General Relativistic Magnetohydrodynamic Simulations of Binary Neutron Star Mergers

Bnmo Giacomazzo Department of Astronomy, University of Maryland, Co llege Park, Maryland, USA Gravitational Astrophysics Laboratory, NA SA Goddard Space Flight Center, Greenbelt, Maryland, USA

Luciano Rezzolla Max-Planck-lnstitut fur Gravitationsphysik, Alhert-Einstein-Institut, Potsdam Germany Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, USA

Luca Baiotti Institute of Laser Engineering, Osaka University, Osaka, Japan

We report on our recent general-relativistic simulations of magnetized, equal-mass neutron star binaries and on the role that realistic magnetic fields may have in the evolution of these systems. In particular, we study the evolution of the magnetic fields and show that they can influence the survival of the hypermassive neutron star produced at the merger by accelerating; its collapse to a hlack hole. We also show how the magnetic field cau be amplified and that this can lead to the production of the relativistic jets observed in short ga1nrna-ra.y bursts.

1 Introduction

The numerical investigation in full general relativity (GR) of the merger of binary neutron stars (BNSs) has produced a series of new and interesting results in the last years.1 Thanks to several numerical improvements, it has been indeed possible to start to investigate the full dynamics of 5 these systems including the formation of tori around rapidly rotating black holes (BHs) z.3A, which could not be modeled via l'\cwtonian simulations. This progress has allowed the beginning of an accurate investigation of whether Bl'\S mergers could indeed be behind the central engine of short 1-ray bursts (GRBs) . 5,7,s Fully GR simulations have shown that the end result of BNS mergers is the formation of a rapidly spinning BH surrounded by a hot torus. Driven by neutrino processes and magnetic fields, such a compact system may be capable of launching a relativistic fireball with an energy of � 1048 erg on a timescale of 0.1 - 1 s. Moreover, BNSs are also one 9 of the most powerful sources of gravitational waves (GWs) that will be detected in the next few years by advanced LICO and advanced Virgo. 10 GR simulations of BNSs have then started to provide accurate templates that can be used to infer properties of the NSs composing the binaries, such as the equation of state (EOS) of NSs, once their GWs will be detected. 11 Here we review some of the main results we published recently 5•8 and that describe the merger of magnetized equal-mass Bl'\Ss. We will show how magnetic fields can affect the dynamics of these systems and their role in powering short GRBs. Table Properties of the eight equal-mass binaries considered: baryon mass of each star; total ADM mass M, l: initial orbital angular velocity !10; mean coordinate radius along the line connecting the two stars; /\!ADM; rnaxixnurn initial magnetic field wherere* is 81 10 or 12. Bo , Binary no (rad/ms) r, (km) Bo (G) 1.445 2.680 1.78 15.0 ± 0.3 0 or 1.97 10* Ml .45-B* x 1.625 2.981 1.85 13.6 ± 0.3 0 or 1.97 10* Ml . 62-B* x

2 Numerical and Physical Setup

All the details of the mathematical and numerical setup used for producing the results presented here are discussed in depth in our previous publications 5•8 and here we limit ourselves to a brief overview. We have used the general relativistic magnetohydrodynamic (GRMHD) Whisky codc 12•13•14, which solves the equations of GRMHD on dynamical curved backgrounds. In particular Whisky makes use of the Cactus framework which provides the evolution of the Einstein equations for the metric expressed in the BSSN formulation. 15 The GRMHD equations arc solved using high resolution shock capturing schemes and in particular by using the Piecewise Parabolic Method (PPM), 16 and the Harten-Lax-van Leer-Einfeldt (HLLE) approximate Riemann solver 17 to compute the fluxes. In order to guarantee the divergence-free character of the MHD equations we have employed the flux-CD approach, 18 but with one substantial difference, namely, that we use as an evolution variable the vector potential instead of the magnetic field. 5 The system of GRMHD equations is closed by an EOS and we have employed the commonly used "ideal-fluid" EOS, in which the pressure pis expressed as p= pc(r 1 , where pis the rest-mass density, - ) f is the specific internal energy and r is the adiabatic exponent . Both the Einstein and the GRMHD equations are solved using the vertex-centered AMR approach provided by the Carpet driver. 19 The results presented below refer to simulations h performed using 6 levels of mesh refinement with the finest level having a resolution of = 0.1500 221 m and covering each of the two :NSs, while the coarsest grid extends up to .!YI0 '.'.:'='. 254.4 375.7 km and has a resolution of h � 4.8 7.1 km. For all the simulations r = Af8 '.'.:'='. /'vf8 '.'.:'='. reported here we have used a reflection-symmetry condition across the z = 0 plane and a 7r syrnrnetry condition across the 0 plane� :i; = Finally, the initial data 2•20 were produced by Taniguchi and Gourgoulhon 21 with the multi b domain spectral-method code LORENE. Since no self-consistent solution is available for mag netized binaries yet, a poloidal magnetic field is added a-posteriori and it is initially confined inside each of the :NSs. 20•5 The main properties of the initial data are listed in Table 1 and we have considered two classes of binaries differing in the initial masses, i.e., binaries Ml . 45-B*, and binaries Ml . 62-B*. For each of these classes we have considered four different magnetiza tions {indicated by the asterisk) so that, for instance, Ml . 62-B12 is a high-mass binary with a maximum initial magnetic field 1.97 1012 G. Bo = x

3 Results

In order to highlight some of the most salient aspects of the binary dynamics we will focus on the high-mass models Ml . 62-B* since they describe all the aspects of a typical BNS merger. The main difference with the low-mass models 1.45-B* is that the latter will collapse to a BH on a

"Stated differently, we evolve only the region {x 2: 0, z 2: O} applying a 180°-rotational-syrnmetry boundary condition across the plane at x = 0. 0http://www.lorene.obspm.fr

70 Figure 1: Snapshots at representative tirnes of the evolution of the high-mass binary with initial maxi1111nn 1:.l - magnetic field of 10 G, i.e., . 1 . The first pa11el shows the initial condition, the second one the moment of the rnerger and the third oneMl 62theB HMNS2 funned after it. The rest-1nass density is visualized using volu111e rendering colors from red to yellow while the white lines are the magnetic field lines inside the stars. The la.st ( ) three panels show instead the evolution of the torus after the collapse of the HMNS to BH white sphere at the ( center . In these last three panels the white lines are the magnetic field lines in the region close to the spin axis ) of the BH and the green lines are the magnetic field lines in the torus. Figure published in Rezzolla et al 2011.

71 _ B=O G 9 B= !06 G B=l010 G B= 1012 G

M162-B* 5 4 0 5 10 1 20 0 10 5 15 t [ms] log(B+ 1) [GJ

Figure 2: Left Panel: Evolution of the 1naximum of the rest-n1ass density nonnalized to its initial value fur the high-mass models. Right panel: Lifetime of the Hl\1NS formed after theµ merger in the high-mass case as a function of the initial magnetic field. The error bar has been estimated from a set of sin1ulations of unmagnetized binary NS mergers at three different resolutions; in particular, we have assumed that the magnetized runs have the same relative error on the delay time of the correspondip.g unmagnetized model. Indicated with a dashed line is the continuation of the delay times to ultra-high ma!':nctic fields of 1017 G. Figure published in Giacomazzo et al 2011.

much longer timescale 4 and, because of the much higher computational cost, we have followed those models only for ::::: lOms after the merger and hence before their collapse to BH. A synthetic overview of the dynamics is summarized in Fig. 1, whieh shows snapshots at representative times of the evolution of the high-mass binary with an initial maximum magnetic field of 1012 G, i.e., Ml . 62-B12. The rest-mass density pis visualized using volume rendering with colors from red to yellow, while the white lines represent the magnetic field lines. The first panel shows the initial conditions with the two NSs having a purely poloidal field contained inside each star. The second panel shows the time when the two NSs enter into contact (t ""' 7ms) and when the Kelvin-Helmholtz instability starts to curl the magnetic field lines producing a strong toroidal component. Zll.5 The third panel shows the H).l[NS formed after the merger, while the three last panels show the evolution of the formed BH and of the torus surrounding it. In the last three panel the green lines refer to the magnetic field in the torus and the white ones to the field lines near the spin axis of the BH. We will comment on these three last panels later in this section. All the high-mass models studied here form tori with masses between ::::: 0.031\!0 (model Ml . 62-BlO) and ::::: 0.09M0 (model Ml . 62-BS) which are sufficientlymassiv e to be able to power a short GRB. The mass and spin of the BH formed from the merger of all the high-mass models are respectively MB11 ::::: 2.9!i10 and JB11/Mbu ::::: 0.8. As mentioned before, we did not evolve the low-mass models until collapse to BH, but our previous studies have shown that such models ran form tori of similar or even larger masses. 1 An important effect of the magnetic field is that by redistributing the angular momentum inside the H).l[l'\S it can accelerate its collapse to BH. While magnetic fields as low as � 108 G 5 0 are too low to affect the dynamics, magnetic fields � 101 G or larger can instead shorten the life of the H).l[l'\S. This is shown in Fig. 2 where in the left panel we show the evolution of the maximum of the rest-mass density p normalized to its initial value for all the high-mass models. The first minimum in the evolution of the maximum of p corresponds to the time of the merger of the two NS cores. The HMNS that is subsequently formed oscillates for few ms before collapsing

72 10 20 30 20 30 l [ms] l [ms]

l"igure Evolution of the maximum of the temperature (top) and of the maximum of the Lorentz Left panel: T factor (bottom).3: The two vertical dotted and dashed lines represent respectively the time of the merger and of the collapse to BH. Evolution of the maximum of the magnetic field in its poloidal (red solid line) Right panel: 'nd toroidal (blue dashed line) components. The bottom panel shows the maximum local !luid energy indicating that an unbound out!low develops and is sustained after BH formation. All the panels refer to the (i.e.. E1oc 1) high-mass model> Figure published in Rezzolla et al Mi . 62-B12. 2011.

to a BH which happens when the maximum of grows exponentially. As one can see from this p

figure the evolution of in the case with a magnetic field � G (blue dot-dashed line) is p lOH almost identical to the unmagnetized case (solid black line). The models with higher values of the magnetic field instead collapse much earlier (magenta long-dashed line, model Ml . 62-Bl2 and red short-dashed line, model Ml . 62-BlO ). In the right panel of Fig. 2 we plot instead the lifetimeof the HMNS formed after the merger of the high-mass models as a function of the initial magnetic field. The fact that this curve exhibits a minimum should not come as a surprise. As said before low magnetic fields do not change the lifetime and so the curve is expected to Td

be flatfor low magnetic fields, i.e., smaller than � 108 G. Stronger magnetic fields, i.e., larger 1 than � 10 6 G, will instead increase the total pressure in the HMNS and extend its lifetime. 20 Intermediate field values instead do not sufficiently contribute to the the total pressure in the HMNS. but arc still sufficient to redistribute its angular momentum and accelerate its collapse to BH. From this figure it is also clear that if we were able to determine the lifetime of the HMNS (by measuring for example the delay between the merger and the collapse to BH in the GW signal). we would then be able to infer approximatively the strength of the magnetic field.

3. 1 Jet formation

By continuing the evolution of the high-mass model Ml . 62-B12 far beyond BH formation we were also able to show for the first time that BNS mergers can generate the jet-like structures that are behind the emission of short GRBs. 8 In particular the last three panels of Fig. 1 show the evolution of the rest-mass density and magnetic field lines in the torus and around the BH formed after the collapse of the HM.\!S. In these panels the white lines are the magnetic field lines iu the region close to the spin axis of the BH and the green lines are the magnetic field lines in the torus. It is evident that after the formation of the BH, the magnetic field lines change from the "chaotic" structure they had in the HMNS (third panel in Fig. 1) to a more ordered structure with a mainly toroidal field in the toms (green lines) and a mainly poloidal field along the spin-axis of the BH (white lines). It has been already shown in the past that such a configurationcan launch the relativistic jets that arc thought to be behind the short GRBs, 22 but it is the first time that it has been shown that such configuration isthe natural result of the

73 Figure 4: the G\V signal shown through the = rn = mode of the polarization, (h _ top part «nd the f 2, 2 + )n, ( ) l\IHD luminosity, L\IIHD: butt.om part computed from the integrated Poyntiug and shown with solid ( ) a._.;; flux a line for the high-mass model The corresponding energy, EMHD , is shown with a dashed line. The Ml . 62-B12. dotted and dashed vertical lines show the times of merger as deduced from the first peak in the evolution of the ( GW amplitude and BH formation, respectively. Figure published in Rezzolla et al ) 2011. merger of magnetized I3NSs. The left panel of Fig. 3 shows on the top the evolution of the maximum of the temperature and on the bottom the maximum Lorentz factor. Soon after the merger and because of the shocks produced the temperature grows to � 101° K and remains almost constant during the lifetime of the HMNS. During the collapse the temperature increases further and reaches maximum 1 values of � 10 2 K in the torus. Such an high temperature could potentially lead to a strong emission of neutrinos. The maximum Lorentz factor after the collapse to BH is associated to matter outflow along the boundaries of the funnel created by the magnetic field lirn"s. In the bottom-right panel of Fig. 3 we also show the maximum local fluid energy and it highlights that this outflow is unbound l) and persists for the whole duration of the simulation. (i.e., E10c > Even if the Lorentz factor of this outflow is still low compared to the typical values observed in short GRBs, much larger values could be obtained by including the emission of neutrinos or via the activation of mechanism, such as the Blandford-Znajek mechanism 2�, which could produce relativistic jets. 22 A quantitative view of the magnetic-field growth is shown instead in the top-right panel of Fig. 3. which shows the evolution of the maximum values in the poloidal and toroidal compo nents. �ote that the latter is negligible small before the merger, reaches equipartition with the poloidal field as a result of a Kelvin-Helmholtz instability triggered by the shearing of the stellar surfaces at merger 20, and finally grows to 1015 G by the end of the simulation. At later times 22 ms), when the instability is suppressed,c:= the further growth of the field is due to (t >� the shearing of the field lines and it inCTeases only as a power-law with exponent 3.5 (4.5) for the poloidal (toroidal) component. Although the magnetic-field growth essentially stalls after t 35 ms, further slower growths are possible24, yielding correspondingly larger Poynting fluxes. Indeedc:= , when the ratio between the magnetic fluxacross the horizon and the mass accretion rate becomes sufficiently large, a Blandford-Znajek mechanism 23 may be ignited; 22 such conditions arc not met over the timescale of our simulations, but could develop over longer timescales. We have also computed the GW signal emitted by this model as well as an estimate of the electro-magnetic emission. In the top panel of Fig. 4 we show indeed the GW signal, while in the bottom part we plot the evolution of the :\1HD luminosity, LMHD: as computed from the integrated Poynting flux (solid line) and of the corresponding energy, EMHD, (dashed line).

74 Clearly, the :'vlHD emission starts only at the time of merger, it is almost constant and equal to "" 1044 erg/s during the life of the IIMNS, and increases exponentially after BII formation, when the GW signal essentially shuts off. Assuming that the quasi-stationary MHD luminosity is � 4x1048 erg/s, the total :'vlHDenergy released during the lifetime of the torus is � 1.2x1048 erg, which, if considering that our jet structure has an opening half-angle of � 30°, suggests a lower limit to the isotropic equivalent energy in the outflow of � 9 x 1048 erg. While this is at the low end of the observed distribution of gamma-ray energies for short GRBs, larger MHD luminosities are expected either through the additional growth of the magnetic field via the on-going winding of the field lines in the disk (the simulation covers only one tenth of taccr), or when magnetic reconnection (which cannot take place within our ideal-MHD approach), is also accounted for. Even if we did not follow the entire evolution of the torus, by measuring its accretion rates we estimated that its lifetime wouldbe "" 0.3 seconds and consequently in good agreement with the duration of short GRBs.

4 Conclusions

We have reported on some of the main results obtained from the firstgeneral relativistic simula tions of magnetized B�Ss with astrophysically realistic magnetic fields.5 ·8 We have shown for the firsttime how the magnetic fieldscan impact the evolution of the H:'vlNSformed after the merger and that fields equal 'or larger than � 1010 G accelerate its collapse to BH. We have shown that all the systems that collapse to BH form tori sufficientlymassive to power short GRBs and that the magnetic field structure around the BH has those characteristics that are necessary in order to launch relativistic jets. A detailed analysis of our results has also shown a good agreement with observations, even if the introduction of more physical ingredients (e.g., neutrino emission and more realistic EOS) will be required in order to increase the accuracy of this model.

Acknowledgments

We thank the developers of Lorene for providing us with initial data and those of Cactus and Carpet for the numerical infrastructures used by Whisky. Useful input from A. Aloy, J. Gra :\11. nat, I. Hinder, C. Kouveliotou, J. Read, C. Reisswig, E. Schrietter, A. Tonita, A. Vicere, and S. Yoshida is also acknowledged. We also thank M. Koppitz for assisting us in the production of Fig. 1. The computations were performed on the Damiana Cluster at the AEI, on Queen Bee through LO�I (www . loni . org), and at the Texas Advanced Computing Center through TERAGRID Allocation No. TG-MCA02N014. This work was supported in part by the DFG Grant SFB/Transregio 7, by "CompStar" , a Research �etworking Programme of the European Science Foundation, by the JSPS Grant-in-Aid for ScientificResearch (19-07803), by the MEXT Grant-in-Aid for Young Scientists (22740163) and by KASA Grant Ko. NKX09Al75G.

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