the code, we cputed the evolution of a uniform density dust cloud, i.e.spherical distribution of matter with zero pressure, starinitially from rest. This is a familiar problem from cosmoogy, and the solution is well known, but the conversion tocoordinates and the matching conditions at the surface o

FIG. 3. Conformal factorA and lapse functionN for anOppenheimer-Volkov solution with aG52 polytropic EOS, andconformal radiusr s /M056.874~solid lines!, compared to our com-puted values att/M05100 for two models: run OV1 started fromequilibrium ~crosses! and run OV3 started from a configuratio10% larger with a velocity damping term used to drive the systtoward equilibrium~triangles!. The agreement is within 1% for alparticles. Units are defined such thatG5c51. All masses, lengthsand times in the OV and dust cloud calculations are made dimsionless by scaling results against the system’s initial gravitatio~ADM ! mass. Note that the conformal radius is not equivalent toareal radius typically used in solving the OV equation.

12403

ey,-

ent

tB

l

-

d

he

dust cloud make the resulting expressions considerably mcomplicated. The complete description of the metric asfunction of time was derived for the case of both maximtime-slicing@66# and polar time-slicing@67#; it is the formercase which can be compared to our results. In this approthe field values and CF time and position variable valuescomputed by solving a set of ordinary differential equatiowhich describe the behavior of the fields in terms of tcomoving time and space variables. Using these, it isimple process of interpolation to derive the fields as futions of CF time and position. We checked our integraticode by comparing our results against the plots in@66#, find-ing perfect agreement.

To construct our initial configuration, we used essentiathe same techniques described above. Particles were laidin an HCP lattice, with masses set proportional to the alytically known value ofr* (r ), which we derive as follows.Note thatr* varies with radius; it isr(r ) that is initiallyuniform.

The total mass of the cloud was set to unity, and the iniradius in comoving coordinates tor s , the parameter used foall figures in @66#. The initial metric in comoving coordi-nates, for a cloud with unit mass and comoving radiusr s , isgiven by

ds252dt21a~t!2~dx21sin2xdV2!, ~82!

n-ale

FIG. 4. Evolution of the maximum density for three runs bason the OV solution described in Fig. 3. Run OV1~solid line! wasstarted from equilibrium, and shows only small variations whiresult from typical uncertainties in SPH summation. Run OV2~dot-ted line! used the same EOS, but was started from a radius 1smaller than the equilibrium value, showing sinusoidal oscillatiowith a period ofT/M05112 and no systematic drift. Run OV~dashed line! was started from the same configuration as run B,with an overdamped drag term to force the system toward equrium, converging rapidly toward the proper value ofM0

2rmax

50.006.

6-14

m

qn,

ha

aera

umr

ev

init

llyxaheed

ourof

lsoch

n is

atiths a

haso-theud.ersgezon

aax-ate

y aula-al

nale an

eedithlu-


a~t50![2/~sinxs!3, ~83!

r s[2/~sinxs!2, ~84!

with the exterior metric described by the Schwarzchild forEq. ~71!, with r 5a(t50)sinx. In terms ofxs , the CF ra-dius is given by

r s51

2~ r s2M01Ar s~r s22M0!!

51

2 S 2

sin2xs

211A 4

sin4xs

24

sin2xsD

51

2

11cosxs

12cosxs. ~85!

Equating interior metric coefficients, we setAdr5adx andAr5a sinx, dividing and then integrating to find

r}S 12cosx

11cosx D 1/2

, ~86!

51

2 S 12cosx

11cosx D 1/2S 11cosxs

12cosxsD 3/2

, ~87!

where the proportionality constant is determined from E~85!. The initial conformal factor can now be written dowsince

A5a sinx

r5

4~11cosx!

~11cosxs!3

, ~88!

and we can simplify the resulting expression by noting tcosxs5(2rs21)/(2rs11) and cosx5(12r2/2r s

3)/(11r 2/2r s

3). Since the matter starts from rest, we know thui50 initially, and thusgn51 everywhere, and we find thinitial rest-mass density profile is given as a function ofdius by

r* 5gnA3S 3M

4p r s3D 5

3

4pr 3 S 111

2r s

11r 2

2r s3D 3

. ~89!

Once the initial particle configuration was set, we calclated the dynamical evolution of the system using the satechniques described above for the OV case. Since the psureless material had no outwardly directed force, the intable fate of the system was collapse to a BH.

To allow for direct comparison with the figures shown@66#, we computed the evolution of a dust cloud with unADM mass and an initial areal radiusr s510, just as theydid. In Fig. 5, we show the evolution of a set of equaspaced Lagrangian tracer particles compared to the esemianalytic solution we computed. This corresponds to tFigs. 8, 9, with two slight differences. Our figures are plott

12403

,

.

t

t

-

-e

es-i-

ctir

in CF coordinates, rather than comoving coordinates, andtracers are equally-spaced in radius, not in incrementsenclosed mass. For comparison with their Fig. 9, we ashow the more familiar areal radius of the cloud, whiroughly satisfies the relationr s' r s21 initially. We see theagreement between our calculation and the exact solutiovery good throughout the evolution, up untilt/M0'38,where a slight discrepancy begins to develop, primarilythe surface of the cloud. This time corresponds closely wthe formation of an event horizon for the system, shown adotted line, starting at the center att/M0538 and movingoutward, reaching the surface of the cloud att/M0543.4,shown as a horizontal line. This late time discrepancytwo sources. The first is that SPH, which by definition prduces a differentiable density field, cannot reproducestep-function density drop at the surface of the dust cloThis explains to a large degree why the outermost tracdiverge furthest from their exact path. In addition, the larfield values and gradients found around the event horipresent a challenge for our field solver. We typically seedramatic increase during this period in the number of relation iterations required to converge to a sufficiently accursolution.

In Fig. 6, we show the evolution of the conformal factorA

FIG. 5. A comparison between the actual paths traced out bset of equally-spaced Lagrangian tracers in run DC10, our calction of a collapsing dust cloud with unit ADM mass and initial are

radius r s510, ~dashed lines! and the exact semianalytical solutioof @66#, shown as solid lines. All radii are shown here in conformcoordinates. We see excellent agreement up until the point wherevent horizon forms at the center of the cloud, att/M0538. Theevent horizon moves outward~heavy dotted line!, eventually en-closing the entire cloud att/M0543.4, shown as a horizontal linin the figure. At this point, when the matter can be properly definas a BH, our field solver stops converging. For comparison wFig. 9 of @66#, we also show, as a long-dashed line, the exact so

tion for the cloud’s surface in comoving~areal! radii, r s(t).

6-15

n

ti

pe.thit

t

o

veo

oo

iet

ee-ortivlare

eingagectu-Mers.of

sult-ely.-ely.undner

talwnse-bital

beorhepe-thenoforanas

icalia-dthe

esi-antas

hismsAsthatthatroxi-ur-B,ia-ot

ing

an

lues

ctunt


and lapse functionN for the dust cloud, again in comparisoto the exact solution, at timest/M050, 20, 30, and 40. Wesee again that our code can reproduce the proper solupastt/M0530, but byt/M0540 shows nontrivial deviationsfrom the proper solution. Att/M0540, the relative error inthe metric fields and position of the Lagrange radii is aproximately 4%, growing to roughly 15% by the time thevent horizon encompasses the entire mass distributionall cases, the deviations from the correct solution takesame form: our computed field values are closer to un~and the previous timestep’s solution! than we would expecfrom the semianalytic solution.

This behavior was confirmed by testing the collapsedust clouds with initial areal radii ofr s55 and r s550. Inboth cases, we find extremely accurate results until the ehorizon forms, at which point our accuracy degrades tnoticeable extent. The effect seems to depend primarilythe formation of an event horizon in the system, and notthe relative change of the density or various field quantitWe have concluded that the relaxation techniques used incode have difficulty in their current form in handling thsteep spatial gradients in the shift function near the evhorizon ~see Fig. 2 of@66#!, and we are working on techniques to better handle this situation. Of particular imptance is altering the relaxation parameters of the iterascheme used by the field solver in the presence of thesefield values and gradients near the event horizon, to corfor the systematic drift away from the expected values.

FIG. 6. The evolution of the conformal factorA and Lapse func-tion N for the dust cloud in run DC10, compared to the exact sotion. We see, att/M050, 20, 30, and 40, the SPH particle valufor the lapse~the four curves with values less than 1.0! and confor-mal factor~values greater than 1.0!, shown as points, and the exasolutions, shown as dashed lines. The agreement is good upt/M0530, but att/M0540 we see some quantitative disagreemesince the field solver breaks down as we near the point wherecloud collapses completely into a BH.

12403

on

-

Iney

f

ntanns.he

nt

-e

rgect

C. Circular orbits of quasiequilibrium configurations

Finally, to test out the overall stability of the code, wevolved several quasiequilibrium binary configurations usour code, but without adding in the radiation reaction drterms. Since the CF formalism is time-symmetric, we expthat any stable quasiequilibrium model should yield a circlar orbit, maintaining a constant binary separation, ADmass, and internal rotation profile, among other parametHere, we chose models at an initial binary separationr 0 /Mch519.91, 20.42, and 22.97 from theM /R50.14, 0.14 equal-mass sequence of TG, denoting the reing calculations as runs QC1, QC2, and QC3, respectivFor NS with ADM masses ofM051.4 M ( , these separations correspond to 33.64, 34.51, and 38.81 km, respectivThe innermost of these represented the limiting case fofor this sequence just before formation of a cusp at the inedge of the two NS.

As a first check of our code, we compared the orbifrequency determined from our dynamical runs to the knovalue for each model taken from the quasiequilibriumquence. We found excellent agreement between the orperiods computed from our runs,T/Mch5422.6, 438.2, and506.4, and those determined by TG,T/Mch5422.8, 438.0,and 519.7, for runs QC1, QC2, and QC3, respectively.

There are several conserved quantities which shouldrespected in the time-symmetric CF formalism, allowing ffurther code tests. In Fig. 7, we show the evolution of tbinary separation for the runs. For each run, the orbitalriod is shown with tick marks. We see in each case thatorbit is stable, with variations in the binary separation ofmore than 4% during the first two orbits. The time scalethe radial variations is similar to the orbital period, but notexact match; this reflects both the effects of GR as wellthe slight degree of time-asymmetry present in the numerimplementation of the CF formalism. We note that the devtions from circularity were largest for run QC3, which hathe largest binary separation. We believe this results fromlarger relative magnitude of the spurious initial velocitithat result from deviations away from equilibrium in the intial condition; while these terms are of essentially constmagnitude in all three runs, the equilibrium velocity field hthe smallest magnitude at the largest separation.

Run QC1 was started from the innermost point along tbinary NS equilibrium sequence, and the binary perforthree complete orbits with no sign of plunging behavior.such, these results can be taken as the first direct proofthe entire equilibrium sequence is stable, and suggestthese configurations should be reasonably accurate appmations to the true physical state of merging binaries. Fther evidence of this claim is presented below in Sec. Vwhen we describe the results of our calculations with radtion reaction effects included. Note that this result is nunexpected given the absence of a turning point~minimumof ADM mass and total angular momentum! along this irro-tational equilibrium sequence. Indeed, while such a turnpoint along an equilibrium sequence ofcorotating binariesmarks the onset ofsecular instability, a truedynamical in-stability is usually associated with a turning point along

-

ntilt,he

6-16

reelibsllyf-

sell

vethol,aroftet

c-h

italtotedese

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ed

iticit

bitoier

tedhetuml

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ularM

, buttic


irrotational sequence@55,68,69#.While this result may appear at first glance to disag

with those of@70#, who find that the ISCO occurs at a greatbinary separation than the termination point of an equirium sequence, we believe that the difference is purelymantic. We define an initial configuration to be dynamicastable if in the absence of dissipative radiation reaction efects, the circular orbit remains stable~no merger occurs!when evolved forward in time. In@70#, a configuration isdescribed as within the ISCO if in full GR~i.e., includingdissipative radiation reaction effects!, a binary starts merging~the surfaces of the two NS come into contact! within oneorbit when evolved forward in time. Clearly, using thedefinitions, the same initial configuration can be dynamicastable ~in the time-symmetric CF sense! at a separationwithin the ‘‘ISCO’’ as defined by@70#.

In order to estimate how well our code respects conserquantities, we show in Figs. 8 and 9 the evolution of bothADM mass and system angular momentum, calculated frthe SPH expressions, Eqs.~42! and ~44!, and the spectrabasis forms, Eqs.~41! and ~43!, respectively. In the formerwe see that the SPH expression for the ADM mass is remably constant over time, with only very minor deviationsrelative magnitude less than a tenth of a percent. The sysangular momentum varies more, but is still conservedwithin 1%, with no sign of a systematic drift in either diretion. We note that much of the variation is correlated with t

FIG. 7. Binary separation as a function of time for the binarin runs QC1, QC2, and QC3, using initial data generated by@27#,evolved forward in time with radiation reaction terms neglectThe calculations use binaries with initial separations ofr 0 /Mch

519.91, 20.42, and 22.97, respectively. Full orbital periods, wT/Mch5422.8, 438.0, and 519.7, respectively, are shown with tmarks. We see that the resulting orbits are nearly circular, wchanges in separation of no more than 4% over the first two orThis is the strongest available evidence that the innermost palong this equilibrium sequence is actually stable against merg

12403

er-e-

y

dem

k-

mo

e

deviations in the binary separation, oscillating on the orbtime scale. We see much more variation on an iterationiteration basis when we look at the same quantities compuusing the spectral basis. This is hardly surprising, since th

s

.

hkhs.nt.

FIG. 8. ADM mass and total angular momentum, calculafrom Eqs.~42! and~44!, runs QC1, QC2, and QC3. We see that tADM mass is conserved almost exactly, and the angular momento within 1%, with the variation occurring primarily on the orbitatime scale.

FIG. 9. ADM mass and total angular momentum, calculatedthe spectral basis from Eqs.~41! and ~43!, for the runs shown inFig. 7. We see roughly the same amount of variation in the angmomentum as was found for SPH summation in Fig. 8. The ADmass shows considerably more variation than the SPH versionremains well within 1% of the original value with no systemadrift.

6-17

aa

yo

eth

f a

a-ehilityni-oed1

yardwa

isome

on

atsacreonyethckasonyior-to

tinsioy

s-nigtlyca

r.

belar-

f athewe

ary

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ns

eldale.ethe

thery,athiralve

m-nal,

ithn-3,

e inly


values are computed at the end of a relaxation routine,we expect some degree of variation on a step-by-step bdepending on the exact phase space path traced out biterative solution. All the same, we see that the angular mmentum is conserved to roughly the same level in the sptral basis as it was when computed using SPH, and whilevariation in the ADM mass is larger, we see no sign osystematic drift.

V. DYNAMICAL CALCULATIONS

While dynamical calculations including the effects of rdiation reaction are the only way to study the coalescencbinary NS systems, they have several additional uses ware often overlooked. Of particular importance is the abito determine the validity of quasiequilibrium models as itial conditions for dynamical calculations, regardlesswhether CF or full GR gravity is used. Thus, we computtwo dynamical runs including radiation reaction. Run RRwas started from the cusp point of the sequence~the sameinitial configuration used in run QC1!, and was used to studthe details of the coalescence process. Run RR2 was stfrom a larger separation~the same initial configuration usein run QC3!, and was used to test out the deviationsexpect from quasiequilibrium prior to reaching the termintion point along the equilibrium sequence.

A. Stable regime

The evolution of the binary separation for run RR2shown in Fig. 10. The dotted curve, showing the result frthe calculation, does not have the behavior one wouldpect. Notably, after the binary separation decreases mtonically until reachingr /Mch521.2 at t/Mch5700, thesystem turns around and expands briefly back to a separof r /Mch521.5 att/Mch5950 before shrinking again. Thidoes not reflect any inherent problem in our radiation retion formalism, since oscillations in the binary orbit wefound for calculations which ignored all radiation reactieffects. In fact, if we ‘‘correct’’ the binary separation blooking at the difference in separation at equivalent timbetween runs RR2 and QC3, which were started fromsame initial configuration, with and without radiation bareaction terms, we see a pattern of monotonic decreshown as a solid line. This seems to indicate that deviatifrom circularity in the orbit of the quasiequilibrium binarconfigurations represent a systematic effect in the evolut

The ‘‘corrected’’ infall curve shows clear signs of an obital eccentricity with a time scale roughly correspondingthe orbital period. This is a natural consequence of starout from an initial condition with zero infall velocity, and habeen seen before in virtually every PN and CF calculat~FR @39,48#!. Its origins are clear: the framework used bGGTMB to construct quasiequilibrium initial conditions asumes a helical Killing vector exists, which enforces an itial circularity in the orbit, rather than the proper infallintrajectory. If calculations could be started from sufficienlarge separations, GW emission would cause the orbit tocularize, but the process works slowly, and breaks downthe binary makes the transition toward a dynamical merge

12403

ndsisthe-

c-e

ofch

f

ted

e-

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ion

-

se

e,s

n.

g

n

-

ir-sIt

is fair to say that no computational scheme can currentlytrusted to remain stable over the period required to circuize the orbit.

B. Coalescence

We have also computed the full dynamical evolution obinary system started from the innermost point alongquasiequilibrium sequence, denoted run RR1. In Fig. 11,show density contours from the system during the binphase, which lasted untilt/Mch5883. The contours areequally spaced logarithmically, two per decade, ranging frdensity values ofM ch

2 r* 51026.521021. We recognize afamiliar pattern from past PN and relativistic calculatio~FR @39,43,48#!, including the development of tidal lagangles as the time scale on which the gravitational fievolves becomes comparable to the dynamical time scThis gives rise to an ‘‘off-axis’’ collision, as matter from thinner portion of each star runs along the trailing edge ofother, forming a turbulent vortex sheet~see FR3!. Some frac-tion of the mass in these flows eventually crosses throughouter Lagrange point on the opposite side of the binaforming the very low-density spiral arm structures seent/Mch5850. In contrast to Newtonian binaries, in whicangular momentum transfer outward leads to massive sparm formation, the very low-density arms formed here ha

FIG. 10. Binary separation over time for run RR2, started froan initial separation ofr 0 /Mch522.97, the same initial configuration that was used for run QC3. The dotted line shows the origi‘‘uncorrected’’ result, including a separation increase fromt/Mch

5700–950. This is primarily due to oscillations associated wnumerical noise and deviations from equilibrium in the initial cofiguration. Correcting for deviations from circularity in run QCwhich ignored radiation reaction~dashed curve!, yields the ‘‘cor-rected’’ result, shown as a solid line. We see monotonic decreasthe separation over time, with an ellipticity induced by our initialcircular orbit.

6-18

vahe

rn

beno

mea

onisrgx

arthtatoToig

rro

nin

ip

ex-

lar5,ushat

e in--gelyma-gersshisula-

nu-

ac-

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mn

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d

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hedtheep-

ent


velocities much smaller than the escape velocity, and themajority of the mass remains gravitationally bound to tsystem.

At t/Mch5883, shortly before the binary field solvefails to converge, we take our matter and field configuratioand transform them into the single-star description descriin Sec. III B. In Fig. 12, we show the particle configuratioat t/Mch5883, which can be described as a bar with twlow-density arms trailing off the edges. While it may seeinappropriate at first to describe the configuration as anlipse, we note that the low-density contours, showndashed lines, do form a much more elliptical pattern thanmight at first expect. In interpreting SPH particle plots, itimportant to remember that low-density particles have lasmoothing lengths, implying that the matter distribution etends well beyond the apparent sharp edge. The boundof the innermost computational domains are shown infigure as heavy solid lines, for both binary and single-sconfigurations. We see that they align rather well, failingoverlap only in low-density regions near the boundary.confirm the validity of the switch, we compare the field slutions for the particles before and after the transition. In F13, we show the relative change in the lapse functionN ~toppanel! and conformal factorA ~bottom panel!, as a functionof thex-coordinate. We see that in all cases the relative eis ,1%.

The evolution of the single-body configuration is showin Fig. 14. We see a strong pattern of differential rotationthe merger remnant, which slowly relaxes from a very ell

FIG. 11. The evolution of the matter run RR1, started from joutside the separation where a cusp develops. We followed thelution through the merger and formation of a remnant. Density ctours are logarithmically spaced, two per decade, ranging fM ch

2 r* 51026.521021. We see the development of significatidal lag angles att/Mch5500, followed by an ‘‘off-center’’ colli-sion. This process leads to the formation of a vortex sheet ansmall amount of matter ejection byt/Mch5850 from matter run-ning along the surface of the other NS.

12403

st

sd

l-se

e-ieser

o-.

r

-

tical shape toward a more spheroidal one. This is to bepected, as our choice of EOS withG52.0 should not be ableto support a long term ellipsoidal deformation@58#. Byt/Mch51220, the remnant has relaxed to a nearly circuprofile, but the differential rotation, as shown in Fig. 1persists. The latter will dissipate slowly on either the viscoor magnetic braking time scales, beyond the scope of wwe can reasonably calculate@71–73#. Differential rotation isexpected to stabilize the star against gravitational collapsthe short-term@74,75#. Quantitatively accurate determinations of the rotational velocity profile in terms of the parameters of the initial system are likely to be crucial for makinprediction as to which systems will or will not collapspromptly to BHs, especially since these systems will likehave very large masses. As we see in Fig. 16, the vastjority of the rest mass of the system ends up in the merremnant itself, with a fraction of a percent of the total maforming a low-density, bound halo around the remnant. Tseems to be the consensus from PN and relativistic calctions of irrotational binaries~FR3 @43#! and even PN andrelativistic calculations of synchronized binaries~FR3 @48#!,which traditionally yielded significantly higher mass ejectiofractions in Newtonian calculations. We note that PN calclations of irrotational binaries yielded an ejected mass frtion of &1% ~FR3!, rather than the 6% quoted by@43#.

To compare with the relativistic results of@43,48#, weshow the evolution of the maximum density as a functiontime in Fig. 17. We see at the earliest times a low-amplitupulsation, resulting from small deviations from equilibriuin the initial SPH particle configuration. This pulsatio

to--

m

a

FIG. 12. Detailed view of the binary to single-star transitioperformed during run RR1 att/Mch5883. We show the positionsof all SPH particles, as well as density contours, shown as daslines, logarithmically spaced two per decade. The surface ofinner computational domain in both the binary and single-star rresentations are shown as heavy solid lines, with good agreembetween the two.

6-19

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FIG. 13. The relative change in the lapse functionN ~top panel!and the conformal factorA immediately preceding and followingthe conversion from binary to single-star representations duringRR1 for every particle, shown as a function of the particle’s potion in thex direction. The maximum error is approximately 0.8%with a mean difference of;0.2%.

FIG. 14. The evolution of the matter in run RR1, after the trasition to a single-star representation, following the same convtions Fig. 11. We see that a dense remnant forms in the center osystem, surrounded by a thin halo. Some ellipticity is seen shoafter the merger, but the system quickly relaxes toward a spheroconfiguration, with maximum density in the center of the systeunlike the toroidal configuration found by@43#.

12403

damps away almost completely by the time the stars pluinward. As the system begins to accelerate rapidly inwprior to the merger itself, the maximum density decreasethe stars are tidally stretched. This effect, seen in a numbecalculations, further indicates that these systems will not

n-

-n-helyal,

FIG. 15. Angular velocity of the merger remnant of run RR1t/Mch51220, shown as a function of cylindrical radius,r cyl

[Ax21y2. We see strong differential rotation, with the higheangular velocity in the center, decreasing monotonically withdius.

FIG. 16. Enclosed mass as a fraction of thetotal mass of themerger remnant in run RR1, att/Mch51220, expressed as a function of ~spherical! radius. All but a few percent of the total massthe system forms the body of the merger remnant, with no mthan a small fraction of a percent ejected from the system.

6-20

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dergo a pre-merger gravitational collapse. This processoriginally suggested in@45#, but is likely to have been anumerical artifact since they used a version of the CF fmalism containing an error in one of the evolution equatioIndeed, the quasiequilibrium sequences in TG show a sdecreasein the central baryonic and energy densities,r* andr, as the binary separation decreases. After contact, we sstrong rise in the maximum density, followed by a raphigh-amplitude oscillation. This result is similar to that seby @48#, although they found a relatively higher maximudensity value during both the peak and the trough ofoscillation. This is almost certainly a result of using differeinitial spin configurations. Irrotational systems, such asone used in run RR1, concentrate relatively more angmomentum at small radii compared to initially synchronizsystems, like run A of@48#. Thus, when a remnant is formefrom an initially irrotational binary, the central density witypically be lower, since there is a greater centrifugal barand less pressure support is needed to stabilize the conration. Our results differ rather significantly from those@43#, who found a smaller-amplitude, more sinusoidal oslation after merger. It is possible that this discrepancy canattributed to the use of the CF approximation rather thanGR. A much more likely explanation, however, is that tdifference results from the numerical methods used toscribe shock heating in the matter. Lagrangian SPH cowere used here and in@48#, whereas Shibata and collabortors @41–43# use an Eulerian grid-based code, which mhave a better ability to resolve shock fronts. It seems cleaexamining the results from run M1414 in@43# that as the NScores converge, the increase in the central density ispressed by the conversion of kinetic energy into heat. Tmerger remnant shows a spike in the internal energy in

FIG. 17. Maximum density as a function of time for run RRWe see the SPH configuration oscillates slightly around equrium, decreasing slowly as the binary plunges toward merger. Ding the merger, we see a sharp decrease, followed by a largeupward and evidence for sharp, nonsinusoidal oscillations.

12403

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very center, 2–3 times the adiabatic value, and a corresping decrease in the density, giving the remnant its toroiprofile. Only calculations using their new shock-capturischeme@76# produce this behavior; previous calculatioyielded remnants with centrally condensed density profi@41,42#. It will be of great future interest to determinwhether or not SPH studies of rapidly rotating collapsimatter configurations can produce these toroidal configutions with ‘‘hot’’ cores, and if so, which aspects of shocphysics are crucial for understanding this process.

In Fig. 18, we show the GW signal for run RR1, calclated from Eqs.~54! and~55!. The waveforms show a familiar chirp signal up untilt/Mch'850, followed by a modu-lated, high-frequency ringdown component. The strengthour signal at peak amplitude matches extremely well withresults of@43,48#, as one would expect from simple dimensional analysis. Our modulated remnant signal, thoughmuch more similar to the results of@43# than@48#, who finda damped ringdown signal of lower amplitude for this modwith no obvious modulation. Previous PN calculations~FR2,FR3! identified the source of the GW amplitude modulatias a combination of differential rotation and ellipticity in thremnant. When the inner and outer regions of the remnhave ellipsoidal deformations which are roughly aligned,we see att/Mch'900 in Fig. 11, the GW amplitude will beat a maximum. When differential rotation drives the innregions into misalignment, as we see att/Mch'1020 in thesame figure, the GW amplitude reaches a minimum. Eve

-r-ike

FIG. 18. The GW signal in theh1 andh3 polarizations for runRR1, as seen by an observer situated along the vertical axis,lowing Eqs. ~54! and ~55!. We see a chirp signal followed bymodulated ringdown spike. The modulation is caused by the alment between quadrupole deformations in the inner regions ofremnant core and those at larger radius. When there is strongalignment, there is destructive interference and the signal amplidrops, as we see att/Mch5920 andt/Mch51220 in Fig. 14. Att/Mch51080 the density contours are more aligned, and theplitude reaches a temporary maximum.

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ally, these effects dissipate away as the remnant relaxeward a more spheroidal configuration. We believe the lackmodulation in the waveforms shown in Fig. 10 of@48#merely reflects the use of a different initial spin configution. As stated above, synchronized initial configuratiocontain relatively less angular momentum at smaller raand the lack of a centrifugal barrier allows the remnantlipticity to damp away much more quickly as the densregions in the NS cores fall into the remnant’s center. Oresults are in broad agreement with those of@43#, who find amodulated waveform immediately after the merger onsimilar time scale as our results, before a longer-livsmaller-amplitude damped modulation eventually appear

Calculating the GW energy spectrum for the mergwaveform is more complicated than simply applying E~57!, since taking the Fourier transform of a signal with nozero initial and final values introduces aliasing of the bouary conditions into the resulting waveform. To correctly drive the proper spectrum, one must also ‘‘attach’’ analysolutions to the beginning and end of the calculated sigrepresenting the portions of the inspiral and ringdophases, respectively, which fall outside the bounds ofnumerical evolution. In the past, the authors~FR3!, and oth-ers@48,63,77#, have modeled the initial inspiral phase via tNewtonian point-mass approximation, but that is clearlyphysically realistic. Indeed, it has been shown in FGRT trelativistic effects should have a significant effect on the Gspectrum even before the dynamical merger. Summariwe know that for an equal-mass binary system, the tomass-energy is given by E5Mt2Mt

2/8r 2521.2Mch

2M ch2 /(20.6r 2), and the~Keplerian! orbital frequency by

f Kep5(AMt /r 3)/2p. In terms of the GW emission frequency,f GW[2 f Kep5(20.6/p)AMch /r 3, we find

EN~ f GW!521.2Mch2p2/3

2M ch

5/3f GW2/3 , ~90!

dEN

d fGW5

p2/3

3M ch

5/3f GW21/3, ~91!

where the latter equation demonstrates the familiar powlaw dependence of the GW energy spectrum.

For the quasiequilibrium sequence from which we taour initial condition, we found that the system ADM macan be given in terms of the GW frequency by a~phenom-enological! fit of the form

E~ f GW!/Mch5EN /Mch20.4905~Mchf GW!

1231~Mchf GW!2. ~92!

Differentiating this equation yields what we term the ‘‘qusiequilibrium energy spectrum,’’ but we require additionassumptions to be made before we can construct thehistory of the inspiral waveform. First, we determine a fit fthe GW frequency as a function of conformal separatifinding that we can approximate the proper functionwithin 0.1% with the form

12403

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21.521.592~r /Mch!22.5

16.325~r /Mch!23.5. ~93!

Next, we assume that the GW signal amplitude is given bslightly modified version of the quadrupole form,

Qxx[2]~ t !52Qyy

[2]~ t !

5~1.02k!20.2p2M chr2f GW

2 cos„uGW~ t !…, ~94!

Qxy[2]~ t !5~1.02k!20.2p2Mchr

2f GW2

3sin„uGW~ t !…, ~95!

where the prefactork is used to match the amplitude of thinspiral signal onto that at the beginning of our calculatsignal. We find thatk50.015 throughout the early phasesour calculated waveform, indicating that it can be weapproximated by the expected quadrupole form. Furthmore, we note that the resulting energy spectrum is estially independent ofk; as k increases, the GW amplituddecreases, decreasing both the energy loss rate and thquency sweep rate in the same proportion, leavingdE/d funchanged. Finally, we assume that the energy loss ratGWs is given by the standard quadrupole expression,

dE/dt50.2 4p2f GW2 Qi j

[2]Qi j[2]&. ~96!

To construct the inspiral waveform, we start from a poalong our calculated waveform and evolve backward in tifrom that point. At every time step, we calculate the instataneous energy loss rate from Eq.~96!. After adjusting thetotal energy, we calculate the new GW frequency by implitly solving Eq. ~92!, and adjust the phase of the GW signappropriately. We find the new binary separation by solvEq. ~93! implicitly, and finally evaluate the waveform viaEqs.~94! and ~95!.

The question of where to match the quasiequilibriuwaveform to the calculated one deserves some attenMatching the two att50 is very much a mistake, becauserepresents a transition from an infalling configuration tocircular one; beforehand, the frequency sweep rated fGW /dtis positive and increasing, while afterward it is reset instataneously to zero. This mismatch in the infall velocity, athus the frequency sweep rate, results in energy ‘‘piling uat the transition frequency, as can be seen in FR3 andsmaller degree in Fig. 12 of@48#. We find that matching theinspiral waveform to the calculated signal att50 reproducesthis error, but that byt/Mch5250, the match in the inspiravelocity is sufficient to leave no measurable trace in thesulting spectrum.

We note that@48# also match their inspiral waveform ttheir calculated one at some time into the calculation, butbelieve that they place too much trust in the behavior ofenergy spectrum near this transition frequency. In their pathey alter the frequency of a Newtonian inspiral waveformmatch their relativistic calculation by adjusting by hand tcoalescence time,t[(dr/dt)/r . We believe this approach to

6-22

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be a mistake. The effect of this change is to use a Newtonwaveform with different physical parameters from thoused in the calculation; in particular, it is equivalent to cculating a Newtonian waveform using the wrongtotal mass,and thus the wrong limit for the spectrum at low frequenciIt is only through the use of an inspiral waveform whofrequency approaches the proper Keplerian limit at low fquencies and therelativistic form at high frequencies thaaccurate energy spectra can be constructed. Indeed, sincinitial condition was taken from the same quasiequilibriusequence used to generate our frequency data, the initialfrequency derived from our calculation matched that ofinspiral waveform within 0.1% without requiring further manipulation.

In Fig. 19, we show as a solid line what we believe tothe first complete and consistent relativistic waveform fobinary NS merger at frequenciesf GW<1.5 kHz. The fre-quencies listed on the upper axis assume our ‘‘standmodel’’ parameters, i.e., each NS has an ADM massM051.4M ( . The two dotted lines show the components whmake up the energy spectrum. At low frequenciMchf GW,0.004, we see the primary contribution is frothe quasiequilibrium inspiral waveform, whereas at highfrequencies it is from our calculated waveform. The shodashed line shows the Newtonian point-mass relation, giby Eq. ~91!, and the long-dashed curve the quasiequilibriuresult, found by numerically differentiating Eq.~92!. We seeexcellent agreement between our calculated waveformthe quasiequilibrium fit, up until frequenciesMchf GW'0.00720.009. This peak represents the ‘‘piling up’’ of energy at the frequency corresponding to the phase of mmum GW luminosity, as the stars make contact and the inrate drops dramatically. The second peak, atMchf GW'0.01020.011, represents emission from the ringdownthe merger remnant. It is likely that we underestimatetrue height of this second peak, since we assume that thesignal after our calculation damps away exponentially. Sit is extremely unlikely that including the ringdown phawill increase the strength of this peak by more than a facof a few, since our chosen EOS will not support a long-livellipsoidal deformation, and it is likely that the ringdowoscillations will smear the GW emission over a small ranof frequencies rather than coherently emitting at a singlequency.

In general, the energy spectrum we calculated here cfirms the general conclusions we put forward in FR3 aFGRT, albeit in a much more consistent way. The GWergy spectrum does show a significant drop away fromNewtonian point-mass form at frequencies significantlylow 1 kHz, in almost the exact same form as we predicfrom quasiequilibrium data alone in FGRT. Nowhere dothe spectrum rise above the Newtonian value, includingpeaks associated with maximum GW luminosity andringdown oscillations. These results suggest that the wsignal amplitude of the peaks above 1 kHz, which lie outsthe Advanced LIGO broadband frequency range, may inhdetections by high-frequency narrow-band interferometerwell. However, combining lower-frequency narrow-band dtectors with broadband LIGO measurements, as suggest

12403

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@34#, appears extremely feasible, and may allow GW msurements to constrain the NS compactness and EOS.

One possible cause for concern with our code is noncservative behavior caused by numerical errors that deveafter t/Mch5500. In Fig. 20, we show the change in thsystem’s angular momentum over time. The dotted curvthe uncorrected result derived from our calculation, whshows two periods of angular momentum generation, thefrom t/Mch56002700, and the second att/Mch58752900. The former is associated with the numerical inacracies discussed in Sec. V A for run RR2. All of our rushow some spurious angular momentum generationslowing of the binary infall during this period. The lattespike occurs immediately before and after the transition fra binary to a single-star description. Correcting for boththese spurious terms yields the solid line on the plot, whstill underestimates by a nontrivial amount the angular mmentum loss we would expect from the quadrupole formu

~ Jk!GW50.4e i jk^Qil[2]Qlk

[3]&. ~97!

Using the quadrupole formula on our results yields a toangular momentum loss fraction which very nearly equ

FIG. 19. GW energy spectrum,M ch22dE/d f , as a function of

the GW frequency,Mchf GW , for run RR1. The dotted lines showrespectively at high and low frequencies, the components conuted by our calculated signal and the quasiequilibrium inspiral coponent. Also shown are the Newtonian point-mass energy spec(dE/d fGW)Newt ~short-dashed line!, Eq. ~91!, and the quasiequilib-rium fit (dE/d fGW)QE derived from Eq.~92!. We see confirmationthat the ‘‘break frequency’’ calculated from a fit ofE( f ) for theequilibrium sequence~i.e., the frequency at which the energy spetrum decreases to a given fraction of the Newtonian level! is repro-duced by a full numerical evolution. On the upper axis, we showcorresponding frequencies in Hz assuming the NS each have aM051.4M ( The two peaks correspond to the phases of maximGW luminosity and ringdown oscillations, respectively.

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that found by@43,48#, with approximately 7% of the system’s angular momentum converted into GW emission. Tdiscrepancy between this amount and what we derive fthe SPH particle configuration, Eq.~44!, can be easily understood. First, we do not see strong angular momentum losthe NS first make contact, since this is where we pushfield solver to its limit. Second, our method for estimatithe instantaneous angular velocity, Eq.~53!, underestimatesthe proper value ofv, yielding a radiation reaction force, Eq~45!, smaller than the correct quadrupole value~which wecan determine after the fact!. This error could be decreasein magnitude by calculating the angular velocity from tchange of position of the NS centers-of-mass in time,such a prescription is difficult to define consistently in tsingle-body regime. Even defining the orbital frequencyterms of the rate of change of the quadrupole tensor, as dby @48#, underestimates the correct angular momentumrate by up to 40% during the GW emission peak.

We see a similar pattern at work in the evolution of tsystem’s ADM mass, shown in Fig. 21, comparing the eergy loss to GWs from the quadrupole approximation fmula ~dashed line!, Eq. ~96!, to the value we find from SPHsummation via Eq.~42! ~solid line!. The quadrupole valueagrees well with other calculations, which typically finDMADM /MADM50.004. Looking at our particle summatiovalue, we see a slow decrease fromt/Mch50 –500, of mag-

FIG. 20. Angular momentum loss for the binary in run RRmeasured using the SPH integral, Eq.~44! ~dotted line!, and theresult after correcting for spurious angular momentum creationing the binary to single-object transition~solid line!. We see that theresult yields an angular momentum loss approximately half thatwould have predicted from the quadrupole formula, Eq.~97!~dashed line!, primarily because our estimate for the system’s agular velocity used in the back reaction force is systematiclower than the GW signal would indicate. The quadrupole resultderive shows that about 7% of the system angular momentumemitted in GWs, in line with previous relativistic estimates.

12403

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nitude 0.2–0.3 % of total value, in good agreement wother calculations and our own quadrupole estimate. Frthat point on, we see a small spurious increase from the pthe numerical errors begin to become significant, followby a sharp decrease of;2% immediately prior to our tran-sition from binary to single-star descriptions. Once we hamade the transition, the ADM mass oscillates slightly, wan overall peak-to-trough amplitude of 1%. Thus, we coclude that while the instantaneous value we measure fthe particles directly is liable to be off by up to 2%, we careconstruct the proper energy loss rate after the calculatioover.

VI. CONCLUSIONS

We have developed and tested a new relativistic 3Dgrangian hydrodynamics code, which should prove usefulstudying a wide variety of physical systems. Here, asinitial investigation, we have performed the first full evolutions of the coalescence and merger of irrotational binaryin the CF approximation to GR. Moreover, these calculatiorepresent the first numerical evolution of coalescing binsystems performed with either a spectral methods field soor the use of spherical coordinates adapted to a binary eronment.

The code has been validated using several tests. Weaccurately reproduce static spherical stellar configurationwell as the known solution for a collapsing pressureless d

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FIG. 21. The evolution of the total ADM mass for the systemrun RR1, calculated using the SPH integral, Eq.~42! ~solid line!.Up until t/Mch5600, we see a slow decrease as energy is emiin GWs, at approximately the rate predicted by the quadrupolemula, Eq.~96! ~dashed line!. Beyondt/Mch5600 numerical inac-curacies lead to oscillations of total amplitude;2%. From thequadrupole estimate, we find that 0.4% of the system’s energradiated away as GWs, in line with previous estimated from retivistic calculations.

6-24

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cloud. In both cases, the CF approximation yields the exsolution in GR, as it will for any spherical configuration. Wfind that we can reproduce these known semianalytic stions to high accuracy, up until the formation of an evehorizon for the collapsing dust clouds.

Our dynamical evolution calculations for quasiequilirium models at a number of different binary separationsdicates that we can successfully integrate forward for sevorbits, with typical errors in conserved quantities of;1%.In doing so, we have demonstrated directly for the first tithat theM /R50.14, equal-mass sequence of TG is stablethe way to its innermost configuration, at which point a cudevelops on the inner edge of each NS.

Our dynamical calculation of a complete binary Nmerger, including radiation reaction effects, demonstrathat our spherical coordinates, spectral method approacrobust enough to follow the system from the point just befthe formation of a cusp through merger and the formationa stable remnant. Some errors were introduced duringperiod into the globally conserved quantities such asADM mass and system angular momentum, but we find tthe field values were computed consistently throughout,that the global dynamics was treated in a quantitativelycurate way.

We find that the merger remnant formed in our calculatis differentially rotating, with a transient quadrupole defomation. This combination of effects produces a GW amptude with a modulated form, similar to what has been sbefore in PN calculations~FR3! and more recent full GRcalculations of the same model@43#. We find that the rem-nant is initially stable against gravitational collapse, as@43#, with the supermassive NS~which has a baryonic masessentially twice that of either NS in isolation! supported bystrong differential rotation. We find that a density maximudevelops rather rapidly in the center of the merger remnas has been seen in all other PN and CF calculations, buthat of @43#, whose full GR merger calculation yielded

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toroidal remnant. We believe the difference results from thuse of a capturing scheme, whereas our runs were perforusing an adiabatic treatment.

By combining our calculated GW signal with a relativistquasiequilibrium inspiral precursor, we have generatedfirst GW energy spectrum from a binary NS merger whichcomplete at all sub-kHz frequencies and consistent throuout. We find that the energy spectrum deviates fromNewtonian power-law relation by more than 50% at frequecies f GW,1 kHz ~the ‘‘break frequency’’!, in very goodagreement with the predictions of FGRT. There are distipeaks in the power spectrum corresponding to the phasemaximum GW luminosity and merger remnant ringdowbut at levels significantly below the point-mass power-lavalue.

In our future work on binary NS systems, we hopeaddress a number of topics, many of which deserve mmore careful study. Based on the excellent agreementtween our calculated GW energy spectrum and that bapurely on equilibrium sequence data, we hope to do a brphase space survey to determine the dependence of‘‘break frequency’’ on both the NS EOS and the systemmass ratio. Beyond this parameter study of NS-NS mergwe also plan to investigate in detail the formation processthe merger remnant, to determine the conditions which mlead to the formation of a quasitoroidal merger remnant. Twill necessarily involve the use of a relativistic artificial viscosity scheme to treat shocks. The density profile ofmerger remnant is likely to influence the final fate of tsystem, and may prove crucial for determining the coindence properties of GW emissions and short-period GRshould they result from compact object binary mergers,has been widely suggested.

ACKNOWLEDGMENT

This work was supported by NSF grants PHY-01334and PHY-0245028 to Northwestern University.

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Mergers of irrotational neutron star binaries in ...coupled, nonlinear, elliptic ﬁeld equations,...

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