Numerical Relativity of Compact Binaries in the 21St Century E-Mail: M.Duez@Wsu.Edu

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REVIEW Numerical relativity of compact binaries in the 21st Recent citations - Towards the nonlinear regime in extensions to GR: assessing possible century options Gwyneth Allwright and Luis Lehner To cite this article: Matthew D Duez and Yosef Zlochower 2019 Rep. Prog. Phys. 82 016902

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Rep. Prog. Phys. Rep. Prog. Phys. 82 (2019) 016902 (46pp) https://doi.org/10.1088/1361-6633/aadb16

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Matthew D Duez1 and Yosef Zlochower2 016902 1 Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, United States of America M D Duez and Y Zlochower 2 Center for Computational Relativity and Gravitation and School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, United States of America

Numerical relativity of compact binaries in the 21st century E-mail: [email protected]

Received 11 January 2016, revised 13 August 2018 Printed in the UK Accepted for publication 17 August 2018 Published 7 November 2018

ROP Corresponding Editor Dr Beverley Berger

Abstract 10.1088/1361-6633/aadb16 We review the dramatic progress in the simulations of compact objects and compact-object binaries that has taken place in the first two decades of the twenty-first century. This includes 1361-6633 simulations of the inspirals and violent mergers of binaries containing black holes and neutron stars, as well as simulations of black-hole formation through failed supernovae and high- 1 mass neutron star–neutron star mergers. Modeling such events requires numerical integration of the field equations of general relativity in three spatial dimensions, coupled, in the case of neutron-star containing binaries, with increasingly sophisticated treatment of fluids, electromagnetic fields, and neutrino radiation. However, it was not until 2005 that accurate long-term evolutions of binaries containing black holes were even possible (Pretorius 2005 Phys. Rev. Lett. 95 121101, Campanelli et al 2006 Phys. Rev. Lett. 96 111101, Baker et al 2006 Phys. Rev. Lett. 96 111102). Since then, there has been an explosion of new results and insights into the physics of strongly-gravitating system. Particular emphasis has been placed on understanding the gravitational wave and electromagnetic signatures from these extreme events. And with the recent dramatic discoveries of gravitational waves from merging black holes by the Laser Interferometric Gravitational Wave Observatory and Virgo, and the subsequent discovery of both electromagnetic and gravitational wave signals from a merging neutron star–neutron star binary, numerical relativity became an indispensable tool for the new field of multimessenger astronomy. Keywords: black holes, numerical relativity, relativistic binaries, compact objects (Some figures may appear in colour only in the online journal)

Contents 1.5. Posing the problem: recasting the field equations as an initial value problem...... 6 1. Introduction...... 2 2. Numerical relativity formalisms and techniques...... 7 1.1. The field equations of general relativity...... 3 2.1. Finite differencing...... 7 1.2. Evolution of matter sources...... 4 2.2. Pseudospectral methods...... 8 1.3. Black holes...... 4 2.3. Making the problem well posed...... 8 1.4. Relativistic stars...... 5 2.4. Breakthroughs in numerical relativity...... 10

1361-6633/19/016902+46$33.00 1 © 2018 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 82 (2019) 016902 Review 2.5. Methods for evolving the fluid equations...... 11 [7, 8] and later by LIGO and Virgo [9], and the simultaneous 2.5.1. Conservative formulation...... 11 observation of gravitational waves and electromagnetic spec- 2.5.2. Excision and punctures in the presence of tra from the merger of a neutron star–neutron star binary by matter...... 12 LIGO, Virgo, and a large team of astronomers in 2017 [10– 2.5.3. More physics: equations of state, neutrinos, 12], the new field of gravitational wave and multimessenger magnetic fields...... 13 astronomy was born. 3. Black hole–black hole binary simulations...... 14 Fundamental to this new science is the ability to infer the 3.1. Early efforts...... 14 dynamics of the sources based on the observed signals, some- 3.2. Post-breakthrough results...... 14 thing that can only be accomplished using detailed theoretical 3.2.1. Recoils...... 14 predictions based on numerical simulations of the nonlinear 3.2.2. Modeling the remnant properties...... 16 Einstein field equations of general relativity both in vacuum 3.2.3. Numerical relativity at the extremes...... 17 and coupled to the equations of magnetohydrodynamics. 3.2.4. Waveform modeling...... 18 Indeed, the body of techniques that emerged based on efforts to 3.2.5. Semi-analytic waveform models...... 20 solve this system, known as numerical relativity, was designed 4. Relativistic stars and disks...... 21 largely with such gravitational wave source modeling in mind, 4.1. Black hole–black hole binaries with accretion..... 21 but it has also been turned to other astrophysical phenomena 4.2. Relativistic stars...... 22 involving strongly-curved, dynamical spacetime. 4.2.1. Radial stability and collapse outcome...... 23 Only rather exotic phenomena involve sufficiently strong 4.2.2. Magnetohydrodynamic evolution...... 23 spacetime curvature to require numerical relativity. Newtonian 4.2.3. Nonaxisymmetric mode instability...... 24 gravity clearly works quite well for main sequence stars, 4.2.4. Stability of self-gravitating black hole accre- planets, and the like. As is well-known, relativity becomes tion disks...... 24 important when speeds approach the speed of light c, so a 5. Black hole formation...... 25 reasonable guess would be to expect important general rela- 5.1. Population I/II core collapse and collapsars...... 25 tivistic effects as the escape velocity approaches c. Then an 5.2. Massive star collapse in the early universe...... 26 object of mass M and radius R will require relativistic treat- 6. Non-vacuum compact binaries...... 27 2 ment if R is close to the gravitational radius rG 2GM/c , 6.1. White dwarf—compact object binaries...... 28 the radius of a nonspinning black hole of mass M≡. The same 6.2. Black hole neutron star binaries...... 28 – condition can be stated in terms of the dimensionless com- 6.2.1. Expectations before numerical relativity paction GM. Strong-gravity objects have high compac- simulations...... 28 C≡ Rc2 6.2.2. Inspiral and merger in numerical relativity: tion (order unity being the standard of ‘high’). Black holes parameter space exploration and gravita- ( 1) and neutron stars ( 0.1) are compact objects C∼ C∼ 4 tional waves...... 29 by this definition. White dwarfs ( 10− ) are a marginal C∼ 6.2.3. Post-merger in numerical relativity: neutri- case—relativity plays a large role in their stability condition nos, ejecta, and MHD...... 30 but not their equilibrium structure—and are usually also clas- 6.3. Neutron star–neutron star mergers...... 31 sified as compact. 6.3.1. Pre-breakthrough simulations...... 31 Formulating the integration of the Einstein equations so 6.3.2. Post-merger evolution: neutrinos, ejecta, that evolutions are stable and the coordinates evolve sensibly MHD...... 32 turned out to be a difficult task. There was some worry that 6.3.3. General relativistic effects: prompt collapse numerical relativity might not be ready when the advanced threshold and post-merger gravitational gravitational wave detectors needed it. Finally, in 2005 [1–3], waves...... 32 the first stable black hole–black hole binary merger simula- 6.3.4. Longer term evolution of remnants: subgrid tions were carried out. There followed a race to produce accu- scale modeling...... 33 rate waveforms for gravitational wave observation efforts, 7. Comparison to observations...... 34 which were already underway. 7.1. Gravitational wave astronomy...... 34 In this review, we describe how numerical relativity has 7.2. GW170817: The age of multimessenger astronomy come to be a robust tool for studying strong-gravity sys- begins...... 34 tems. We also review some of the major accomplishments of 8. Conclusion...... 35 numerical relativity to date. To provide an appropriate scope, Acknowledgments...... 36 we focus on applications to compact binaries and black hole References...... 36 formation, processes where general relativity is essential and whose astrophysical importance is clear. 1. Introduction The article is organized as follows. In the rest of the introduction, we provide background on general relativity, black With the discovery of gravitational waves from merging holes, and relativistic stars. In section 2, we cover methods black hole–black hole binaries by the Laser Interferometric for evolving the Einstein field equations and coupled mat- Gravitational Wave Observatory (LIGO) in 2015 [4–6], the ter sources. Particular attention is given to the historical subsequent observations of other black hole mergers by LIGO ‘breakthrough’ discoveries that enabled stable evolutions of

2 Rep. Prog. Phys. 82 (2019) 016902 Review multiple-black-hole spacetimes. The next sections review sim- Here, we will not make a distinction between the comp ulation results, covering the period before the breakthroughs onents of a tensor and a tensor itself. For our purposes a tensor and after. Section 3 is devoted to black hole black hole binary L of type ( p, q) is a set of p q functions, denoted by – × mergers. The following three sections describe simulations of α α α 1 2··· p Lβ β β , (5) phenomena with matter, especially neutron stars. Finally, in 1 2··· q section 7, we return to our original motivation and consider which, under a change of coordinates from some coordinate what has been learned by the confrontation of numerical rela- system x to another y, transform as tivity predictions with actual LIGO-Virgo observations. α α α α α α ∂y 1 ∂y 2 ∂y p 1 2··· p L β β β (y)= 1 2 q ∂xµ1 ∂xµ2 ··· ∂xµp ··· 1.1. The field equations of general relativity ν ν ν (6) ∂x 1 ∂x 2 ∂x q µ µ µ 1 2··· p Lν1ν2 νq (x). The theory of special relativity introduced the notion of space- × ∂yβ1 ∂yβ2 ··· ∂yβq ··· time. In that theory, spacetime is a geometrically flat 4-dimen- Thus the metric g is a tensor. Associated to g is its matrix sional (4D) manifold. General relativity extends this notion µν µν inverse gµν (i.e. gµσg = δµ, where δ indicates the usual to non-flat manifolds. In general relativity, the Newtonian σν ν Kronecker delta function). notion of a gravitational force is replaced by geodesic motion The metric, g , in special relativity is intrinsically flat. By in a curved spacetime. Unless acted on by other (non-gravi- µν this, we mean any of several equivalent statements: there is a tational) forces, objects whose size is much smaller than the coordinate transformation such that the new metric is every- local spacetime radius of curvature travel along geodesics. where identical to η , as discussed above, parallel geodesics Throughout this paper, we will use geometric units. In µν will remain parallel, and the intrinsic curvature of the metric, these units, the speed of light, c, and Newton s constant G, ’ as measured by the Riemann curvature tensor, vanishes eve- are taken to be 1. A consequence of this is that distances, rywhere. We briefly describe how geodesics and the Riemann time intervals, masses, and energy all have the same units. curvature tensor are calculated. By convention, the unit of each of these is denoted by an A geodesic is the generalization of a straight line in arbitrary mass M3. Euclidean space. In Cartesian coordinates, the tangent vector In the section below, we will provide an extremely brief to a straight line is a constant. This can be expressed as overview of the field equations of general relativity. For a comprehensive overview, we suggests consulting [13] for a d tµ = xµ(λ), (7) very accessible introduction to general relativity, and [14, 15], dλ and [16] for a more advanced treatment. The material below d was synthesized from these references. tµ(λ)=0, (8) The geometry of a spacetime can be entirely described by a dλ line element ds2. In Minkowski spacetime, in Cartesian coor- where tν is the tangent vector to the line and xµ(λ) are the dinates, the line element takes the form Cartesian coordinates of each point of the line. Equation (8) can be re-written as ds2 = dt2 + dx2 + dy2 + dz2. (1) − µ 2 ∂t Note that ds is not necessarily positive. For timelike paths, tν = 0. (9) the proper time along the path is given by the integral of ∂xν dτ = √ ds2. Equation (1) can be written as However, even in flat space, this equation does not hold true − 2 µ ν in arbitrary coordinates. To fix this, we replace the ordinary ds = ηµν dx dx , (2) derivative ∂/∂xν with the covariant derivative . The equa- ∇ν where xµ =(t, x, y, z) and the components of the symmetric tion for a geodesic in arbitrary coordinates, as well as flat and tensor η are given by diag( 1, 1, 1, 1). Here we used two non-flat metrics, is µν − standard conventions, the timelike coordinate is listed first and d repeated Greek indices are summed over. By convention, the xµ(λ)=tµ(λ), (10) dλ index of the timelike coordinate is 0, while the spatial coordinates have indices 1, 2, 3. In arbitrary coordinates, the line µ ν t µt = 0. (11) element becomes ∇ Finally, α, the covariant derivative associated with the met- 2 µ ν ∇ µ1µ2 ds = gµν dy dy , (3) ric gµν, is defined by its action on arbitrary tensors Lµ1µ2···, which is given by ··· where yµ is some new set of coordinates and µ1µ2 µ1µ2 α β αLν ν ··· = ∂αLν ν ··· ∂x ∂x ∇ 1 2··· 1 2··· g = η . (4) µ1 βµ2 µ2 µ1β µν µ ν αβ +Γ L ··· +Γ L ··· + ∂y ∂y αβ ν1ν2 αβ ν1ν2 (12) ··· ··· ··· β µ1µ2 β µ1µ2 Γ L ··· Γ L ··· , αν1 βν2 αν2 µ1β 3 For example, solar mass intervals of distance and time are about 1.5 km − ··· − ··· −··· 6 and 5 10− s, respectively. where ×

3 Rep. Prog. Phys. 82 (2019) 016902 Review

1 photons, but not necessarily neutrinos—form a nearly perfect Γσ = gσα (∂ g + ∂ g ∂ g ) , (13) µρ 2 µ αρ ρ αµ − α µρ fluid, meaning the mean free path is very small compared to the system s scale, making the collection a fluid, and viscosity and ∂ is shorthand for ∂/∂xα4. The components of Γσ ’ α µρ and heat transport are small enough to be ignored. This fluid are collectively known as the Christoffel symbols. Unlike the will have a stress tensor metric and the tensors constructed from the Christoffel sym- gas bols below, the components of the Christoffel symbols do not Tµν =(ρ0 + u + P)uµuν + Pgµν , (19) transform according to equation (6). where ρ , u, P, and u are the rest mass density, internal energy, The Riemann curvature tensor is constructed from the met- 0 µ pressure, and 4-velocity. (Note ρ must be distinguished from ric and has the following form5 0 the total energy density ρ = ρ0 + u.) The 4-velocity has R σ = ∂ Γσ ∂ Γσ +Γα Γσ Γα Γσ . only three independent components, with the Lorentz factor µνρ ν µρ − µ νρ µρ αν − νρ αµ (14) W αut given by the normalization condition u u = 1: The Riemann curvature tensor can be further split into a trace- ≡ · − 2 = + γij (20) free part, known as the Weyl tensor Cµνρσ, and the Ricci tensor W 1 uiuj. ρ Equations (18) and (19) must be supplemented by the rest Rµν = Rµρν . (15) mass conservation equation Finally, the Einstein tensor G , is given by µν µ µ(ρ0u )=0 (21) 1 αβ ∇ Gµν = Rµν gµν g Rαβ. (16) and also an equation of state (EoS) − 2 Regardless of coordinates, the Riemann curvature tensor is P = P(ρ0, T, Xi) (22) identically zero in special relativity. General relativity extends the notion of spacetime to include non-flat metrics, where u = u(ρ0, T, Xi), (23)

Gµν = 8πTµν , (17) where T is the temperature and Xi are composition variables. For a detailed exposition of relativistic hydrodynamics, and Tµν is the stress energy tensor, a measure of the total including its numerical treatment, see the book by Rezzolla energy and momentum flux from matter and non-gravitational and Zanotti [634]. interactions and radiation. Because the Einstein equations do not constrain the components of Cµνρσ, even in vacuum there can be non-trivial curvature. Note that equation (17) is the 1.3. Black holes standard covariant form of the Einstein equations. Perhaps one of the most interesting predictions of general relativity is the existence of black holes. Black holes are 1.2. Evolution of matter sources regions in spacetime where the curvature is sufficiently strong that light, and therefore any physical signal, cannot escape. In general relativity, the curvature of spacetime possess its own Astrophysically, black holes form as stellar objects collapse. dynamics, and indeed one of the most important phenomena Despite all the microphysics that goes into the dynamics of treated by numerical relativity, the merger of two black holes, stellar objects, once equilibrated, a black hole can be com- is a vacuum problem. That is, Tµν = 0. Numerical relativity pletely described by two parameters: its mass and spin6. In is also used to study phenomena involving matter flows in geometric units the magnitude of the spin angular momen- strongly-curved dynamical spacetimes. Two problems where tum S is bounded by the mass m, where S < m2. Typically one such relativistic effects should be particularly important are defines a specific spin a, where a = S/m and a dimensionless compact object mergers involving neutron stars and the for- spin χ, where χ = S/m2. mation of black holes by stellar collapse. If the black hole is non-spinning, it is known as a Matter and energy constitute the stress-energy tensor Tµν Schwarzschild black hole [17, 18], and if S is non-zero, it is that is the source term in Einstein equations. In general, Tµν known as a Kerr black hole [19]. In both cases the black hole will be a sum of stress tensors for the gas, electromagnetic, spacetimes are named after their discoverers. and neutrino fields. The energy and momentum conservation A black hole has no material surface, of course, but there is equations a boundary separating the region from which it is impossible µν µT = 0 (18) ever to escape (the black hole interior) to the outside universe. ∇ This boundary is called the event horizon. The event horizon provide evolution equations for the matter. is a null surface, i.e. one must move at the speed of light to In the non-vacuum systems we will consider, the parti- stay on it. In order to determine if a true event horizon exists, cles including nucleons, nuclei, electrons, positrons, and — one needs to know the entire future of the spacetime (otherwise there is always the possibility that an observer can escape 4 Readers familiar with differential geometry will notice that in equation 13 we are limiting ourselves to coordinate (holonomic) bases, which are suf- 6 More correctly, an equilibrated black hole is completely described by its ficient for numerical relativity. mass, spin, and charge. However, astrophysical black holes are expected to 5 σ Note that while the components of Rµνρ transform according to equa- have effectively zero charge because accretion from the interstellar medium σ tion (6), the components of Γ µρ do not. should rapidly discharge them.

4 Rep. Prog. Phys. 82 (2019) 016902 Review the supposed black hole at a later time). In practice, numer ical relativists find event horizons by evolving a cluster of null geodesics or a null surface backwards in time starting at the very end of their simulations. (For a review, see [20].) We will see that some methods of numerically handling black hole interiors (excision methods) require some knowledge of the horizon location during the simulation. For these purposes, numerical relativists use the apparent horizon. Apparent horizons are two-dimensional (2D) surfaces that may exist at each time in a numerical simulation. They are defined to be surfaces from which outward-pointing null rays do not expand. This very unusual situation can only occur in the vicinity of a black hole, but finding such surfaces only requires information about the metric and extrinsic curvature at a given time. For a stationary black hole, the apparent horizon will coincide with the event horizon; in a dynamical spacetime, it will be inside the event horizon. Black holes can form binaries, as demonstrated by LIGO’s recent detection of gravitational waves [4, 5]. In such a case, Figure 1. The evolution of an unstable non-rotating star in the black holes are not truly in equilibrium, but each can still numerical relativity. The inset shows the TOV equilibrium sequence. For each central density, there is a unique equilibrium, be described reasonably well as Kerr or Schwarzschild black and the inset plots the baryonic mass M0 against the central density holes, at least when the binary components are well sepa- ρc. Equilibria to the left of the turning point (solid curve) are stable, rated. Such a binary can then be described by several intrinsic to the right (dashed) are unstable. A star on the unstable branch parameters, such as the mass ratio of the two black holes, the (open circle) will (under tiny perturbations) migrate to lower spin magnitudes and orientations of the two black holes, and density and oscillate about the stable equilibrium of the given mass (asterisk). This evolution is shown in the main plot. It illustrates the orbital eccentricity. the type of experiments that can be done in numerical relativity. The initial state would probably not occur in any astrophysically realistic scenario. For the evolution, various pieces of physics can 1.4. Relativistic stars be turned on or off to study their effect. The solid line shows a simulation that allows shock heating, while the dotted line shows Much astrophysical thinking is guided by idealized equilibria, a simulation where this has been artificially turned off, forcing the such as the spherically symmetric star and the thin accretion star to evolve adiabatically. Reprinted figure with permission from disk, and this remains true in the study of compact object sys- [23], Copyright (2002) by the American Physical Society. tems. Neutron stars are extremely compact objects, containing a little over a solar mass (rG 4km) within a radius of ∼10 Rotation in equilibrium stars is constrained, though, by the km, and so must be studied using≈ general relativity; they will mass-shedding limit, at which fluid on the equator is in geo- be our primary type of relativistic star. desic (‘Keplerian’) orbit. Faster rotation at the equator would For spherical neutron stars of an assumed barotropic equa- centrifugally eject mass. This restricts the degree of possible tion of state P = P(ρ0), the Tolman–Oppenheimer–Volkoff uniform rotation particularly severely, so that the maximum (TOV) [21, 22] equations of hydrostatic equilibrium yield mass only increases by around 20%, the most massive con- a sequence of equilibria, one for each central density ρc. At figurations being found close to (but not exactly on) the mass- max a critical central density ρcrit, the mass reaches a maximum shedding limit [24, 26]. Stars with masses between M TOV max value M TOV = M(ρcrit), and only the configurations on the and this higher limit are called supramassive. A sufficient con- ascending (dM/dρc > 0) side, which generally turns out to dition for instability of uniformly rotating stars (on the secu- be ρc <ρcrit, are stable. A star on the unstable side will either lar timescale on which uniform rotation is maintained) can collapse to a black hole or undergo large radial oscillations be determined via locating the turning point in the constant about the lower density configuration of the same mass. (See angular momentum sequence [27], similar to TOV sequences. figure 1.) (The actual instability onset occurs slightly on the ‘stable’ max Can neutron stars exist with M > M TOV? If the neu- side [28].) The point of onset of dynamical instability can be tron star is spinning, this provides some additional support determined by numerical simulations. against gravity. Codes exist for generating the resulting 2D Stars with mass above the supramassive limit are called (axisymmetric) equilibria (e.g. [24, 25]). The matter is usu- hypermassive. Such equilibria can exist with the help of ally be taken to be a perfect fluid with purely azimuthal flows differential rotation, providing rotational support while evad- [ur = uθ = 0, uφ = utΩ], and a rotation law for Ω must be ing the mass-shedding limit by keeping the rotation rate sub- specified. Rotation might be uniform (Ω=constant) or differ Keplerian near the equator. Numerical relativity confirms [29] ential (Ω varies through the star). Viscosity (or similar angular that such stars can persist stably for multiple dynamical time- momentum transport mechanisms) will tend to produce uni- scales, as we discuss in section 4.2. form rotation, but differentially rotating equilibria can persist One might also look to thermal support—hot nuclear on timescales shorter than that of viscosity. matter—to increase the maximum mass, effectively changing

5 Rep. Prog. Phys. 82 (2019) 016902 Review the equation of state to give more pressure support. Effects of thermal support have been studied for uniformly rotating neutron stars by Goussard et al [30] and for uniformly and differentially rotating neutron stars by Kaplan et al [31]. The latter suggest an approximate turning point method for assessing stability which has been numerically confirmed by Weih et al [32] and used by Bauswein and Stergioulas to explain some numerical relativity findings on the threshold mass for prompt collapse of a neutron star–neutron star binary merger remnant to a black hole [33]. The distinctions introduced above between normal, supra- Figure 2. The standard 3 + 1 coordinate system. Here each point massive, and hypermassive neutron stars have played a large on a given spatial slice is specified with a spatial coordinate xi. i role in the interpretation of neutron star neutron star binary The points labeled with the same value of x on two different – hypersurfaces are connected by a curve (denoted by x = const) that merger simulations. When two neutron stars merge, the is offset from the curve normal to the spatial slices (these would be resulting object will either collapse to a black hole or settle vertical lines in the plot).The shift vector βi measures how skewed to a dynamical equilibrium state in roughly a dynamical time- the curves of constant xi are from the curves normal to the spatial scale ∼10−1 ms. An equilibrium remnant could be described surfaces and the lapse function α measures how far in proper time as a type of relativistic star. If two 1.4 M neutron stars merge, one slice is from another for observers traveling along the normal max directions. In general relativity, there is complete freedom in this remnant could easily have a mass in excess of M TOV. specifying both α and βi as functions of both space and time. However, the remnant will also be spinning rapidly and differ entially, and it will have acquired a great deal of heat from the that Latin indices take on the values 1, 2, or 3 of the space- i merger shock. Thus, normal, supramassive, and hypermassive like dimensions) . A point labeled x0 on one spatial slice and remnants are all possible, depending on the stars’ masses and another with the same label on a different slice may be skewed max µ the unknown value of M TOV. However, while differentially with respect to the unit normal direction n (which must be rotating stars are in equilibrium on a dynamical timescale, timelike). Here, the two points are shifted with respect to each they evolve on the secular timescale of effects that transport other by a spatial vector βi. If a particle moves from spatial angular momentum (∼10 ms). Similarly, the equilibrium will slice t0 to t0 + dt along the normal, it will experience a proper be adjusted by loss of thermal support on the neutrino cool- time interval of α dt, where α is known as the lapse function. ing timescale (∼s). Except in the unlikely event that it sheds The lapse function and shift vector βi are freely specifiable enough mass to drop below the supramassive limit, a hyper- as a functions of both the spatial and time coordinates. The massive remnant will ultimately collapse on one of these time- choice of lapse function determines how far neighboring spa- scales. Thus, the main outline of the post-merger evolution tial hypersurfaces are from each other. seems to depend on one parameter, the mass of the binary, On each t = const hypersurface, there is an induced met- and one EoS-related number, the neutron star maximum mass. ric γij . The induced metric is simply the spatial components of the spacetime metric γij = gij. On the other hand, the 4D metric can be specified by providing the lapse function, shift 1.5. Posing the problem: recasting the field equations as an vector, and spatial metric. The 4-metric has the form initial value problem g β β β Returning to the Einstein equations themselves, similar to 00 1 2 3 how the 4-vector Aµ in electromagnetism is not unique due β1 γ11 γ12 γ13 gµν =   , to gauge freedom, the metric that satisfies equation (17) (and β2 γ21 γ22 γ23   any relevant boundary conditions) is not unique. In gen- β3 γ31 γ32 γ33   eral relativity, the gauge freedom comes in the form of the   where g = α2 + γ βiβ j and β = γ β j. While the sur- freedom to choose coordinates arbitrarily. In order to get a 00 − ij i ij face normal has components nµ =(1, β1, β2, β3)/α and unique solution, we need to impose gauge conditions. Many, − − − n = α t. but not all, formulations of the Einstein equations for numer µ − ∇µ The 3-metric γ and its matrix inverse γij can be used to ical simulations use what is known as a 3 + 1 decomposition ij define a covariant derivative, Christoffel symbols, Riemann [34] (see also recent texts on numerical relativity [35 37]). – tensor, and Ricci tensor. The formulas for these are nearly In a 3 + 1 decomposition, the coordinates are constructed by identical to those presented in section 1.1, with the excep- using a family of non-intersecting, spatial hypersurfaces7 The tion that indices only take on the values 1, 2, and 3. We will basic setup is illustrated in figure 2. The spacetime is split into distinguish these 3-dimensional (3D) tensor (and tensor-like spatial hypersurfaces labeled by a coordinate t. On each spa- objects) from their 4D counterparts by either using different tial slice, coordinates xi are specified (we use the convention symbols (e.g. using the symbol Di to indicate the 3D covariant 7 In ordinary Euclidean geometry, a surface can be obtained by consider- derivative), or by prepending a superscript 3 surrounded by ing the level sets of some function of space f (x, y, z) (i.e. the points where (3) parentheses (e.g. Rij). f (x, y, z)=const). A hypersurface is the generalization of this to higher µ dimensions. A spatial hypersurface is one where all possible curves on the Using the surface normal n , we can define a spatial tensor hypersurface are spacelike. Kij known as the extrinsic curvature, by projecting the tensor

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i 2 ( µnν + ν nµ)/2 onto the slice. The resulting tensor is related ∂tΠ β ∂iΠ= Φ, ∇ ∇ − −∇ (32) to the time derivative of γij by ∂tγij = 2αKij + Diβj + Djβi, i − ∂tΦ β ∂iΦ=Π, where Di is the covariant derivative associated with γij . − Finally, with these choices, the ten Einstein equa- where the spatial coordinates will be denoted by x, y, z. We tions become six evolution equations for γij and four con- will explore numerical techniques for solving equation (32) straint equations. This is analogous to the way the Maxwell in the next section. equations split into evolution equations for E and B and two constraint equations for divE and divB. 2.1. Finite differencing The resulting field equations, usually known as the Arnowitt–Deser–Misner [34] (ADM) equations, but are actu- To solve equation (32), we consider a discrete grid labeled ally a reformulation of the standard ADM equations by York with 3 integer indices (i, j, k), where the values of x, y, and z [38], are given by (see, e.g. [35–37]) at a point (i, j, k) are given by (x0 + i dx, y0 + j dy, z0 + k dz). Furthermore, we denote the values of a function f (x, y, z) on ∂tγij = 2αKij + Diβj + Djβi, (24) − this grid by fi,j,k. We then approximate spatial derivatives using (3) k these points. For example, ∂tKij = DiDjα + α( Rij 2KikK j ) − − 2 2 2 1 ∂x f (x, y, z)=(fi+1,j,k + fi 1,j,k 2fi,j,k) /dx + (dx )(33), 8πα(Sij γij(S ρ)) (25) − − O − − 2 − is an approximation to ∂2f using a three point stencil. By k k k x + β DkKij + KikDjβ + KkjDiβ , using more points in the stencil, this derivative can be made more accurate in the sense that the error will scale with higher (3) 2 ij 16 πρ = R + K KijK , (26) powers of . Modern numerical relativity codes tend to use − dx sixth-order to eighth-order finite differencing [39 41]. With i ij ij – 8πS = Dj(K γ K), (27) these approximations, equation (32) becomes − (3) (3) where Rij and R are the Ricci curvature tensor and Ricci ∂ Π = ( i,j,k Π + i,j,k Φ ) t i,j,k Cl,m,n l,m,n Dl,m,n l,m,n , (34) scalar associated with γij and the source terms are given by l,m,n µν ρ = nµnν T , (28) ∂ Φ = ( i,j,k Π + i,j,k Φ ) t i,j,k El,m,n l,m,n Fl,m,n l,m,n , (35) i ij µ l,m,n S = γ n Tµj, (29) − where the coefficients C, D, E, F and the values of (l, m, n) Sij = Tij, (30) in the sums are determined by the finite difference stencil used. Thus, the partial differential equation (32) becomes a ij S = γ Sij. (31) set of coupled ordinary differential equations for Πi,j,k and Equations (24) and (25) form the evolution equations, while Φi,j,k. These equations are then typically solved using standard equations (26) and (27) are the Hamiltonian and momentum Runge–Kutta techniques [42]. A major drawback of the above technique is that it is constraint equations, respectively. In equations (24)–(31) Latin indices are raised and lowered with the spatial metric extremely computationally wasteful. For convenience here, i i ij ij we will assume that dx = dy = dz = h. The smaller the grid γij , i.e. βj = γijβ and β = γ βj. The tensor γ is the matrix spacing h the smaller the error, but the maximum value of inverse of γij . Greek indices are raised and lowered with the µν h one could use and still have an acceptably accurate solu- full metric gµν and its inverse g . tion generally varies quite strongly over space (e.g. many points are needed to resolve black holes, but few needed far 2. Numerical relativity formalisms and techniques away). Furthermore, the regions where high resolution (i.e. small values of h) are needed tend to be quite small. Thus, if There are two major classes of techniques used for numerical one used a uniform grid capable of resolving the entire space, simulations of the Einstein equations. These are finite-differ- essentially all the calculation time would be spent evolving ence methods, typically coupled to adaptive mesh refinement the overresolved regions. This problem can be ameliorated techniques, and pseudospectral methods. To understand how to some extend by choosing special coordinates [43–45] that these methods work, we will consider some toy problems. concentrate gridpoints in certain regions. The state-of-the art Most of the finite difference codes are based on modifications technique for overcoming this inefficiency is the use of adap- to the ADM system (see section 2.3). These equations are in tive meshes [46–50]. In an adaptive mesh code, a coarse grid a form with mixed second and first derivatives. Basically, the covers the entire computational domain, with increasingly system is such that only first time derivatives occur, but first finer grids placed in location where high resolution is required and second spatial derivatives occur. (Of course, auxiliary (see figure 3). evolution variables can be introduced so that the system only Evolutions also involve a discretization in time, and the has first spatial derivatives, but at the cost of introducing addi- smaller the timestep dt, the more steps are needed to cover tional constraints.) A good toy problem to illustrate how such a given time interval, and the more expensive the simula- equations are evolved is thus tion. Explicit time integration methods are subject to the

7 Rep. Prog. Phys. 82 (2019) 016902 Review The choice of finite difference versus spectral methods affects which method of handling black holes is easier to implement. A black hole interior presents a major challenge to any numerical technique because of the curvature singularity it harbors (see, e.g. Wald [59] for a discussion on the inevitability of forming curvature singularities). Fortunately, the singularity is concealed behind an event horizon. The region inside the horizon cannot affect the exterior solution, so numerical simulations need not evolve it accurately. They only need to keep it from causing the simulation to crash. One way to do this is to simply not evolve a region inside the horizon, i.e. to excise this region. Spectral methods can do this naturally, because even near the inner edge Figure 3. Schematic of how mesh refinement works in one of the grid, no points are needed from the other side of the dimension. Shown are the values of a function at discrete points excision boundary to take derivatives. A boundary condi- along the x axis. A coarse grid (black) covers the entire domain tion physically should not be needed (because no informa- and progressively finer grids (blue and red) cover the parts of the tion flows out of a region where all characteristic speeds domain where the function varies rapidly. go inward), and none is required. The other method, the Courant–Friedrichs–Lewy stability condition [51], which puncture method, described in detail below, involves allow- limits dt on a mesh to be less than around h/vs, where vs is ing singularities in the computational domain. In the appro- the maximum signal speed, which for spacetime evolution is priate gauge, these singularities are sufficiently benign that the speed of light. The effect on dt is a price to be paid for finite difference methods can handle them. It would be smaller h. more difficult to evolve a puncture stably with a spectral code [56]. 2.2. Pseudospectral methods The other major techniques used in black-hole simulations 2.3. Making the problem well posed fall under the category of pseudospectral methods [52–56]. As will be discussed later, the ADM equations by themselves In spectral methods the evolved fields are expressed in terms proved to be unstable for many strong-field problems, includ- of a finite sum of basis functions. An example of this would ing black-hole mergers. With all the difficulties encountered be to describe a field on a sphere in terms of an expansion in trying to implement the ADM equations in the 1990s, empha- spherical harmonics. These methods have the advantage that sis changed to developing new 3 + 1 systems and analyzing if the fields are smooth8 then the error in truncating the expan- their well-posedness [60–62, 62–73]. Informally speaking, a sion converges to zero exponentially with the number of basis hyperbolic system of equations is well posed if the solution functions used in the expansion. In pseudospectral methods, depends continuously on the initial and boundary data. Ill- values of functions are stored at special gridpoints, the colo- posed systems can have solutions that grow without bound cation points, corresponding to Gaussian quadrature points of even for very small evolution times. the basis functions [57]. Codes can then transform between To understand how reformulation of the basic evolution spectral and colocation-point representations via Gaussian equations can make or destroy well-posedness, consider a quadrature. This is especially useful for computing products simple vector wave equation and other pointwise operations which are much simpler using gridpoints. In pseudospectral form, spectral methods can be ∂E = B, thought of as a particular limit of finite differencing, the limit ∂t ∇× that uses the entire domain as its stencil so as to make the highest-order derivative operator [58]. The order of this opera- ∂B = E, tor will then increase with the number of colocation points, ∂t −∇× giving the method faster convergence than a fixed polynomial (36) order. The payoff is that differentiations and interpolations subject to become more expensive much more quickly than fixed-stencil finite difference methods as resolution is increased. Also, E = E = 0, C ∇· exponential convergence is lost for functions that are only B = B = 0. smooth to finite order. Perhaps the best known numerical rela- C ∇· (37) tivity code that uses pseudospectral techniques is the Spectral This system is well posed in the sense that the solution Einstein Code [52–54], or SpEC, used by the SXS (simulating (E(t), B(t)) depends continuously in the initial data. Any con- extreme spacetimes)9 collaboration. straint violation (failure of E or B to be zero) will be pre- served (not grow or decrease)C by theC evolution system. 8 More precisely the fields are smooth on the real axis and can be analytically continued into the complex plane. This system can be transformed into two separate identical 9 www.black-holes.org/ second-order equations for E and B of the form

8 Rep. Prog. Phys. 82 (2019) 016902 Review many well-posed formulations have been proposed but turned ∂2A + A = 0. (38) out not to be an immediate panacea for unstable black hole ∂t2 ∇×∇× simulations. Some of the most influential of these formations This latter system is not quite equivalent to the origi- are the Bona-Masso family [75–78], the NOKBSSN family nal system in that constraint violations now grow lin- [79–81], Z4 family [82–89], the Kidder–Scheel–Teukolsky early in time. A better system is obtained by noting that family [90, 91], and the generalized-harmonic family [73, 92, 93]. A = 2 A + A . Using this, and the ∇×∇× −∇ ∇ ∇· One of the key improvements in numerical simulations assumption that A = 0 we get prior to the breakthroughs of 2005 was the introduction of ∇· the so-called NOKBSSN formulation of the Einstein equa- A = 0, (39) tions in 3 + 1. This system, which is named after its devel- where opers Nakamura, Oohara, Kojima, Shibata, Baumgarte, and 2 Shapiro [79–81], modifies the standard ADM equations in ∂ 2 = . (40) several crucial ways. Firstly, the spatial metric γij is split ∂t2 −∇ into an overall conformal factor eφ and a conformal metric 4φ If we solve equation (39) with a small divergence, the norm γ˜ij , where e γ˜ij = γij and the determinant of γ˜ij is unity. This of the divergence will remain bounded. More generally, if the conformal metric has its corresponding Christoffel symbols constraint equations (37) are satisfied, then any solution to (3) ˜k (3) Γ ij and Ricci tensor R˜ij. Second, the three combinations equation (39) is also a solution to ij (3) ˜k (3) k ij γ˜ Γ ij = Γ˜ (k = 1, 2, 3), as well as the K = γ Kij are promoted to evolved variables. Third, the remaining evolved A + κ A . (41) ∇ ∇· extrinsic curvature variables are trace-free conformal extrin- 4φ If κ is chosen larger than one, small violations of the A = 0 sic curvature variables A˜ij = e− [Kij (1/3)Kγij]. Finally, − constraint can blow up arbitrarily quickly. Thus, with∇· seem- the momentum constraint equations are used to modify the ingly inconsequential changes, we can turn a system from one evolution equations for (3)Γ˜k, which introduces a constraint with a minor blowup in the constraints to either one with a damping quality to the system. These changes, in conjunction catastrophic blowup in the constraints, or no blowup at all. with a particular choice of gauge conditions, namely the use However, with the addition of an auxiliary field, we can do of certain Bona-Masso [75] type lapse conditions (known as even better. Consider the system [74] 1+log slicing) and Γ-driver shift conditions [94] led to the first genuinely stable, fully nonlinear implementations of the ∂E Einstein equations for systems without symmetries, at least = B + ψE, ∂t ∇× ∇ for non-black-hole spacetimes. The factoring out of the conformal factor φ proved to be particularly advantageous for ∂B = E + ψB, collisions of black holes [95, 96]. However, the state of the ∂t −∇× ∇ (42) art for black-hole evolutions in the early 2000s only allowed ∂ψ for head-on collisions and grazing collisions [95, 96], where E = E ψ , ∂t −∇· − E the black holes merge well before completing one orbit. The NOKBSSN system also proved to be stable using higher- ∂ψ B = B ψ . order finite-differencing methods [44] for head-on collisions, ∂t −∇· − B as well. For this system, the constraints satisfy A key technique used in many of these early evolutions was the fixed-puncture formalism [97, 98]. In this formalism, ∂ CE = 0, a two-sheeted Einstein–Rosen bridge associated with a single CE − ∂t (43) black hole is mapped into a single sheet with a singularity at ∂ the center. As shown in figure 4, the standard Schwarzschild CB = 0. CB − ∂t spacetime can be recast as a puncture by taking a spatial slice that passes through the bifurcation sphere10. On either side of Solutions to equations (43) decay exponentially in time. the sphere, the slice extends infinitely far. Next we introduce Thus if numerical effects introduce constraint violations, a coordinate , which is related to the usual Schwarzschild these will be damped away. The main message here is that coordinate r byR even for a physically motivated evolution system, the addition or removal of terms nominally equal to zero can make M 2 r = 1 + . (44) the difference between a solvable and an insolvable system. R 2 Furthermore, for a constrained system, like the equations of R electromagnetism and general relativity, enlarging the system The spatial metric then takes in the form of equations by adding new fields can suppress unphysical constraint violations. 10 In the extended Schwarzschild spacetime, also called the Kruskal exten- The standard ADM formulation is now known to be ill- sion, there is both a black hole and a white hole. The bifurcation sphere is posed in the nonlinear regime [68, 69]. On the other hand, the point where the two horizons meet(see figure 4).

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Figure 4. (Left) A spacetime diagram of a Schwarzschild black hole with the (θ, φ) angular coordinates suppressed. Each point represents a sphere of radius 4πr2. Note that there are two curves corresponding to each value of r. Radially ingoing and outgoing light rays travel along 45◦ lines in this diagram. The event horizon(s) correspond to the r = 2M diagonal curves. The dotted line represents a spatial slice with a two-sheeted topology. (Right) The spatial surface corresponding to the dotted line shown with one spatial dimension (z) suppressed. Points on the dotted line correspond to circles here. The horizons correspond to the thick circle and the r = 5M and r = 10M curves each map to one circle inside the horizon and one outside. The central point maps to the r = of Region III. ∞

evolution of ψ4 on a carefully chosen black-hole background ds2 = ψ4 d 2 + 2dΩ2 , (45) is then used to evolve the gravitational radiation to infinity. R R where ψ = 1 + M/(2 ). Here = 0 and = both cor- R R R ∞ 2.4. Breakthroughs in numerical relativity respond to r = and the metric is singular at = 0. This singularity is not∞ the curvature blow-up singularityR at the Perhaps one of the most significant breakthroughs in numer center of the black hole, that singularity is in the future of ical relativity occurred when Pretorius, who had previously this slice, rather the singularity is entirely gauge and results developed an adaptive-mesh-refinement (AMR) code based from us stuffing an entire asymptotically flat universe into the on the generalized harmonic system [93], included constraint sphere = M/2. This singularity is further only present in damping techniques [60] developed for the Z4 system by R the NOKBSSN function φ, the components of NOKBSSN Gundlach et al [88]. The key development there was that the conformal metric γ˜ij are nonsingular. In the fixed puncture general relativistic action can be extended to include terms approach, there are singularities of this type associated with that vanish when the constraints are satisfied, but act to damp each black hole and the gauge conditions are chosen so that these constraint violations when they are nonzero. When these singularities do not move. Each of these singularities is applied to the generalized harmonic system used by Pretorius, called a ‘puncture’. the constraint violations for a black hole–black hole binary Keeping the puncture fixed has several advantages. First, remained bounded and Pretorius was thus able to perform the the singularity in the conformal factor can be handled analyti- first successful evolution of an orbiting binary [1]. Pretorius cally. Second, by keeping the black holes fixed in coordinate presented his initial results at the Banff International Research space, one can use the much simpler fixed excision techniques. Station Workshop on Numerical Relativity in April 200511. Using these techniques, Brügmann, Tichy, and Jansen [99] The actual system Pretorius evolved was the Einstein equa- and later Diener et al [100] were able to evolve a quasicircular tions coupled to a scalar field. The initial data consisted of binary for roughly one orbit. However, they were still not able two scalar boosted stars with supercritical densities. The stars to get the merger waveform using these techniques. collapsed into two black holes that then orbited and merged. One approach that was able to get the merger waveform Figure 5 is a reproduction of figure 3 in of Pretorius’ paper from these early simulations was the Lazarus method [43, [1]. It shows the very first merger waveform from an orbiting 101–105], which used the numerical simulation to generate black hole–black hole binary ever published. initial data for a subsequent perturbative evolution of the radi- Just four months after Pretorius submitted his ground- ative scalar ψ4. (Measures of gravitational radiation, including breaking paper, a new breakthrough was announced that ψ4, are described in section 3.2.4.) The key to the success of became known as the moving punctures approach [2, 3]. This the Lazarus approach was that when two black holes are close new method was developed independently by the groups then enough, even though they have not merged yet, the exterior at the University of Texas at Brownsville (UTB) and NASA spacetime can be well described by black hole perturbation theory (i.e. the close-limit approximation [106]). A subsequent 11 http://bh0.physics.ubc.ca/BIRS05/

10 Rep. Prog. Phys. 82 (2019) 016902 Review These include the BAM code [49] developed at Jena, the Maya–Kranc code developed at Penn State [107], as well as the original codes, LazEv developed at The University of Texas at Brownsville, and Hahndol, developed at GSFC. More recently, the publicly available EinsteinToolkit [41, 108] code included an open-source implementation, known as McLachlan. LazEv, Maya–Kranc, and McLachlan all used the Cactus Computational Toolkit [109], originally developed at the Albert Einstein Intsitute in Golm, Germany. There were several significant differences between Pretorius’ techniques and the moving punctures approach. Unlike in the moving punctures approach, the system Pretorius developed used excision to handle the black hole singularities, compactified the computational domain to include spatial infinity, and of course, used the generalized harmonic system with constraint damping. This system was sufficiently unlike the other techniques used by numerical relativists at the time that it was only slowly adopted. Figure 5. A reproduction of the waveform shown in figure 3 of Figure 6 shows reproductions from the breakthrough [1] courtesy of the author. This was from the first fully nonlinear papers using the moving punctures approach. The waveforms, numerical simulation of the last orbit, merger, and ringdown of a with various analyses, and the horizons are shown. black hole–black hole binary. It was generated by Pretorius using Soon after the announcement of the moving punctures the generalized harmonic coordinate approach. The curves show breakthrough, simulations were reported where the binary the waveforms as calculated at various radii and translated in time. Reprinted figure with permission from [1], Copyright (2005) by the completed more than one full orbit [110] and then multiple American Physical Society. orbits [111]. The latter, in particular, compared the merger waveforms from simulations starting at various separations Goddard Space Flight Center (GSFC) and first demonstrated and found that the merger waveform was insensitive to ini- publicly in the Numerical Relativity 2005: Compact Binaries tial conditions. This was followed shortly afterwards by the 12 workshop at NASA GSFC . Notably, the moving punctures discovery of the orbital hangup effect [112] for spinning approach allowed groups worldwide to evolve black hole– binaries. This effect either delays or accelerate the merger black hole binaries. It is based on an extension of the stand- depending on whether the spins of the two black holes are ard NOKBSSN system, with several important changes. (1) (partially) aligned or counteraligned with the orbital angular The singular conformal factor is replaced by a non-singular momentum, and proved to be important for parameters esti- 4φ function χ = e− (although evolutions with φ itself are also mation of LIGO sources [5]. Some of the important discover- used [3]) which is evolved fully numerically. (2) The gauge ies that proceeded from these simulations will be described conditions explicitly allow the punctures to move. Previously, below in section 3. the shift condition was chosen so that the punctures could not move. (3) The standard 1 + log lapse condition 2.5. Methods for evolving the fluid equations ∂tα = 2αK, (46) − 2.5.1. Conservative formulation. Many of the early numer would require K to be singular at the puncture in order for the ical relativity hydrodynamic simulations used a formulation lapse to change from a zero value when the puncture is on a introduced by Wilson in 1972 [113]. In Wilson’s scheme, the given point to a non-zero value when the puncture has passed. evolution variables are This singular behavior is removed by changing the lapse condition to an advection equation (ρ = W√γρ0, E = ρ , Si = ρ hui), (48) a variable for rest-mass density, internal energy density, and ∂ α β α = 2αK. (47) t − · ∇ − momentum density, respectively. Often, there are numerical Finally, within the evolution equations for Γ˜i there is a singu- advantages to evolving entropy rather than internal energy, so lar term on the puncture location proportional to the lapse. By some codes (e.g. [114, 115]) specializing to Gamma-law EoS choosing an initial lapse that is identically zero on the punc- Γ evolved a variable e = W√γ(ρ ) . The equations can be ture, this singularity is also removed. With these changes, the 0 finite differenced in a conservative form: fluxes are calculated at gauge naturally evolves such that the black holes orbit each cell interfaces; the same flux added to one grid cell is removed other and inspiral in coordinate space. The pathologies seen from its neighbor, and no truncation error accrues to the total with the fixed-puncture approach vanished with these new rest mass. However, the variables evolved (in particular E) are dynamic punctures. Furthermore, because the moving punc- not those that are physically conserved, so an explicit artificial tures approach was similar to existing codes, groups around viscosity must be added to correctly account for shocks. the world were able to rapidly develop their own versions. Shock handling is accommodated more naturally if one 12 https://astrogravs.gsfc.nasa.gov/conf/numrel2005/ evolves the physically conserved variables, meaning that one

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Figure 6. A reproduction of the waveform, black-hole trajectories, and horizons calculated in the moving punctures breakthrough papers [2, 3] courtesy of the authors. The top panels are from [2], while the lower ones are from [3]. Reprinted figure with permission from [2], Copyright (2006) by the American Physical Society. Reprinted figure with permission from [3], Copyright (2006) by the American Physical Society. The two papers were published in the same volume of Physical Review Letters. The top-left panels show the real and imaginary parts of the ( = 2, m = 2) mode of the waveform at various resolutions, as well as a convergence study. The top-right panel shows the individual horizons, first common horizon, and puncture trajectory. The bottom-left panel shows a comparison of waveforms at different extraction radii and resolution with the prediction of the Lazarus approach. The bottom right panel also shows the individual horizons, first common horizon, and puncture trajectory. should evolve the total energy density rather than internal out to be sufficient to recover W and T, but this will involve energy density. The resulting equations can be solved using some sort of root-finding process. established high-resolution shock capturing methods. This is the path followed in what has come to be called the Valencia 2.5.2. Excision and punctures in the presence of matter. If formulation [23, 116, 117], which all current numerical rela- the black hole interior is removed via excision, matter must tivity hydrodynamics codes essentially follow. The evolution be able to flow into the black hole and‘ disappear’ in a stable equations take conservative form way. For problems involving collapse to a black hole, one must maintain accuracy inside the collapsing object until an ∂tU + F = S (49) ∇· apparent horizon is located, after which a region inside this where the conservative variables U are horizon can be excised. Scheel et al [119] did this in 1D for spherical collisionless matter. Next, Brandt et al [120] intro- U =(ρ = W√γρ , Xiρ , (50) 0 duced a code for evolving nonvacuum black hole spacetimes using an isometry inner boundary condition at the apparent τ = √γα2T00 ρ , S = √γαT0 ). (51) − i i horizon. Techniques for matter excision using horizon pen- Conservative shock-capturing hydrodynamics codes in etrating coordinates were introduced roughly simultaneously numerical relativity have achieved at best 3rd-order conv by Duez et al [121] and Baiotti et al [122]. Optimal methods ergence [118]. for one-sided differencing of the fluid equations near the exci- After computing U at a new timestep, it remains to recover sion boundary are investigated by Hawke et al [123], although the original (‘primitive’) variables such as ρ0 and ui. It turns these initial simulations were less sensitive to this than to the

12 Rep. Prog. Phys. 82 (2019) 016902 Review gauge choices needed to keep the coordinates horizon pene- matter is very degenerate and in beta equilibrium, so the EoS trating. Excision is still the method used for simulations in the is effectively one-dimensional (1D): P = P(ρ0). generalized harmonic formulation (e.g. SpEC). A particularly These 1D EoS are conveniently parameterized as piece- elegant grid structure for hydrodynamic excision is provided wise-polytropes, for which the density is divided into inter- by the cubed-sphere arrangement [124]. vals, and each interval has its own polytrope law. For example, In NOKBSSN, it is much simpler to use moving puncture in the ith interval, covering the density range ρi 1 <ρ0 <ρi, − gauges and avoid explicit excision. One might worry that mat Γi the pressure is P = κiρ0 . The polytropic indices Γi and trans erial inflow into the puncture would cause numerical problems, ition densities ρi are free parameters. Γ0 and κ0 covers the but fortunately this turns out not to be the case. Numerical low-density range where the pressure, dominated by relativ- experiments showed that puncture simulations can handle stel- istic electrons, is known. The other κi are set by requiring P lar collapse to a black hole [125] and spherical accretion into to be continuous at ρi. Fortunately, only a few free parameters a black hole [126] with no code changes except a small extra needed to adequately cover the range of plausible EoS [128]. dissipation in the metric evolution (in [125]) and a means of At the end of inspiral, tidal disruption breaks beta equilib- resetting fluid variables near the puncture where conservative rium, as the matter decompresses faster than charged weak to primitive variable recovery fails (in [126]). Shortly after this interactions can adjust Ye. Also, the occurrence of shocks realization, Shibata and Uryu carried out the first NOKBSSN heats the matter so that it is no longer degenerate. A number black hole–neutron star merger simulations [127]. of studies add a Gamma-law thermal piece to the pressure to allow shocks to heat the gas. Any cold EoS can by thus aug- 2.5.3. More physics: equations of state, neutrinos, magn mented as follows: etic fields. Information about the properties of the matter enters through the equation of state. An extremely simple = cold(ρ0)+ th (52) but nevertheless useful equation of state is the polytropic law Γ P = Pcold(ρ0)+(Γth 1)ρth (53) P = κρ = (Γ 1)ρ0, where κ and Γ are constants. Higher − 0 − Γ means stiffer EoS. A notable feature of this EoS is that it where now th comes from the energy density evolution. The is barotropic; there is no explicit temperature dependence, thermal Gamma law may not capture important parts of the meaning the matter must be degenerate or the temperature true 3D EoS. This has been tested in the context of neutron must itself be a function of density (as, for example, in an star–neutron star binary mergers by Bauswein et al [129]. isentropic gas). In many cases, we may wish to allow an ini- They find that the thermal Gamma law approximation can tially polytropic gas to pick up added thermal pressure and alter the post-merger gravitational wave frequency by 2%– internal energy via shock heating. In this case, one uses the 8%, post-merger torus mass by 30%, and delay time to col- more general Gamma-law EoS P = (Γ 1)ρ0, where ε is lapse to a black hole by up to a factor of 2. In addition, only now given not by the polytropic law but by− the energy density 3D EoS provide the physical temperature information needed evolution equation. For adiabatic evolution, the polytropic law for neutrino calculations. should be maintained. One gets a surprising amount of mile- The evolution of the lepton number and Ye are given by weak age out of this simple EoS family. Nonrelativistic ideal degen- nuclear processes such as electron and positron capture, which erate Fermi gases have Γ=5/3; relativistic ideal degenerate emit neutrinos that travel some distance, as well as the reverse Fermi gases have Γ=4/3; stars with both radiation and gas absorption processes. Also, neutrino cooling is the dominant pressure with a constant fraction of the total from each have source of cooling in most simulations with neutron stars, and Γ=4/3. Neutron stars are not polytropes, but much of the neutrino absorption above the neutrinosphere can be an impor- early numerical relativity work involving neutron stars mod- tant driver of winds. Newtonian simulations, especially in eled them as Γ=2 polytropes. the supernova context, have long concerned themselves with Ultimately, an accurate treatment of dense matter is these effects, and around 2010 they began to be incorporated needed. For a general astrophysical gas, there may be many into numerical relativity simulations. At first, neutrino emis- composition variables Xi, each in need of its own evolution sion effects were approximated by local sink terms for the equation. However, when dealing with high densities and energy and lepton number (‘neutrino leakage’) [130–134]. temperatures above ∼MeV, the matter can be assumed to be in Effective emission rates differ in optically thick and optically nuclear statistical equilibrium, in which case there is only one thin regions; the neutrino optical depth can be computed by an composition variable, the proton fraction or electron lepton inexpensive iterative procedure [134, 135]. The current state- number fraction Ye = np/(np + nn). This variable evolves due of-the-art for numerical relativity is neutrino transport in an to charged-current weak nuclear interactions, which do not energy-integrated moment closure approximation [136–139], always have time to equilibrate. Thus, for our equation of state, which is impressive progress in so short a time but still far from we are left with functions of three variables, e.g. P(ρ, T, Ye). a full solution to the 6D Boltzmann equation. Unfortunately, they are unknown functions for densities much A final major piece of realistic matter numerical relativity above nuclear saturation, so numerical relativity simulations simulations is the electromagnetic field evolution. Neutron star of neutron stars must explore the range of equations of state interiors have plenty of free charges and very high electrical consistent with known nuclear and astrophysical constraints. conductivity, so the magnetohydrodynamic (MHD) approx For the problem most relevant for gravitational waves, com- imation is valid in most regions. Thus, we must add the Maxwell EM pact binary inspirals, the situation simplifies. The nuclear stress tensor for the electromagnetic field T µν to the total

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stress tensor Tµν, and so magnetic terms appear in τ and Si. initially stationary black holes for a short time. Later, with The evolution of the magnetic field Bi is given by the induction faster computers and improved algorithms, Smarr et al were equation—in words, that magnetic field lines are attached to [162–164] able to simulate head-on collisions through merger. (‘frozen into’) fluid elements. The electric field is set by the It was not until 1993 that computers were powerful enough to MHD condition that the electric field in the conducting fluid calculate accurate waveforms from such mergers [165]. rest frame vanish: αE = (v j + β j)Bk. At all times, the In the 1990s the National Science Foundation of the United i − ijk magnetic field should satisfy the constraint B = 0. In prac- States supported a large collaboration, the Binary Black Hole tice, magnetic monopoles are avoided by constrained∇· transport Grand Challenge Alliance, with the goal of advancing numer (staggering magnetic and electric variables so that the change ical relativity to the point where evolutions of orbiting black of B exactly vanishes [140]), by evolving a vector poten- holes became feasible. There were several important develop- tial∇· [141] (which, with appropriate staggering, is equivalent to ments enabled by the grand challenge. These include the full constrained transport [142, 143]), or by divergence cleaning 3D evolutions of boosted single black holes using excision (extending Maxwell’s equations so that monopoles damp and [166], perturbative techniques to extract gravitational wave- propagate off the grid [74, 144, 145]). forms from numerical simulations [167], stable evolutions of An interesting limit of the MHD equations occurs in mag- single black-hole spacetimes using characteristic techniques netospheres, where the Maxwell piece of Tµν dominates, so [168], and the development of toolsets for parallel simula- that τ and Si become essentially the electromagnetic energy tions. The alliance [169], and independently Brügmann [170], density and Poynting flux, respectively. Magnetospheres differ were able to evolve grazing collisions of black hole–black from vacuum electromagnetism because enough free charges hole binaries in 3D. However, these simulations crashed after remain to prevent electric potential differences along field a short evolution time ( 50M). lines (E B = 0). These conditions are expected to obtain in As mentioned above, the Lazarus approach [43, 101–105], the region· around neutron stars and the polar jet region around could be used to extend these simulations and generate wave- accreting black holes. Specialized codes have been developed forms from closely separated binaries. to evolve the relativistic force-free equations, evolving either These grazing collisions were performed using the ADM the electric and magnetic fields [146–148] or the magnetic system of equations evolved as a standard Cauchy problem. field and Poynting flux [149]. Lehner et al [150] introduce a Using characteristic evolution techniques, the PITT Null code scheme for evolving the full MHD equations in high-density [168, 171, 172] (developed at the University of Pittsburgh) regions and the force-free equations in low-density regions. could evolve highly distorted spacetimes for arbitrary lengths This scheme was successfully used to study the collapse of of time. For these long evolutions, the PITT code used a coor- magnetized neutron stars, but for future applications a single dinate system based on outgoing null (i.e. lightlike) geodes- set of equations able to handle both fluid and field-dominated ics. In a two black-hole spacetime, these geodesics would regimes was desirable. This was done by Palenzuela [151] form caustics in the vicinity of the two black holes, making in the context of a resistive MHD code. Resistivity and the the associated coordinate system singular. Thus the PITT force-free limit might sound like different issues, but in fact code could not evolve the interior of a black hole–black hole the inhibition of flow by charged particles across field lines binary. However, this code has since proven to be very useful in a magnetosphere can be modeled as an anisotropic resis- for evolving the exterior region in which gravitational waves tivity [146], so by allowing sufficiently general Ohm’s laws, propagate away from the central system. An interior Cauchy Palenzuela’s code can both handle resistivity inside stars and evolution an exterior characteristic evolution can be combined impose the force-free limit outside. Also, Paschalidis et al to produce highly-accurate waveforms in a technique known [152] have adjusted their MHD code to extend to the force-free as Cauchy-characteristic extraction [172–178]. i limit, showing that MHD and force-free (B , Sk) evolution just differ in the primitive variable recovery. These codes have been used to study magnetosphere interaction in the late inspiral of 3.2. Post-breakthrough results neutron star neutron star [153 155] and black hole neutron – – — With the breakthroughs of 2005, there was rapid progress in star binaries [156], which has been suggested as a mechanism our understanding of the physics of black hole black hole to create precurser signals to short duration gamma ray bursts. – binaries.

3. Black hole–black hole binary simulations 3.2.1. Recoils. One of the most remarkable results that came from these simulations is that the merger remnant can recoil at There have been several recent reviews of the history of thousands of kilometers per second. Determining just how fast numerical relativity [35, 36, 157–160]. Here we will briefly the remnant can recoil took several years and required many cover some of the major highlights. hundreds of individual simulations. Perhaps the most straightforward way to conceptualize why the emitted power due to an inspiral can have (instanta- 3.1. Early efforts neously) a preferred direction is to consider the case of une- Attempts at numerical simulations of black hole–black hole qual-mass black holes. The asymmetry of the system leads to binaries date back to the 1960s with the pioneering work of a small excess of radiation along the direction of the linear Hahn and Lindquist [161], who were able to simulate two momentum of the smaller black hole. For perfectly circular

14 Rep. Prog. Phys. 82 (2019) 016902 Review orbits, this effect would average out to zero over an orbit, but since the binary will also be inspiraling, the cancellation will not be exact and a net recoil will be generated. The net recoil only becomes significant during the fast plunge phase. Initial measurements of the recoil concentrated on analyzing individual configurations [179], or several configurations, but at very close separations [180]. The study of recoils started in earnest with Gonzalez et al [181]. Theirs was the first to do what was previously unheard of, a large number of relatively long-term accurate simulations. In the case of [181], they performed over 30 individual simulations and determined that the recoil very nearly obeys the simple form 2 1 ula V = 16 Aη √1 4η(1 + Bη), where A = 750 km s− − and B = −0.93, η = q/(1 + q)2 is the symmetric mass ratio, and q = m1/m2 is the usual mass ratio. Based on the formula provided by Gonzalez et al, the maximum recoil generated Figure 7. 1 A reproduction of figure 2 of [181] courtesy of the by unequal mass binaries is 175 11 km s− . As we will ± authors. The results of the first large-scale numerical relativity see, this contribution to the recoil can be vastly swamped by study. Shown are the measured recoils for over 30 binary contributions due to the spins of the black holes themselves. simulations, the estimated errors (the region between dotted Figure 7 shows the results of their study. curves), a fit (red curve), and various older approximations for Soon after, other groups showed that the maximum recoil recoils. The horizontal axis is the symmetric mass ratio defined as η = /( + )2. Reprinted figure with permission from for spinning binaries, where the spins are aligned and antia- m1m2 m1 m2 [182], Copyright (2007) by the American Physical Society. ligned with the angular momentum, is much larger. In [182] and [183], it was shown that the maximum recoil for an small spins, the recoil depends sinusoidally on the polar orien- equal mass, spinning binary with one black hole spin aligned tation (i.e. V sin θ). However, for larger spins, the recoil is ∝ with the orbital angular momentum and other antialigned substantially larger for smaller angles, as shown in figure 11. 1 1 is ∼475 km s− . A still larger recoil of V 525 km s− Of critical importance for modeling the superkick is the max ∼ for a mass ratio of q 0.62 was found in [185] when they dependence on mass ratio. It is perhaps surprising that even extended the analysis ≈of aligned/counteraligned spin binaries though the central supermassive black holes in galaxies with to unequal masses. a bulge can range in mass from under a million to tens of The recoils induced by unequal masses and aligned/ billions of solar masses, roughly ∼93% [195–197] of galac- counteraligned spins is always in the orbital plane of the binary tic mergers are expected to produce supermassive black hole (which, by symmetry, does not precess). [186] performed a set merges with mass ratios in the range 1/10 < q < 1. If the fal- of simulations that showed that the out-of-plane recoil, which loff of the recoil with mass ratio is steep, then even a 10:1 is induced by spins lying in the orbital plane, can be much binary may have a negligible recoil. Based on post-Newto- larger. These superkicks [186–190] were found to be up to nian theory, the group formerly at UTB, now at the Rochester 1 4000 km s− when the spins were exactly in the orbital plane. Institute of Technology (RIT), argued that this dependence The superkick configuration is quite interesting, not only should vary as q2 [186, 198]. This was first put to the test in because of the large recoil, but also because of the direction of [199], where the authors found the recoils fall faster than q2, at the recoil. Figure 8 shows the basic setup. The spins are anti- least in certain symmetric configurations. A follow-up series aligned with each other and in the in the orbital plane. Such of papers by the RIT group [200, 201] modeled the spin and a system will not precess and the orbital angular momentum mass ratio dependence for more generic configurations and will always point in the z direction. Furthermore, the system found a leading q2 dependence on the recoil for these more has π-rotation symmetry about the z-axis. This means the generic configurations. recoil cannot lie in the orbital plane. What actually happens Modeling the recoil for generic configurations is compli- in this case is the binary bobs up and down long the orbital cated by the need to perform hundreds of simulations. Even axis at ever increasing speeds until it merges. This bobbing is for a fixed mass ratio and zero eccentricity, the recoil depends controlled by the orientation of the spins, as shown in figure 9. on six spin degrees of freedom. In order to tackle this prob- The net effect is quite unexpected. The magnitude and direc- lem, the dimensionality of the free parameters needs to be tion of the recoil depends sinusoidally on the azimuthal orien- reduced. The basic procedure developed by the RIT group tation of the spins (see figure 10). Originally, it was thought in a series of papers [191, 200, 201] starts with a family of that these in-plane spins maximized the recoil, however, it was simulations related by a rotation of the azimuthal direction later found out in [191, 193, 194] that, due to the hangup and of the spins at the initial separation (the spins of both black other nonlinear-in-spin effects [112], having partially miss- holes are rotated the same amount). This leads to a 1-param aligned spins actually leads to a substantially larger recoil (up eter family of configurations with (typically) a nearly sinusoi- 1 to 5000 km s− ). The basic setup of this hangup-kick con- dal dependence of the recoil. The amplitude of this sinusoidal figuration is very similar to the superkick, with the exception dependence is then measured as a function of the polar spin that the out-of-plane components of the spins are aligned. For orientations. This procedure works well when the azimuthal

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Figure 8. A sketch of the superkick configuration. Spins are entirely in the plane and anti-aligned.

spins are antialigned, but is of unknown utility for completely generic configurations. Another method used there was to Figure 9. The bobbing of a binary studied in [191]. The coordinate consider the subspace where only one of the black holes is distance of the binary from the original orbital plane is shown spinning (either the smaller or larger one). This again reduces for three azimuthal variations of the hangup-kick configuration. Reprinted figure with permission from [192], Copyright (2013) by the dimensionality of the problem. Even so, the number of the American Physical Society. individual simulations required to generate the latest models for the recoil was 200. these types of binaries have been studied for their use in wave- The discovery that merging black holes can recoil at thou- form modeling with examples in the SXS [213, 214], Georgia 1 sands of km s− sparked many searches for recoiling super- Tech [215, 216] and RIT [217, 218] catalogs. More recently, massive black holes. These searches are ongoing. For example, because of the apparent lack of observed highly-recoiling QSO 3C 186 was recently proposed as a candidate recoiling AGN, the RIT group began a systematic study of the spin 1 black hole with recoil of order 2000 km s− [202]. An active and mass ratio dependence of the recoil for aligned/counter galactic nucleus (AGN) is thought to be a candidate for a recoil- aligned binaries in [185, 219]. ing supermassive black hole if there is a red/blue shift between broad line and narrow line emissions. Gas tightly bound to the 3.2.2. Modeling the remnant properties. One of the impor- central black hole would have much higher velocity disper- tant tasks required in order to make the wealth of informa- sion (which is a function of the kinetic energy of the gas) than tion from numerical simulations useful for astrophysics was to gas further out. If a binary merges and the remnant recoils, model the radiated energy-momentum and the corresponding gas close to the remnant will remain bound and recoil with the final mass, spin, and recoil of the remnant black hole from remnant, while gas further out is left behind. This then would black hole–black hole binary mergers in terms of the initial lead to two different redshifts for gas that remains bound to parameters of the binary. Developing these models required the central black hole and the rest. To date, no source has been thousands of computationally expensive simulations. definitively shown to be a recoiling remnant black hole. For a In developing these models, two different techniques were recent history of these searches, see [203]. initially used, but current models now combine aspects of If large recoils are common, then why are there not more both. The first technique used post-Newtonian theory [181, candidates? Since only gas-rich mergers lead to luminous 183, 186, 188, 189, 198–200, 207, 210], or other physical signals that can be detected electromagnetically, it may well arguments [212, 220, 221] to determine the functional form be that a recoiling AGN cannot be luminous. Newtonian and and free parameters of an approximate model for the relevant post-Newtonian simulations appear to indicate that accretion quantity, and the other used ad-hoc expansions [112, 209, will tend to align or counteralign the black hole spins with 211, 222]. The work of [209, 211] pioneered the technique the orbital angular momentum [204–206]. Depending on the of using symmetry arguments to limit the degrees of free- degree of alignment, this may essentially suppress the super- dom in the models. Their construction only assumed that the kick style recoils. There has therefore been a resurgence of remnant can be described by the spin vectors of each black interest in modeling recoils for spin-aligned systems. hole and the mass ratio. The model then must obey the fol- The modeling of recoils from binaries with spins aligned lowing two symmetries. If F( χ 1, χ 2, q) is a formula for the and counteraligned with the orbital angular momentum began remnant, mass, spin vector, or recoil, then F must obey soon after the breakthroughs in numerical relativity. The first F( χ 1, χ 2, q)=F( χ 2, χ 1,1/q), i.e. the physical outcome of a such simulations were performed in [182] and [183], with merger cannot depend on the labels (1, 2) of the two black the first systematic studies of the recoil from such binaries holes. Second, if F must transform appropriately under par- in [207] and [208]. And the process of generating empirical ity. One, however, need not use the variables ( χ 1, χ 2, q). models for the remnant masses, spins, and recoils from such Inspired by post-Newtonian expressions, the RIT group has mergers were first performed in [198, 209–212]. In addition, made extensive use of the variables (∆ , S, δM) as well the

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Figure 10. (Left) The measured recoil for an equal-mass, superkick configuration [190] with spins χ = 0.9 and various azimuthal orientations. Note that very large and very small recoils are both possible. Reprinted figure with permission from [191], Copyright (2011) dP by the American Physical Society. (Right) The radiated power dΩ per unit solid angle for a configuration studied in [192]. Note the large excess of power directed upwards, which leads to a downward kick. Reproduced from [193]. (c) IOP Publishing Ltd. All rights reserved.

185, 201, 219, 226–228]) with the goal of reducing systematic biases (which may arise from different groups concentrating on different regions of parameters space).

3.2.3. Numerical relativity at the extremes. State of the art numerical relativity codes now routinely evolve binaries with mass ratios as small as q 1/10 [216, 229–233], moderately- to-highly precessing systems [186, 191, 200, 201, 234–242], and binaries with moderate spins. However, much smaller mass ratios, and spins close to 1 are still quite challenging. Prior to the work of [243] it was not even possible to construct initial data for binaries with spins larger than ∼0.93 [244]. This limitation was due to the use of conformally flat initial data13. Conformal flatness of the spatial metric is a convenient assumption because the Einstein constraint system take on par ticularly simple forms. Indeed, using the puncture approach, the momentum constraints can be solved exactly using the Figure 11. The measured maximum recoil for equal-mass binaries in a hangup-kick configuration as a function of the polar orientation Bowen-York ansatz [245]. There were several attempts to gen- of the spins [193, 194]. erate data for highly-spinning black hole–black hole binaries, while still preserving conformal flatness [246, 247], but these variables η and S0 [185, 190–191, 200, 201, 219, 223–225]. introduced negligible improvements. Lovelace et al [243] These are defined as were able to overcome these limitations by choosing the initial data to be a superposition of conformally Kerr black holes S = m2 χ , (54) 1 1 1 in the Kerr–Schild gauge. Using these new data, they were soon able to evolve binaries with spins as large 0.97 [248], and 2 S2 = m2 χ 2, (55) later spins as high as 0.994 [249]. While spins of 0.92 may seem reasonably close to 1, the 2 S =(S1 + S2)/m , (56) scale is misleading. The amount of rotational energy in a black hole with spin 0.9 is only 52% of the maximum. Furthermore, ∆=(S2/m2 S1/m1)/m, (57) particle limit and perturbative calculations show even more − extreme differences between spins of 1 and spins only slightly δm =(m m )/m, (58) 1 − 2 smaller. For example, Yang et al [250] studied an analog to turbulence in black-hole perturbation theory. For spins close to 1, S0 = S +(1/2)δm∆. (59) there is an inverse energy cascade from higher azimuthal (m) 2 modes to lower ones for modes that obey = 1 χ − . Any expansion in terms of one set of variables can be reex- | − | pressed in terms of an other. However, since the goal is to This give hints that a more informative measure of the spin is model the remnant with accuracy, one wants to use variables actually 1/. Similarly, analysis of Kerr geodesic [251, 252] that minimize the number of free parameters required to fit and particle-limit calculations of recoils [253, 254] indicate the known data. that the dynamics of nearly-extremal-spin black holes cannot State of the art models for remnant properties now combine 13 Initial data are said to be conformally flat if the spatial metric associated results from simulations of many different groups (see [184, with the data is proportional to the flat space metric.

17 Rep. Prog. Phys. 82 (2019) 016902 Review and numerical relativity predictions for the orbital frequency was found for the D 50M cases. The longest simulations to merger published to date was in [261], where a binary was evolved for 175 orbits. For reference, an equal-mass binary at a separation of D = 100M will complete over 2000 orbits before merging and requires about 8.2 106M of evolution × time. (If the mass of each black hole in the binary is 30M , then the merger from D = 100M would take about 40 min.)

3.2.4. Waveform modeling. One of the major goals of numer ical relativity is to produce accurate waveforms for gravitational wave data analysis. The actual process of obtaining the waveform from a numerical simulation can be involved. See [262] for a review of modern techniques for extracting the gravitational waveform from a numerical simulation. In figure 13, we show an example waveform from a recent black hole–black hole binary simulation [239, 263]. In order Figure 12. The merger of a 100:1 mass ratio binary [231]. The to use physical units, we consider a binary that has a total smaller black hole is a factor of nearly 4 times smaller than mass of 20M . The plot shows the waveform from the last might be expected by a mass ratio of 1:100. Reprinted figure with 48 orbits for an equal-mass binary in a precessing configura- permission from [232], Copyright (2011) by the American Physical Society. tion. There are two distinct phases of the waveform. The longest phase is due to the slow inspiral and eventual plunge. In be elucidated with any degree of certainty using lower spin the figure, this corresponds to the start of the waveform until simulations. about 1.92s. Small oscillations in the amplitude are apparent. Recently, the group at RIT also introduced their version of These are due to precession of the orbital plane. The overall highly-spinning initial data, also based on the superposition ramp up of both the amplitude and frequency is due to the of two Kerr black holes [255, 256], but this time in a puncture inspiral and its associated increase in the orbital frequency. gauge14. The main differences between the two approaches is Following a brief transition between 1.915s and 1.925s, the how easily the latter can be incorporated into moving-punc- waveform changes to a damped sinusoid. This phase is due to tures code. They compared their results to the SXS results for the rapid equilibration of the now single black hole. both spins of χ = 0.95 [255] and χ = 0.99 [257], and found Remarkably, on September 14, 2015 the twin LIGO obser- very good agreement. vatories detected the gravitational waveform from the inspiral The other type of extreme simulation concerns small mass and merger of two black holes [4, 5]. The resulting waveform ratios. Because current numerical relativity codes use explicit is shown in figure 14. algorithms to evolve the spacetime, the Courant–Friedrichs– In this section we will review the history of fully non-lin- Lewy condition, which determines how large a timestep can ear numerical simulations of black hole mergers to generate be relative to the spatial discretization, severely limits the run and verify the waveforms. The current generation of numer speed when one of the black holes is much smaller than the ical relativity codes calculate the gravitational waveform of a other. Basically, the number of timesteps required near the merger simulations by calculating the Regge–Wheler–Zerilli smaller black hole is set by the size of that black hole, not by perturbations (which can be related to the strain h), the Bondi its dynamics. This, coupled to the fact that the inspiral for a News function N, or Weyl scalar ψ4 (as well as combinations small-mass-ratio binary is much slower than for a similar mass of the above). With the exception of the calculation of N using one, means that such simulations are extraordinarily expen- Cauchy-characteristic extraction [174, 175, 178, 264, 265], the sive. To date, the smallest quasi-circular inspiral evolved so waveform is calculated at a series of finite radii and extrapolated far had q = 1/100 [231, 232]. to r = along an outgoing null (lightlike) paths using some ∞ One promising method to overcome these limitations is to form of either polynomial extrapolation, or perturbative expan- use semi-implicit techniques [258, 259], but these have not yet sion [266]. In a suitable gauge, ψ = N¯˙ = h¨, where an overbar been shown to work for small-mass-ratio binaries. 4 denotes complex conjugation and a dot represents a time deriva- Finally, in [260] another extreme was explored: that of tive. Note that gravitational wave detectors measure h directly, binaries at far separations. There the authors used fully non- while the emitted power is directly related to N. The points at linear numerical relativity to model several orbits of binaries r = along outgoing null rays are collectively known as future separated at D = 20M, D = 50M, and D = 100M and com- null ∞infinity. While these points are formally outside the space- pared the orbital dynamics to post-Newtonian and Newtonian time, they are quite useful for defining gravitational radiation. predictions. Very good agreement between post-Newtonian For an isolated source15, the Bondi News function N is 14 Recall that initial data in a puncture gauge is constructed by mapping the directly related to the radiated power-per unit solid angle in two infinitely large spacelike hypersurfaces of an Einstein–Rosen bridge into the gravitational radiation by a single spacelike hypersurface with a singular point. Initial data for a binary would then have two such singular points. 15 More precisely, for an asymptotically flat spacetime.

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Figure 13. The waveform (real and imaginary components of the (2,2) mode of the rescaled strain) from a precessing binary with total 16 mass 20M [239, 263]. To obtain the physical strain, the values plotted need to be rescaled by a factor of 9.7 10− /D, where D is the × distance to the binary in units of kiloparsecs.

ingoing radiation). This class of tetrads, known as quasi-Kin- dP 1 nersley tetrads [105, 270 272] is unique up to an overall phase = (NN¯) . (60) – dΩ 4π factor and normalization. Since there are no analytically known waveforms from N itself is defined on future null infinity, and the only gauge the mergers of black holes, in order to test the correctness of freedom in N is associated with the supertranslation freedom numerically derived waveforms one needs to both carefully N(τ, xA) N(τ + σ(xA), xA + τωA(XA)), where τ is an audit the codes and compare results generated from different → affine parameter (a generalization of proper time along light- code bases. The first such comparison [273] was performed like curves), xA denotes angular coordinates, and σ(xA) and ωA early on with waveforms generated by the inspiral of an are constant functions of angle. equal-mass, low-spin binary16 obtained using the LazEv code However, as standard Cauchy codes cannot include future [2, 44] developed at Brownsville and Rochester Institute of null infinity (but see [267–269] for an approach which may Technology, the Hahndol code [48, 274, 275] developed at allow evolutions that include future null infinity), calculations NASA-GSFC, and Pretorius’ original code [1, 93]. For that of N involve a matching procedure, where data from a Cauchy test, each group evolved similar binaries, but at slightly dif- code is used as boundary data for a characteristic evolution. ferent initial configurations. The results from this first com- This matching requires the specification of unknown data parison are shown in figure 15. Later on, as more groups from the edge of the Cauchy domain to null infinity. This developed their own codes, more large-scale comparisons induces spurious radiation [178], which can be controlled by were performed. For the Samurai [276] project, comparisons moving the matching procedure to farther radii. were made between the SpEC [277, 278] code developed by In Chu et al [233] the authors compared their extrapola- the SXS collaboration, the Hahndol code, the MayaKranc tions of the Regge–Wheeler–Zerilli perturbations to the code [279] developed at Penn State / Georgia Tech, the CCATI News calculation. They found that errors due to both gauge code [207] developed at the Albert Einstein Institute, and the effects and extraction at finite radii lead to mismatches of BAM code [39, 49] developed at the University of Jena. One 4 a ∼5 10− . It is interesting to note that this mismatch may of the major differences between the Samurai project and × be geometrical in nature and arising from the difficulty in [273] was the use of simulated LIGO noise data to determine defining gravitational radiation at a finite distance from the if the differences between the waveforms generated by the source. various codes is, in practice, detectable. The other main technique for extracting radiation involves the calculation of the Weyl scalar ψ4. Calculations of ψ4 have the advantage that there is a simple, well defined procedure 16 The binary evolved by Pretorius had a very small dimensionless spin of for calculating the Newman–Penrose scalars ψ4 in a class of 0.02, while the binaries evolved by the LazEv and Hanhdol codes were tetrads where ψ4 represents the outgoing radiation (and ψ0 the nonspinning.

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Figure 15. The first comparison of numerical relativity waveforms produced by different codes and different groups. The figure is a reproduction of figure 1 of [273] and shows a comparison of waveforms generated by the groups at NASA GSFC, UT Brownsville and RIT, and Frans Pretorius. Note that the initial binary configuration used by Pretorius was a corotating configuration, which has a moderate spin, while the other two configurations were nonspinning. Hence the waveform amplitude and frequency are not expected to be identical to the other two waveforms. Reproduced from [274]. © IOP Publishing Ltd. All rights reserved.

One of the first systematic studies of how sensitive the wave- Figure 14. These plots show the signals of gravitational waves form is to the parameters of the binaries in the non-precessing detected by the twin LIGO observatories at Livingston, Louisiana, case was performed in [635], where they studies the effects of and Hanford, Washington. The signals came from two merging spins on waveform detectability. black holes with masses 30 and 35 times the mass of our sun, respectively, lying 1.3 billion light-years away. The top two plots show data received at Livingston and Hanford, along with the 3.2.5. Semi-analytic waveform models. Because fully predicted shapes for the waveform. These predicted waveforms nonlinear numerical simulations of black hole mergers are show what the waveform from two merging black holes with computationally expensive, many semi-analytic and empiri- these masses should look like according to the equations of Albert cal approaches have been developed in order to model the Einstein’s general theory of relativity, along with the instrument’s waveform of black hole mergers using information from pre- ever-present noise. Time is plotted on the X-axis and strain on the Y-axis. Strain represents the fractional amount by which distances viously modeled simulations [214, 227, 228, 230, 233, 235, are distorted. As the plots reveal, the LIGO data very closely match 237, 238, 248, 261, 276–278, 280–298, 299–316]. As demon- Einstein’s predictions. The final plot compares data from both strated in [317], the systematic errors associated with wave- detectors. The Hanford data have been inverted for comparison, forms models used by the LIGO Scientific Collaboration led due to the differences in orientation of the detectors at the two to negligible errors in the inferred parameters of the source sites. The data were also shifted to correct for the travel time of the gravitational-wave signals between Livingston and Hanford (the of GW150914. signal first reached Livingston, and then, traveling at the speed of One conceptually straightforward method for constructing light, reached Hanford seven thousandths of a second later). As a waveform model would be to interpolate known numerical the plot demonstrates, both detectors witnessed the same event, waveforms as a function of the source parameters. The actual confirming the detection. (Courtesy Caltech/MIT/LIGO Laboratory. details of how this interpolation is performed is quite sophis- Image Credit: Caltech/MIT/LIGO Lab). ticated and requires first representing waveforms in a reduced Such comparisons took on more urgency with the detec- basis. The resulting models [242, 310, 316, 318] have been tion of gravitational waves in 2015 [4, 5]. In the process of shown to be accurate, at least under the conditions tested [242]. verifying that detection two groups, the SXS collaboration Another class of models, usually referred to as phenom- and the group at RIT, generated waveforms as part of their enological models [281, 286, 292, 309, 311, 319], are based work for the LIGO scientific collaboration. Despite the two on an expansion of the waveform in Fourier space. The ampl codes sharing no common routines, and using different initial itude and phase of the waveform are expanded as algebraic data generation techniques, different evolutions techniques, functions of frequency. The coefficients in these expressions and different waveform extraction techniques, the dominant are then modeled as functions of the binary’s parameters by ( = 2, m = 2) modes produced by the two codes agreed to fitting to existing waveforms. better than 99.9% [280]. Yet another method for modeling the waveform from a One of the main goals of waveform modeling is to infer binary, when the two black holes are still far apart, is based on the properties of the source based on the observed waveforms. a series expansion in the black-hole separation and velocity,

20 Rep. Prog. Phys. 82 (2019) 016902 Review known as the post-Newtonian expansion. For a review of post- accretion onto the black holes would be mostly frustrated. As Newtonian theory, see [320]. the binary inspirals, eventually the inspiral timescale becomes 17 Since post-Newtonian theory is expected to be accurate smaller than the disk’s viscous timescale , presumably caus- when the binary separation is large, and become increasing the disk to decouple from the binary, its inner edge unable ingly more inaccurate during the inspiral, a natural ques- to keep up with the shrinking binary. Newtonian 2D (vertically tion is to determine where the post-Newtonian waveforms summed) [335, 336] and 3D [337–339] simulations confirm differ substantially from the numerical ones. The first direct the evacuation of the region around the binary, but find that comparisons of the waveform predictions from numerical gas is efficiently carried in narrow accretion streams from the relativity and post-Newtonian theory were performed by inner disk to the black holes. Meanwhile, perturbations to the Buonanno, Cook, and Pretorius [321], shortly thereafter by disk caused by the merger itself, with the associated mass loss the NASA-GSFC group [319, 322], with the group at Jena and kick of the central system due to gravitational waves, have following soon after that [283, 284, 323]. These early com- been investigated by artificially reducing the mass and adding parisons were between post-Newtonian and numerical relativ- linear momentum to the central object around an equilibrium ity predictions of the waveforms for non-precessing systems. disk [340–344]. All of this suggests that high luminosity can The first comparison of precessing waveforms was done in be maintained after decoupling and through merger. [235]. All of these comparisons were for relatively short Numerical relativity studies of fluids near black hole–black waveforms and in a regime where post-Newtonian theory hole binaries began shortly after the moving puncture revo is not particularly accurate. The longest comparison to date lution. It is not clear that this had to be the case. Some of the between post-Newtonian and numerical relativity waveforms most advanced recent works consider disks around inspiral- was a 175 orbit evolution performed in [261]. Other studies ing binaries without dynamically evolved spacetimes. For of much more separated binaries (where the evolutions were example, Noble et al [345] used a 2.5 post-Newtonian-order not taken to merger) showed good agreement in the dynamics approximation to the binary spacetime further than 10M from between post-Newtonian and numerical relativity for several the binary combined with a 3.5 post-Newtonian approximation orbits with separations of 100M and 50M for an equal-mass, to the binary orbital evolution, while Gold et al [346] simply nonspinning binary [260]. rotated their conformal thin sandwich initial data. Nevertheless, While current post-Newtonian waveforms are not par the first numerical relativity treatments did include spacetime ticularly accurate during the late-inspiral phase, there are evolutions through merger. These early studies by Bode et al models inspired by post-Newtonian theory that reproduce [347–349] and Farris et al [350] considered binaries immersed numerical waveforms with greater accuracy. The Effective in low-angular momentum gas (advection-dominated / Bondi- One Body formalism [324–328] recasts the problem of the like inflow) and calculated electromagnetic luminosity from evolution of the binary as an effective field theory for a sin- bremsstrahlung and synchrotron emission. gle particle. This formalism contains free parameters which Clearly, magnetic field effects might have important effects can be modeled using numerical relativity simulations. The on these inflows. Prior to MHD simulations, Palenzuela et al resulting formalism EOB-NR can reproduce gravitational [351] and Moesta et al [352] performed force-free simula- waveforms, at least for non-precessing binaries, with great tions of the effect of a black hole–black hole binary merger accuracy. Furthermore, these waveforms are produced at a on nearby magnetic field lines (presumed to be anchored to fraction of the cost of the original numerical simulations a circumbinary disk outside the computational domain). An [288, 289, 291, 303, 329–334]. interesting finding of these simulations, shown in figure 16, is the appearance of dual jets by a sort of binary system generalization of the Blandford Znajek effect; in this case energy 4. Relativistic stars and disks – is extracted from the orbital motion of the binary, rather than the spin energy of a black hole (the latter being the classic 4.1. Black hole–black hole binaries with accretion Blandford–Znajek effect). (It should be noted, though, that Astronomers expect that the environment of supermassive while multiple studies confirm the presence of dual jets, they black hole–black hole binary mergers will often be gas-rich; also show that the emission is predominantly quadrupolar therefore, there is hope for an electromagnetic counterpart to [352, 636].) the (low-frequency) gravitational wave signal. In a thin accre- Newtonian MHD simulations by Shi et al [353] found tion disk around a single black hole, angular momentum flows that magnetohydrodynamic disks (as opposed to previous outward (an effect of MHD turbulence), causing gas to slowly disks with alpha viscosity) accrete more rapidly and experi- spiral inward, releasing energy radiatively as it falls deeper ence stronger tidal torques. Soon afterward, numerical rela- into the gravitational potential. (Thus, the more compact tivity MHD simulations were carried out for the low-angular the object, the more efficiently accretion onto it can release momentum plasma case by Giacomazzo et al [354] and for the energy.) A black hole–black hole binary near merger might circumbinary disk case by Farris et al [355]. Using numerical be accompanied by gas orbiting the binary itself, forming a relativity MHD and a post-Newtonian black hole–black hole ‘circumbinary disk’. Early 1D studies of circumbinary disks predicted that gravitational torques from the binary would 17 The viscous timescale is the timescale on which angular momentum clear out a region of radius about twice the orbital separation transport moves gas inward. The name comes from the common practice of (for binaries with mass ratio around unity), suggesting that modeling this transport process with a viscosity.

21 Rep. Prog. Phys. 82 (2019) 016902 Review

(a) (b)

(c)(d)

Figure 16. Electromagnetic energy flux at different times in the force-free magnetosphere surrounding a black hole–black hole binary merger. The collimated part is formed by two tubes orbiting around each other following the motion of the black holes. A strong isotropic emission occurs at the time of merger, followed by a single collimated tube as described by the Blandford–Znajek scenario. From [352]. Reprinted with permission from AAAS. (a) 11.0 M h. (b) 3.0 M h. (c) 4.6 M h. (d) 6.8 M h. − 8 − 8 8 8 binary metric, Noble et al [345] showed that neither binary of further surprises as future simulations incorporate long torques nor decoupling reduce the overall accretion rate by evolutions as well as radiation transport with associated ther- a large factor. Gold et al [346, 356] have carried out numer mal effects. ical relativity MHD simulations varying the binary mass ratio between 1:1 and 1:10, confirming that magnetic fields boost 4.2. Relativistic stars accretion rate, increase shock heating, and produce dual jets merging into a single jet at large distances. After decoupling Numerical relativity is the main tool for studying rapidly but prior to merger, the jets coalesce into a single jet. The rotating relativistic stars, where it is used to test the stabil- merger leads to a one-time boost in the jet’s magnetic field ity of equilibrium configurations and the nonlinear evolution strength and outflow velocity, which the authors hope can pro- driven by instabilities. Because the outcome of these insta- vide a signature of black hole–black hole binary merger (as bilities often involve black holes, relativistic stars will be a opposed to a single black hole disk flare). notable part of our story. For a full treatment of this topic, see Computational cost has forced nearly all simulations to the Living Review article by Paschalidis and Stergioulas [359, date to either excise the inner region containing the binary 360], the book on this subject by Friedman and Stergioulas or, for those cases that do track flow into the separate black [361], and also chapter 14 of Baumgarte and Shapiro [35]. holes, evolve for less than a viscous timescale. To observe vis- Collapsing star simulations did not have to wait for the cously settled accretion flows, Farriset al [357] performed black hole problem to be solved. Simulations up to the turn long-term pre-decoupling 2D Newtonian evolutions resolving of the century could use singularity avoiding slicings such the inner region. They find that individual‘ mini-disks’ form as maximal slicing and its approximates to follow collapse around each black hole. Bowen et al [358] have studied these a short while past apparent horizon formation before grid mini-disks in their post-Newtonian spacetime, inserting disks stretching effects (increasing distortion of slices needed to around each hole and evolving them to an overall steady state. keep them from intersecting the singularity) destroyed the The discovery of these ‘mini-disks’ illustrates the possibility run’s accuracy. Evolutions to late times after collapse, and 22 Rep. Prog. Phys. 82 (2019) 016902 Review

evolutions of systems like black hole—neutron star binaries, 4.2.2. Magnetohydrodynamic evolution. These hypermas- which have a black hole throughout, had to wait until general sive neutron stars, although dynamically stable, are presum- black hole spacetimes could be stably evolved. ably driven to collapse on a secular timescale by processes that transport angular momentum outward, robbing the core of its 4.2.1. Radial stability and collapse outcome. Supramassive rotational support. The ultimate source of angular momentum and hypermassive stars can be created using 2D stellar equilib- transport is most likely turbulence driven by the magnetoro- rium codes (see [360]). Knowing that these equilibria exist, we tational instability. To study secular evolution from first prin- next consider whether they are dynamically stable, and if not ciples requires magnetohydrodynamic (MHD) simulations. whether the instabilities are of a kind to destroy the equilibrium Such simulations must specify an initial state for the magn on a dynamical timescale or if they just introduce some small- etic field. One injects a small magnetic field into the equilibrium state, usually not chosen to be an MHD equilibrium scale ‘churning’ with effects on a secular timescale. Stability concerns the behavior of initially small perturbations. Numer but thought of as a ‘seed’ of the more physical magnetic field ical error provides perturbations on its own, but it is resolu- that will grow through shear and turbulence. MHD simulation-dependent, so stability studies often seed perturbations. A tions suggest that the magnetic field in a star with random popular method for studying radial stability in stars is pressure initial state tends to settle to a helical mixture of poloidal and depletion, a slight reduction of pressure below the equilibrium toroidal field [369]. Equilibrium fields can be constructed in requirement. This can be done in a way that preserves the con- relativity (e.g. [370–373]), but simpler seed fields are more straints and respects the equation of state by simply holding τ often used, e.g. poloidal fields moving along isodensity contours constructed from an azimuthal vector potential and Si fixed and slightly increasing ρ [362]. To study the stability of nonaxisymmetric modes, these modes can be seeded 2 n Ab (ρ0 ρcutoff) forρ0 >ρcutoff by nonaxisymmetric perturbations of the density. A = − , φ 0 otherwise Because no black hole is involved (at least until after (61) instability has clearly manifested itself), numerical relativity where ϖ is the cylindrical radius; Ab, n, and ρcut are freely could begin addressing these questions even before the break- specifiable constants. The expectation was that, for magne- throughs numerical relativity in 2005 that allowed for the torotationally unstable systems, no memory of the seed field evolutions of orbiting black hole–black hole binaries. Already would long survive. Black hole-torus simulations give some in 2000, simulations by Shibata et al [363] showed that the support to this assumption for the interior of the torus but find dynamical instability point for uniformly rotating supra- that the appearance and strength of polar jets is very sensitive massive n = 1 polytropes nearly coincides with the secular to seed field geometry [374]. instability point. Despite the rapid rotation, when pressure- Differential rotation with dΩ/dr < 0 (the usual case for depleted unstable stars collapse to black holes, they leave differential rotation) is unstable to the magnetorotational almost no disk. A follow-up study of supramassive polytropes instability (MRI) [375]. The fastest growing MRI mode has with various polytropic index n < 2 near their mass-shedding size λMRI vA/Ω, where vA B/√ρ0 is the Alfven speed, ∼ 1 ∼ limit also found almost no disk mass around the post-collapse and growth timescale ∼Ω− . For realistic B fields, resolving Kerr black holes [364]. These simulations used singular- λMRI can be a serious computational challenge. Numerical rel- ity avoiding slicings and could not evolve long after horizon ativity MHD studies sometimes avoid this problem by using formation. Subsequent studies using excision after apparent magnetar-strength seed fields. The MRI has been identified in horizon location confirmed the no-disk result for uniformly relativistic disk (e.g. [376–379]) and star [380] simulations. rotating stars with stiff polytropic EoS [121, 122]. Based on The effect of the MRI is to initiate turbulence and thus on angular momentum distribution at the maximum mass con- a secular timescale to dissipate energy as heat and transport figuration, it is expected that this result carries over to realistic angular momentum outward. The MRI’s role in driving disk neutron-star EoS [365]. (In section 5, we will see that the story accretion is the subject of a vast amount of work; the distinc- is quite different for non-compact stars with soft EoS.) After tive role of numerical relativity has been to study its effect on moving puncture gauge conditions were discovered, it was differentially rotating neutron stars. possible to evolve to a final black hole state and confirm that Axisymmetric simulations of magnetized hypermassive the metric matches the spinning puncture form [366]. neutron stars were undertaken by Duez et al [367] using a Numerical relativity provides a straightforward way to test Γ=2 EoS and initial angular velocity about three times the dynamical stability of hypermassive neutron stars: just higher at the center than at the equator. These simulations evolve for several dynamical times (perhaps with some initial were notable for the narrative of this article as being among perturbation). By this test, Baumgarte et al [29] demonstrated the first astrophysically interesting numerical relativity sim- max the stability of a model with mass around 1.6M TOV. It is ulations to prolong evolution past collapse using excision. easy to construct differentially rotating compact stars with (Shortly afterward, both groups in the collaboration—Uni- spin angular momentum on either side of the Kerr limit, so versity of Illinois at Urbana-Chanmpaign (UIUC) and Kyoto inducing their collapse by sufficient pressure depletion pro- University—switched to moving punctures.) A combination vides a test of cosmic censorship; unsurprisingly, it passes of magnetic winding/braking and MRI turbulence transports [121, 368]. The super-Kerr systems undergo a centrifugal angular momentum outward, causing the envelope to expand bounce, and the fluid forms a torus which then fragments due and the core to contract. After about 102 ms, the core under- to nonaxisymmetric instabilities. goes collapse on a dynamical timescale. The collapse leaves

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Figure 17. The evolution of a hypermassive star under the influence of a seeded magnetic field. The upper 4 panels show snapshots of the rest-mass density contours and velocity vectors on the meridional plane. The lower panels show the field lines for the poloidal magnetic field at the same times as the upper panels. The thick solid (red) curves denote the apparent horizon which appears when the central region collapses. Reprinted figure with permission from [368], Copyright (2006) by the American Physical Society.

a massive torus surrounding the black hole, a promising setup differentially rotating or toroidal stars at T/ W well below the | | for a short-duration gamma ray burst [381]. Snapshots from dynamical bar mode threshold [386–389]. The instability has this evolution are shown in figure 17. The same collaboration been seen in numerical relativity rotating stellar core collapse then performed similar studies for other differentially rotating simulations [390] and so may be an important gravitational neutron star initial configurations [382]. Use of a more realis- wave source from a galactic supernova. As the mechanism tic EoS produces a qualitatively similar outcome, but collapse was not at first understood, the instability was called the‘ low is averted (unsurprisingly) if the star is not hypermassive. A T/ W instability’ or the ‘one-armed spiral instability’, names non-hypermassive star with angular momentum too high for that| sometimes| persist. We now know that it is caused by a a uniformly rotating star settles to an equilibrium uniformly corotation resonance [391]. The corotation radius rc of a mode rotating star plus torus configuration. is the radius where Ω(rc)=Ωp. The mode has positive energy for r > rc and negative energy for r < rc, and thus energy can 4.2.3. Nonaxisymmetric mode instability. Nonaxisymmetric be transferred outward at rc to strengthen the mode on both i(Mφ ωt) modes, which have the form δ e − , are interesting sides. The corotation instability takes many crossings to grow, ∝ as gravitational wave sources. Since the background equilib- so the mode must be trapped in a region containing rc. In stars, rium is often differentially rotating, there is a clear conceptual a minimum of the vortensity can act like a trapping potential difference between the fluid’s rotation and the perturbation [392], and this can be produced by a toroidal density structure mode, which rotates with constant pattern speed Ωp = ω/M or extreme differential rotation. Subsequent numerical simu- everywhere. In fact, rotating perfect fluid stars are generi- lations are consistent with this model [393, 394]. Bar mode cally unstable because of gravitational waves via the Chan- growth from both the low-T/ W [394] and the high-T/ W | | | | drasekhar–Friedman–Schutz instability, although in most [395] instabilities has also been simulated in the presence of realistic cases this is suppressed by viscosity or other effects. magnetic fields, where magnetic tension fails to suppress the (See [360] and references therein.) instabilities for any realistic field strength. It is well-known that for rotating stars, the fundamental L = M = 2 (bar) mode becomes unstable for sufficiently high 4.2.4. Stability of self-gravitating black hole accretion T/ W : around 0.14 for a secular instability and around 0.27 disks. Given the ability to evolve matter in dynamical black for| a dynamical| instability. The unstable bars grow to nonlinear hole spacetimes, we can carry out a similar analysis to that amplitude and lead to the shedding of high angular-momentum of relativistic stars, this time for self-gravitating tori around material. It is thus hard to imagine T/ W > 0.27 stars persist- black holes. Once again we require axisymmetric constraint- ing in nature. The dynamical bar mode| instability| (often called satisfying equilibrium initial data, with a central black hole the ‘high T/ W instability’) has been confirmed in numerical introduced either by a horizon inner boundary condition [396] | | relativity simulations of differentially rotating stars [383–385]. or a puncture [397]. Then we just watch perturbations evolve. Numerical relativity investigations also found unexpected Self-gravity can lead to instabilities in black hole-torus unstable growth of low (but nonzero) M modes in strongly systems. Even in Newtonian physics, a disk will break apart if

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it violates the Toomre stability criterion. A potential axisym- around n = 3 (the marginal stability limit for Newtonian poly- metric dynamical instability that has received much attention tropes). Scenarios can be divided by the mass of the progeni- from numerical relativists is the runaway instability [398]. tor star. Population I and II stars with masses ∼101 102M – In this scenario, an equilibrium torus filling its Roche lobe form iron cores of mass ∼M , where n = 3 comes from the will be unstable because a small amount of accretion into the dominance of relativistic degenerate electrons to the pres black hole increases the black hole’s mass, pushing the Roche sure. The first generation of stars, the metal-free Population 2 lobe into the torus. The mass transfer into the black hole is III stars, may have had masses ∼10 –103M . If metal-free then unstable and destroys the torus in a dynamical timescale. gas is unable to cool and does not fragment into Pop III stars, Analysis of stationary configurations suggest that the disk’s 105M supermassive stars may form. These very mas- self-gravity enhances the instability [399], but disks are stabi- sive stars are radiation-pressure dominated and (because of lized by a positive radial angular momentum gradient in the convection) isentropic, leading them to also take the form of disk and by black hole spin [400]. The first relativistic simu- n = 3 polytropes. Like black hole spacetimes, polytrope sys- lations, by Daigne and Font [401, 402], neglected torus self- tems can be scaled to any mass. However, this scale invari- gravity, treating evolution of the metric by allowing the mass ance is broken when one takes into account EoS stiffening and spin of the Kerr black hole to increase by accretion. These and nuclear reactions, which depend on the actual density and confirmed that even rather small angular momentum gradients not just the dimensionless compaction. Numerical relativity prevent the instability. However, only live-metric simulations is needed to determine the collapse outcome: the mass and could properly include torus self-gravity. These were first spin of the black hole and the properties of any accompany- undertaken by Montero, Font, and Shibata [403], and they did ing disk. Even before these simulations could be carried to not find a runaway stability in any of their models. However, post-collapse equilibrium, it was possible to guess from a trick simulations by Korobkin et al [404] did demonstrate the exist- introduced by Shapiro and Shibata [413] that these collapses ence of the instability in disks particularly prone to it. would be more likely to form massive disks. The post-collapse There are also nonaxisymmetric instabilities present in disk is roughly the matter with high enough initial angular black hole-torus systems analogous to those found in rotat- momentum to orbit outside the innermost stable circular orbit ing stars. Before they were identified in stars, corotation insta- (ISCO)18 of the black hole to be formed, which can be of order bilities were already known to exist in nearly-constant angular 10% of a supermassive star progenitor. momentum disks (the Papaloizou–Pringle instability [405, 406]) and in disks with vortensity maxima (the Rossby wave instability [407]). Self-gravity is not an essential feature of 5.1. Population I/II core collapse and collapsars the instability, and in fact self-gravity tends to suppress the Stellar mass black holes are thought to originate in the core Papaloizou–Pringle instability [408]. collapse of massive stars for cases where, for some reason, Torus self-gravity can trigger nonaxisymmetric instabili- the process of permanently expelling the gas around a prot ties not present in nonself-gravitating disks. These instabili- oneutron star fails. (The case of successful supernova explo- ties have been studied systematically by 3D simulations in sion and neutron star formation has been the subject of much Newtonian physics [409] and numerical relativity [410]. The numerical work, the discussion of which would take us too latter study, by Korobkin et al, constitutes one of the first far afield.) If the progenitor has sufficient angular momentum, notable applications of cubed-sphere multipatch technology the newly formed black hole may be surrounded by an accre- to numerical relativity. Simulations show that moderate self- tion torus. This is the explanation of long-duration gamma gravity triggers ‘intermediate mode’ instability [408], spon- ray bursts in the collapsar model of Woosley and MacFadyen taneous elliptic deformations of the disk that, in fact, can [414–416]. Formation of a black hole-torus system may occur be considered the disk analogue of the high-T/ W instabil- in several ways. The inner iron core may originally collapse | | ity [411]. An interesting effect of self-gravity on the m = 1 to a protoneutron star, but a supernova may fail to occur (the Papaloizou–Pringle mode is momentum transfer between the shock stalls and does not sufficiently re-energize), so that the disk and black hole, leading to an outspiraling motion of the star eventually collapses under its accumulating mass; this black hole [410]. The nonlinear development of this instabil- is a Type I collapsar [416]. A mild explosion may occur, but ity was explored in numerical relativity by Kiuchi et al [412], enough material falls back onto the star to trigger collapse—a who suggest it may be a significant source of gravitational Type II collapsar [417]. Finally, the inner core might collapse waves (e.g. from a GRB central engine or the aftermath of a directly to a black hole—a Type III collapsar [418]. supermassive star collapse). The first 1D numerical relativity stellar collapse simulations were carried out in 1966 by May and White [419] using 5. Black hole formation 18 In general relativity, the effective potential associated with orbits (i.e. timelike geodesics) around a black hole is similar to the effective potential in In section 4.2, we considered the collapse of uniformly rotat- Newtonian gravity, but with additional attractive terms proportional to 1/r3. ing compact, stiff stars, finding that they tend to collapse to Because of this term, there is a region from the black-hole horizon to about Kerr black holes with no significant leftover material to form three times the Schwarzschild radius of the black hole were there are no stable circular orbit. At the boundary of this region is the innermost stable an accretion disk. Much more common astrophysically is the circular orbit, or ISCO. A particle following an inspiraling quasicircular formation of black holes from non-compact stars with soft EoS orbit will plunge into the black hole once it crosses this ISCO.

25 Rep. Prog. Phys. 82 (2019) 016902 Review the formulation of Misner and Sharp [420]: a Lagrangian self-consistent black hole formation, and evolution long past method with, however, a slicing that does not avoid singulari- black hole formation to study the dynamics of the torus. With ties. Thus, simulations could not be continued long after black the ability to stably form and evolve black holes in numer hole formation, but this was enough to determine that a black ical relativity, this became possible. The first such simula- hole in fact forms, rather than a neutron star. This problem can tion was carried out by Sekiguchi and Shibata [443]. They be overcome using a retarded time coordinate, which avoids evolved a high-entropy core from collapse through a second the black hole interior [421, 422]. The same basic methods past black hole formation using a finite temperature equa- have continued to be used for subsequent spherical collapse tion of state and neutrino leakage. A range of initial j were simulations with increasingly sophisticated microphysics used; low-j cores produced geometrically thin shocked disks, and neutrino transport [423–426], including detailed studies while high-j cores produce thick tori. The first 3D numerical of the neutrino signals from failed supernovae solving the relativity collapse simulations were performed by Ott et al Boltzmann transport equation [427–429]. 1D general relativ- [444]. Octant symmetry and eleven levels of adaptive mesh ity simulations find that black hole formation tends to happen refinement made it possible to follow the collapse in 3D, but for high progenitor mass (which, due to stellar winds, may be the simulation was only followed for ∼0.1 s after black hole much lower than the zero age main sequence mass, so black formation. Core collapse is followed by a bounce, but the hole formation is more likely for low-metallicity stars, which accretion shock stalls, and the protoneutron star collapses to suffer less mass loss). Prompt collapse may only occur for a black hole. Several key quantities in these simulations are very high mass, low metallicity (perhaps only Population III plotted in figure 18. The dimensionless spin peaks at 0.75 for stars) [426, 430, 431]. 2D simulations are needed to study the most rapidly rotating case and then rapidly decreases as rotating collapse. Some such simulations were carried out lower-j material is accreted. The collapse-bounce-collapse beginning with Nakamura [432] but lacked realistic initial sequence of events leads to a distinct gravitational wave conditions and EoS. In 2005, on the eve of the numerical signal. In addition to this collapse waveform, gravitational relativity revolution in black hole treatment, Sekiguchi and waves may be produced by inhomogeneities in the collaps- Shibata [433] attempted greater realism using a set of two- ing matter, self-gravitational instabilities in the torus, and component piecewise polytrope EoS. The paper limited itself anisotropic neutrino radiation. (See [445, 446] and refer- to the criterion for prompt black hole formation because sub- ences therein.) sequent evolution could not be followed. A difficulty for 3D collapsar simulations is the long (multi- While waiting for numerical relativity, 2D post-collapse second) timescale on which rapid accretion occurs. The inclu- simulations were being used to study the evolution of the sion of MHD and neutrino transport will make modeling the post-black hole formation torus and the possible initiation of hyperaccretion phase even more challenging. a gamma ray burst. Lacking the true post-collapse configuration, these first simulations had to insert a black hole into 5.2. Massive star collapse in the early universe a collapsing flow by hand. For Newtonian simulations, the black hole is a Newtonian or pseudo-Newtonian point-mass Black hole formation from massive stars in the early universe addition to the gravitational potential and an inner absorbing is interesting primarily for explaining the ‘seeds’ from which boundary [416, 417, 434–436]. For relativistic MHD simula- supermassive black holes grew, but also as sources of electro tions, a fixed Kerr metric was used [437–440]. A key input magnetic and gravitational wave signals. parameter is the initial angular momentum. To produce a 1D simulations of Population III stars indicate that stars promising torus, this is usually chosen to be large enough for with mass less than around 260M end their lives in pair- circular orbit well outside the nascent black hole’s ISCO but instability supernovae, while more massive stars collapse small enough that the disk is compact and can lose energy directly to black holes [418, 431]. Nakazato et al [447] study efficiently by neutrinos. However, Lee and Ramirez-Ruiz the effects of neutrino emission on Pop III star collapse using [441] find promising behavior even for somewhat lower angu- a relativistic Boltzmann transport code. Effects of rotation lar momenta; shocked gas on the equator forms a dwarf disk are indirectly addressed by the 2D Newtonian simulations of which accretes rapidly due to general relativistic effects even Ohkubo et al [448], who, in a manner reminiscent of early without magnetic fields or viscosity. Follow-up simulations collapsar simulations, insert a point mass by hand into the by Lopez-Camara et al [436] suggest that low-j collapsars collapsing star. Rather than adding rotation and viscosity, the might differ from high-j collapsars by the former not produc- effects of a disk are modeled on larger scales by injecting a jet ing an accompanying supernova. Hydrodynamic simulations, through the inner boundary. The jet drives an explosion, and such as the original study by MacFadyen and Woosley [416] nucleosynthesis outputs are calculated. add an alpha viscosity and tend to find that the polar regions Supermassive stars began to be simulated in the 1970s. free-fall into the black hole while inside an accretion shock At the time, interest was primarily driven by the prospect of a thick torus forms and viscous heating-driven outflows are explaining active galactic nuclei in terms of these objects, launched. Subsequent simulations with MHD for both high-j which do radiate at their Eddington luminosity. These stars [434] and low-j [438, 442] cases differ primarily in the quick are radiation-pressure dominated and presumed isentropic, appearance of magnetically-driven polar jets. so they are nearly n = 3 polytropes. (Gas pressure makes n Clearly, numerical relativity simulations were needed slightly less than 3, but this deviation decreases with increas- which include the collapse of a realistic rotating stellar core, ing mass.) Although they are not compact, the instability

26 Rep. Prog. Phys. 82 (2019) 016902 Review angular momentum distribution that it should be around 10% of the star’s mass. The Illinois group returned to this problem in 2007 with black hole excision to determine the post-collapse state [458]. Their simulations vindicated the earlier angular-momentum based predictions of a massive disk (several percent of the total original rest mass) and a black hole with spin about 70% of the Kerr limit. This work also included magnetic fields, which do not affect the collapse but do affect the disk evo lution. Insertion of a dynamically unimportant dipole field into the pre-collapse star leads to jet formation after collapse, with enough Poynting luminosity to potentially power an ultra-long GRB detectable at high redshifts [459]. Montero et al [460] carried out simulations with detailed microphysics and hydrogen and helium burning, finding that a metallicity of 10−3 is needed to cause thermonuclear explosion rather than black hole formation for mass-shedding supermassive stars of mass ∼5 105M . Finally, Shibata and collaborators have × Figure 18. The postbounce evolution of the center of a collapsar in included the deviation of Γ from 4/3 [461] and nuclear reac- 3D numerical relativity. Different models correspond to different tions [462], finding collapse outcomes similar to the Γ=4/3 choices for the progenitor spin. Top: Maximum density ρmax and studies [458]. central ADM lapse function αmin as a function of postbounce time in all models. After horizon formation, the region interior Initial data sets for the above simulations assume the star to it is excluded from min/max finding. Bottom: black-hole mass is able to maintain uniform rotation and entropy. Collapse of and dimensionless spin a as a function of postbounce time. All differentially rotating supermassive stars has been simulated models follow the same accretion history once a black hole forms in 2D by Montero et al [460] and in 3D by Saijo and Hawke and settles down. Reprinted figure with permission from [445], [463]. The latter study monitors quasi-periodic gravitational Copyright (2011) by the American Physical Society. waves coming from the post-collapse system even after the hole s quasinormal ringing damps. When a nearly extremal of >105M stars is triggered by general relativity. These ’ black hole is formed, the gravitational wave grows to fairly earliest simulations were 1D, assuming spherical symmetry. high amplitude after collapse for reasons which remain mys- Appenzeller and Fricke [449, 450], using post-Newtonian terious. Zink et al [464] studied the collapse of differentially gravity but including nuclear reactions, found prompt col- rotating toroidal supermassive stars, finding that in this case lapse to a black hole if M > 106M ; for lower masses nuclear the star is subject to strong nonaxisymmetric instabilities. burning of hydrogen explodes the star. Later simulations These lead to the fragmentation of the star into self-gravitat- found that nonzero metallicity (albeit high given the context: ing, collapsing parts, in some cases leading to the formation Z 10 2) catalyzes CNO burning and triggers explosion − of a supermassive black hole black hole binary system [465]. [451∼]. Spherically symmetric full numerical relativity simula- – The above all assume that supermassive stars are able to tions of high-mass supermassive stars found prompt collapse thermally relax to isentropy, which should be true if they are to a black hole [452, 453]. convective. For an alternative scenario, leading to a stellar In fact, this is another system for which we expect rota- mass black hole surrounded by a much more massive enve- tion to be extremely important. As a supermassive star cools lope, see Begelman [466, 467]. and shrinks, it probably spins up to the mass-shedding limit, thereafter following a mass-shedding sequence as it cools till it hits a radial instability. Equilibrium sequences of rapidly 6. Non-vacuum compact binaries rotating supermassive stars in general relativity have been produced by Baumgarte and Shapiro [454] for n = 3 and Non-vacuum compact object binaries (that is, binaries made by Shibata et al [455] for 2.94 n 3. Saijo et al [456] of two compact objects, at least one of which is not a black evolved an n = 3 supermassive star from it is critical point hole) inspiral due to gravitational radiation just like black using 3D post-Newtonian physics, finding that the collapsing hole–black hole binaries. Tidal deformation of the star(s) con- star remains axisymmetric (i.e. nonaxisymmetric modes do stitute an additional time-varying quadrupole, subtly affecting not have time to grow even though T/ W passes the critical the inspiral and associated gravitational waveform. For most value for bar formation) and that nearly| all| the mass falls into of the inspiral, this small tidal effect can be adequately mod- the black hole despite its rotation. Around the same time, 2D eled using post-Newtonian theory, and to good approximation numerical relativity simulations were carried out by Shibata the size and structure of the star(s) only affect the wave- and Shapiro [457], tracking the collapse until about 60% form via their dimensionless tidal deformability parameters 2 5 of the mass was inside the apparent horizon. Because they Λ=(2/3) k2 (c R/GM) , where k2 is the apsidal constant could not evolve long past black hole formation, the final disk while R and M are the star’s radius and mass, respectively. mass remained uncertain, but the authors estimated from the Λ can be thought of as a measure of a star’s response to an

27 Rep. Prog. Phys. 82 (2019) 016902 Review external tidal field. Thus, one may hope to use gravitational For Intermediate mass black hole-white dwarf tidal dis- waveforms from compact neutron star binaries to constrain ruptions, the disparity of length scales is removed and tidal the Λ, and hence the EoS, of neutron stars [647]. An impor- disruption happens in the strong gravity regime. In this case, tant application of numerical relativity, not discussed in this we would have in mind nearly parabolic encounters in dwarf review, is to test–and if necessary improve–these models of galaxies or globular clusters, rather than quasicircular inspi- tidal effects on waveforms during inspiral. Interested readers ral and merger. Rosswog et al [473] carried out Newtonian are referred to a sample of the papers on the topic [637–641]. SPH simulations, including nuclear burning, of such events, Material effects become dramatic at the end of inspiral, looking especially at cases where tidal compression triggers which will involve a collision or, more often, a tidal disrup- explosive nuclear burning. Because the code was Newtonian, tion. The latter happens when the tidal force on a star from the black hole had to be approximated by a Paczynski–Wiita its companion exceeds the star’s own self gravity. This can be potential. The effects of black hole spin could only be studied illustrated by a simple Newtonian order-of-magnitude calcul with numerical relativity simulations, which were carried out ation. Suppose the disrupting star has mass M, radius R and is by Haas et al [474], although without nuclear reaction effects. at a separation d from its companion of mass m. (In all cases Both sets of simulations predict a residual accretion disk and we consider, this companion will be more massive and more accompanying soft x-ray flare lasting about a year. compact, often a black hole.) Then the self-gravitation and 2 3 tidal accelerations are M/R and mR/d , respectively, and they 6.2. Black hole–neutron star binaries match when Because of their potential as gravitational wave sources and 2/3 d/m (R/M)(m/M)− . (62) short gamma ray burst progenitors, most non-vacuum numer ∼ Tidal disruption is likely marked by a sharp decrease in the ical relativity work has focused on black hole–neutron star gravitational wave amplitude as the binary loses its quadru- binary and neutron star–neutron star binary mergers. In addition, these mergers are of interest as possible sources of short- polar shape. The subsequent fate of the disrupted star’s mass must be determined by simulations. Most attention has been duration gamma ray bursts, r-process nucleosynthesis, and given to the two observationally important possibilities of gas kilonovae. (For reviews of multimessenger astronomy, see forming an accretion disk around the remaining binary object [642, 643].). In this section, we review black hole–neutron and gas being ejected from the system. star binary merger simulations, a key application of numer ical relativity dynamical black hole-handling technology. For a fuller treatment, see the Living Review by Shibata and 6.1. White dwarf—compact object binaries Taniguchi [475]. Neutron star-white dwarf mergers and stellar mass black hole– white dwarf mergers are not easily amenable to numerical 6.2.1. Expectations before numerical relativity simula relativity because of the disparity of length scales between the tions. Black hole–neutron star binaries were historically the two objects. Also, white dwarfs are not in nuclear statistical last to be simulated in numerical relativity, but simple argu- equilibrium. Usually, isotope abundances in a white dwarf can ments and Newtonian simulations gave some idea what to be considered fixed, but a merger event may trigger nuclear expect. reactions, which would then provide a new energy reservoir If a neutron star disrupts inside the ISCO of its compan- and must be explicitly tracked. Because the white dwarf dis- ion black hole, no massive disk or ejecta is expected. What’s ruption happens on scales much larger than the neutron star more, the gravitational wave in these cases should be nearly or low-mass black hole, one might ask if a Newtonian white indistinguishable from that of a black hole–black hole binary dwarf plus point mass treatment is sufficient. Such calcul system with the same masses. Defining dISCO = κMBH and ations have been done in SPH [468], indicating that the dis- using equation (62), we conclude that tidal disruption is likely rupted white dwarf shears into an accretion disk, a possible for binaries with setup for a long-duration gamma ray burst. R M 2/3 Paschalidis et al [469, 470] have attempted to use numer NS >κ BH . (63) M M ical relativity to study white dwarf-neutron star mergers. To NS NS make simulations feasible, the white dwarf is replaced by a That is, disruption is favored by low neutron-star compac- pseudo-white dwarf , only ten times bigger than the neu- tion C M /R , low mass ratio q M /M , and high ‘ ’ ≡ NS NS ≡ BH NS tron star rather than 500. The merger outcome is a Thorne– dimensionless black-hole spin χ (to reduce κ). Zytkow-like object which will cool to a hypermassive star and What happens when the neutron star fills its Roche lobe eventually collapse to a black hole. The authors acknowledge and mass transfer begins? The question was first addressed that simulations with more complete microphysics are still in Newtonian simulations by Lee and Kluzniak [476–478], needed. 1D disk calculations by Metzger and collaborators with the black hole treated as a point mass and the neutron [471, 472] indicate that, at least for systems with mass ratio star treated as a polytrope. These simulations found that mass not close to one, heating from nuclear reactions may unbind transfer is stable for stiff EoS but unstable for soft EoS [477, most accreting matter, and they judge it unlikely that enough 478], a difference that carried over to rival nuclear theory- mass accumulates on a neutron star to trigger collapse in those based EoS as studied by Janka et al [479] and Rosswog et al cases. [480]. In the case of unstable mass transfer, the neutron star is

28 Rep. Prog. Phys. 82 (2019) 016902 Review destroyed in a single mass-transfer event. Stable mass transfer, on the other hand, is episodic, yielding an unmistakably different gravitational wave signal. Replacing the Newtonian point mass with a Paczynski–Wiita potential makes mass transfer less stable, so that tidal disruption happens in one pass even for stiff realistic EoS [481, 482]. Newtonian simulations also found massive ejection of unbound matter during mergers. SPH simulations around Kerr black holes supported expectations that prograde black hole spin is favorable to disk formation [483].

Figure 19. A schematic figure of three types of gravitational-wave 6.2.2. Inspiral and merger in numerical relativity: parameter spectra from black hole–neutron star mergers. Spectrum (i) is for space exploration and gravitational waves. The first general the case in which tidal disruption occurs far outside the ISCO, and relativisitic simulations of black hole systems came soon after spectrum (ii) is for the case in which tidal disruption does not occur. the moving puncture revolution. A head-on collision with a Spectrum (iii) is for the case in which tidal disruption occurs and neutron star falling into a black hole was successfully mod- the quasinormal mode (QNM) of the black hole is also excited. eled (using excision) by Loffler et al [644], and soon after Shi- The filled and open circles denote ftidal, the frequency at neutron star tidal disruption, and fQNM, respectively. Reprinted figure with bata and Uryu carried out binary merger simulations starting permission from [491], Copyright (2011) by the American Physical from roughly circular orbit (using moving punctures) [127]. Society. Simulations by the UIUC, SXS, and LSU/BYU/LIU groups quickly followed [484–486]. The former two of groups used during the inspiral, and no merger or ringdown wave is seen. the NOKBSSN formalism with moving punctures; the latter Disruption close to the ISCO gives a case with intermediate two used the generalized harmonic formulation with explicit features: inspiral and merger waves but reduced ringdown excision. These early simulations used simple, polytropic wave. Information about the neutron star is contained in the EoS, and in some cases improper treatment of low-density gravitational wave cutoff frequency. Shibata et al emphasize material led to underestimates in the disk and ejecta masses. that this cutoff frequency is not identical to the gravitational They all found complete neutron-star disruption in a single wave frequency at tidal disruption, but rather is somewhat mass transfer event. When tidal disruption occurs outside the higher. Presumably, this is because the star persists for a time ISCO, the neutron star deforms into a tidal stream, with inner as a clump of matter as it inspirals past the tidal disruption material streaming toward the black hole and outer material radius. streaming outward. Of the matter falling toward the black BH-polytrope simulations also found a strong dependence hole, most will fall into the black hole. Soon after flow into of the post-merger disk mass on the black-hole spin. Etienne the black hole commences, material with sufficient angular et al [491] found that much more massive disks could be momentum wraps around the black hole, causing the tidal formed for neutron stars disrupted by black holes with pro- stream to crash into itself. The resulting shock heats the gas, grade spin, which is perhaps to be expected, since such black which begins setting into an accretion disk very close (tens holes have smaller ISCOs. An extreme case—mass ratio of 3 of km) to the black hole. Of the nuclear matter expanding and prograde black hole spin at 97% of the Kerr limit—was outward, some is bound and eventually falls back onto the simulated by Lovelace et al [492]; not even half of the rest disk, while the rest (the ‘dynamical ejecta’) is unbound and mass is promptly accreted in this case. Retrograde spin, on escapes permanently. the other hand, makes disruption outside the ISCO less likely Using all the numerical relativity results available at the and disk masses lower. The black hole spin orientation has time, Foucart devised an analytic fit to the post-merger disk been varied by Foucart et al [493, 494] and by Kawaguchi mass (defined as the rest mass of bound material outside the et al [495], with the general findings that large spin misalign- black hole 10 ms after merger; recall that matter is continu- ments remove the increase in disk mass seen in prograde spins ously falling onto the disk and into the horizon) as a function and lead to disks initially misaligned with the black hole spin. of C, q, and χ which confirms the expectation that, all else With high enough prograde black-hole spin, Foucart et al being equal, lower C, lower q, or higher χ increases disk mass [494] were able to observe tidal disruption even in systems [487]. Using their own set of simulations, Kawaguchi et al with mass ratios in the 5–7 range [494] where astrophysi- [488] devised a similar analytic fitting formula for the mass cal black hole–neutron star binary systems are thought most and asymptotic speed of unbound ejecta. likely to lie [496]. Using numerical relativity simulations, Shibata et al [489] Most contemporary black hole–neutron star binary sim- found gravitational waves from black hole–neutron star ulations use more realistic EoS. Because the matter only binary mergers to always fall into one of three categories, heats up as the gravitational wave signal turns off, waveform illustrated in figure 19. When the neutron star falls into the studies have sensibly concentrated on piecewise-polytropic black hole before being disrupted, the wave is similar to black parameter studies of EoS effects [490, 495, 497]. In a truly hole–black hole binary waves. When the neutron star disrupts impressive effort, Kyutoku and collaborators carried out 134 well outside the ISCO, the waveform cuts off at this point merger simulations, varying both EoS and binary parameters

29 Rep. Prog. Phys. 82 (2019) 016902 Review

2 [490, 497, 498]. The equation of state was modeled as a ejecta for cases with tidal disruption is often large ∼10− – 1 two-piece piecewise polytrope, which, since the low-density 10− M and highly asymmetric concentrated on one side in — EoS is known, has two free parameters. They choose to sys- the orbital plane—with sufficient momentum to impart a kick 2 −1 tematically vary the high-density Γ and P1, the pressure at of 10 km s on the remnant black hole in some cases [494, 14.7 −3 a fiducial density ρﬁdu = 10 g cm . The reason for using 509]. Additional matter is ejected into weakly bound orbit P1 rather than the transition density is because P1 is found and will fall back onto the central remnant later. This fallback to closely correlate with the neutron star radius and tidal material was studied using Newtonian SPH by Rosswog [511] deformability, making it a good candidate for an equation of and then in numerical relativity by Chawla et al [486]. From state parameter that can be measured by gravitational wave the distribution of fallback times, these studies predict a late- signatures such as the cutoff frequency. These simulations time fallback accretion rate following a t−5/3 power law. have been used to calibrate analytic black hole–neutron star The possible importance of black hole–neutron star dynam- binary waveform models, valid in the range 2 < q < 5, cov- ical ejecta for r-process nucleosynthesis was pointed out by ering the full inspiral and merger [498–501]. Fisher matrix Lattimer and Schramm in 1976 [512]. Newtonian simulation and Bayesian analysis of the analytic model shows that confirm that black hole–neutron star binary mergers pro- LIGO detections can hope to significantly constrain the tidal duce large ejecta masses of neutron-rich material that should deformability and P1, especially given dozens of realistic undergo r-process nucleosynthesis and produce an optical/ detections [498, 501]. near-IR transient [513]. Indeed, so efficient are these merg- Using piecewise polytrope EoS, East et al [502] simulated ers in producing r-process elements that Bauswein et al [514], eccentric black hole–neutron star binary encounters. Here use their own ejecta predictions from SPH conformally flat could finally be seen cases of episodic mass transfer in general general relativistic simulations and the galactic abundance of relativity, as well as instances of the ‘zoom-whirl’ phenom r-process material to constrain the rate of black hole–neutron enon observed in black hole–black hole binary simulations. star mergers. Such events may well occur at interesting rates in dense stellar The SXS simulations confirm expectations that ejecta environments such as globular clusters [503]. If so, the richer is very neutron rich, so as it decompresses it is expected to dynamical possibilities of eccentric merger deserve more undergo r-process nucleosynthesis and produce the second attention. and third r-process peaks. This is indeed what Roberts et al [515] find tracking nuclear reactions in the ejecta, although a 6.2.3. Post-merger in numerical relativity: neutrinos, ejecta, weak first peak can be seeded by neutrino irradiation by the and MHD. Post-merger evolution requires finite-temperature central black hole-disk system. Higher Ye outflow may be pro- EoS, which have been employed in simulations by the SXS vided by winds from the accretion disk. Recently, Fernandez collaboration [132, 139, 504, 505] and by Kyutoku al [506]. et al [516] have produced models including both effects. These simulations fail to find episodic mass transfer even for Using outgoing ejecta and disk profiles from the SXS merger stiff EoS. It is found that, when β equilibrium violation is simulations as initial data, the disk was evolved to late times allowed during tidal disruption (on these timescales, the lep- using 2D Newtonian hydrodynamics with an alpha viscosity ton number advects), tidal streams are narrower, and ejecta to model angular momentum transport. As the disk evolves, velocities lower than β-equilibrium EoS would predict [505, neutrino cooling decreases to the point of insignificance, and 507]. Lepton number equilibrium in the post-merger disk is by ∼200 ms the disk has reached an advective state (viscous re-established about 10 ms after merger under the action of heating balanced by advection of hot material inward rather intense neutrino emission (L 1053 erg s−1, preferentially than by radiative cooling) with strong convection and mass ν ∼ in electron antineutrinos), during which time the disk’s aver- outflow. In most cases, the dynamical ejecta dominates, but if age Ye rises to about 0.1. (For stiff EoS, Ye might be as high as the disk outflow and dynamical ejecta should happen to have 0.2 [506].) As the disk cools, the equilibrium Ye decreases and comparable masses, a solar-like distribution of r-process ele- a re-neutronization is seen in the disk. ments would follow. Whether the neutrino emission is sufficient to power a Radioactive decay of r-process ejection might power a gamma-ray burst remains uncertain. Just et al [508] simu- detectable signal, most likely in the near infrared, called a kil- late black hole tori using an energy-dependent M1 neutrino onova or macronova [517]. Tanaka et al [518] use ejecta from transport scheme, and find that, at least for favorable cases, numerical relativity black hole–neutron star merger simula- neutrino-antineutrino annihilation can power a relativistic out- tions as input to a (photon) radiation transfer code to predict flow sufficient for a low-energy short duration GRB. (black light curves. The simulations did not include nucleosynthesis, hole–neutron star binary mergers are more promising in this but assumed the solar r-process pattern. They find that black regard than neutron star–neutron star binary mergers, because hole–neutron star kilonovae can often appear as bright or the latter have more baryon loading from dynamical ejecta on brighter than neutron star–neutron star kilonovae, because the the poles.) former can produce more ejecta, and that the former will tend After some early false negatives, tidal ejection of unbound to be bluer than the latter. The same group applied these mod- matter was robustly identified in numerical relativity black els to the purported kilonova associated with GRB 130603B, hole–neutron star mergers [494, 509, 510]. In relativistic showing the observed near-infrared excess is consistent with codes, unbound matter is usually identified as that with spe- either a soft EoS neutron star–neutron star binary merger or cific orbital energy e = −ut − 1 > 0. The mass of dynamical a stiff EoS black hole–neutron star merger [519]. Kawaguchi

30 Rep. Prog. Phys. 82 (2019) 016902 Review et al [488] returned to black hole–neutron star kilonovae with massive remnant rotating rapidly and differentially, and for analytic models for the heating and radiative diffusion but a stiff EoS the remnant is subject to bar mode deformations, large suite of merger simulations. Lanthanide-free disk wind leading to a sustained post-merger gravitational wave signal. might create a bluer signal [520], but in the models studied Post-Newtonian simulations were the next logical step. by Fernandez et al, this was all obscured by the dynamical The first such simulation was carried out using smoothed par- ejecta [516]. ticle hydrodynamics by Ayal et al [540]. Unfortunately the 1 The most dramatic, and least-understood, post-merger pro- post-Newtonian-order terms are not always small compared to cesses are magnetically driven. Early simulations with con- Newtonian terms, indicating that truncating at this level is not fined poloidal fields found that the field in the post-merger a valid approximation, and post-Newtonian studies sometimes disk quickly wound into a toroidally dominated configuration, resorted to artificially reducing the post-Newtonian terms (e.g. with no observable jets [486, 521] even when the MRI could [541]). The next advance was to the conformally flat approx be resolved [522]. Later simulations by Paschalidis et al [523] imation to general relativity. Here surprises seemed to arise found that jets can more easily emerge from a field initially when Wilson, Mathews, and Marronetti [542, 543] reported extending outside the neutron star. Such magnetospheric the neutron stars in their simulations collapsing individually fields might also trigger observable signals that preceed a to black holes before merging. However, they used an EoS GRB from the merger [156, 524]. In fact, even confined seed with fairly low neutron star maximum mass and were found fields may be more promising than they originally seemed. to have an error in one of their equations [544]. Pre-merger Extremely high-resolution studies (∆x 100 m) by Kiuchi collapse is no longer considered likely, but the conformal et al [525] find winds driven from inner≈ disk heating, whose flatness approximation has turned out to be a useful and reli- strength increases with resolution, with convergence not yet able tool for neutron star–neutron star binary modeling (e.g. achieved, can pin magnetic flux to the black hole with associ- [545–547]). ated Blandford–Znajek jets. Post-merger disk evolution turns Parallel to efforts toward incorporating relativity were out to be a difficult, multi-scale problem, and crucial proper- efforts to include realistic microphysics in Newtonian simu- ties like the rates of wind outflow, magnetic energy outflow, lations. Grid-based simulations by Ruffert et al [548–550] and neutrino annihilation energy deposit remain poorly con- and SPH simulations by Rosswog et al [551–553] began the strained. For the time being, the long-term evolution of the use of finite-temperature equations of state and inclusion of disk is being investigated using Newtonian alpha-viscosity neutrino effects (in a leakage approximation) for neutron models (e.g. [526, 527]). star–neutron star binaries. These simulations highlighted the potential of neutron star–neutron star binary mergers as GRB central engines. The study of neutron star neutron star binary 6.3. Neutron star–neutron star mergers – mergers with finite-temperature EoS was continued in confor- 6.3.1. Pre-breakthrough simulations. The story of neutron mally flat gravity by Oechslinet al [547]. Neutrino absorption star–neutron star binary merger simulations differs from that effects were studied in Newtonian physics using flux-limited of other major numerical relativity problems in that many diffusion by Dessart et al [554] and through ray tracing and neutron star–neutron star binaries evolve well past merger phenomenological extensions to leakage by Perego et al without encountering black hole formation, so full numerical [555]. Using these very different methods, both groups find relativity merger simulations began earlier for neutron star– strong neutrino-driven winds ejected from the merged rem- neutron star binary systems, and the ability to evolve space- nant, winds that could play important roles in generating times with black holes had a less dramatic effect. Neutron r-process elements and kilonovae and in baryon loading the star–neutron star binary simulations have recently received environment of a potential GRB. a thorough review by Baiotti and Rezzolla [528]. They are Newtonian simulations were also able to investigate the also the subject of a Living Review by Faber and Rasio [529]. ejecta and its potential for r-process nucleosynthesis before Techniques for constructing initial data are described in the the first numerical relativity simulations [556–558], and later review by Tichy [530]. Readers can find in these sources a numerical relativity simulations have found results reasonably more detailed presentation of the large subject of compact close to the earlier Newtonian studies. neutron star–neutron star binaries. Neutron star–neutron star mergers in full numerical relativ- Simulations of neutron star–neutron star binary merg- ity were first carried out by Shibata and Uryu at the turn of the ers were first undertaken in Newtonian physics with mostly century [114, 559, 560]. These initial simulations modeled the simple polytropic equations of state. Nakamura and Oohara neutron stars as Γ=2 polytropes. Their most significant dis- in a series of papers performed the first neutron star–neu- covery was that the remnant does collapse to a black hole, but tron star binary merger simulations using finite differencing only if its mass exceeds a certain threshold. Less massive sys- [531–534]. SPH simulations were performed by Rasio and tems form dynamically stable differentially rotating neutron Shapiro [535, 536]. Further simulations of both types fol- star remnants (which in some cases are hypermassive). Other lowed [537–539]. Gravitational waves had to be studied in the groups added AMR [50] and high-resolution shock-capturing quadrupole approximation, and the possibility of black hole techniques [561]. Binary polytrope simulations with other formation could not be addressed. However, these Newtonian sophisticated numerical relativity codes followed (BAM simulations showed some features that would be confirmed [562], WhiskyTHC [118], SpEC [362]). All found similar by numerical relativity simulations: the stars merge into a results. As the next step in microphysical realism, Shibata and

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collaborators tried cold nuclear-theory based EoS (augmented hole–neutron star binary case, namely that ejecta can be hot- with a Gamma-law thermal component to capture shock heat- ter and more neutrino-processed. Due to n + e+ ν + p ⇒ e ing) [563], confirming the possibility of nonaxisymmetric and ν + n p + e− reactions, a portion of ejecta with e ⇒ post-merger structure. Since then, the piecewise-polytrope Ye 0.3 0.4 can be created. The wide range of ejecta Ye can parameterization has often been used to systematically vary produce∼ all− three r-process peaks without the need for a sub- the cold EoS, or even just as a cheap way to approximate a sequent disk wind [138]. Numerical relativity has also been given cold EoS [564–567]. used to study systems with pre-merger neutron star spin [573, With the introduction of moving punctures, Baiotti et al 574, 584–586] and orbital eccentricity [579, 585, 587, 588]. [561] Kiuchi et al [568] revisited neutron star–neutron star MHD simulations of neutron star–neutron star binary binary mergers with various EoS, following high-mass cases mergers have struggled with the difficulty of resolving the beyond black hole formation to measure the post-collapse MRI in high-density regions and, even more challenging, the black hole and disk properties. For the APR EoS, equal-mass growth of the magnetic field in Kelvin–Helmholtz vortices. binaries with prompt black-hole formation leave very little Early studies [589–591] demonstrated the ability of numer 4 2 torus mass (∼10− M ), but about 10− M disks remain for ical relativity MHD codes to follow neutron star neutron star – binaries with mass ratio around 0.8. Rezzolla et al [569, 570] binary mergers but could not resolve these effects, although have produced analytic fitting formulae relating post-collapse they could resolve the MRI in a post-collapse torus [592] and torus mass to the pre-merger binary parameters. showed the possibility of jet formation from the black hole- torus system (leading perhaps to a short duration gamma ray 6.3.2. Post-merger evolution: neutrinos, ejecta, MHD. Now burst). A series of unprecedentedly high-resolution (as low as the main challenge was not metric evolution but microphysics. δx = 17.5 m) by Kiuchi et al [593, 594] succeeded in resolv- After two neutron stars merge, thermal, neutrino, and magn ing these effects well enough to demonstrate amplification of etic effects become important. Numerical relativity simula- the average field by a factor of 103, but even this is taken as a tions with finite-temperature EoS and neutrino leakage were lower limit. carried out by Sekiguchi et al [131, 571] for the stiff Shen EoS Although this small-scale amplification cannot be ade- and a hyperonic EoS. A survey of neutron star–neutron star quately resolved, numerical relativity MHD studies continue to binary mergers with a large number of finite-temperature EoS study large-scale processes during merger. For example, a few has since been performed (in conformally flat general relativ- studies investigate the difference in merger scenarios between ity) by Bauswein et al [572]. Kastaun and Galeazzi [574] car- ideal and resistive MHD [153, 155, 645]. Most recently, a ried out a further set of mergers with finite-temperature EoS series of papers beginning with Endrizzi et al [595–597] has (LS220 and SHT) and studied the structure of the hypermas- surveyed binary properties, EoS, and seed field effects on sive remnants in detail. Contrary to the widespread presump- magnetized neutron star–neutron star binaries. Among their tion that hypermassive stars ‘cheat’ the mass-shedding limit findings is a characteristic large-scale field structure appear- by having rapidly rotating cores with strong centrifugal sup- ing in many cases. For cases with black hole formation, these port and more slowly rotating envelopes, the authors find that simulations did not observe polar magnetic jets, but Ruiz et al their hypermassive remnants have slowly rotating (pressure- [598] does see them for stronger seed fields (allowing better supported) cores and extended, quasi-Keplerian envelopes. MRI resolution) and longer integration times. Simulations of low-mass neutron star–neutron star binary mergers (i.e. with non-hypermassive remnants) by Kastaun 6.3.3. General relativistic effects: prompt collapse threshold et al [575] and Foucart et al [576] show similar features. The and post-merger gravitational waves. Having outlined the composition of low-density regions and outflows is strongly development of neutron star–neutron star binary merger simu- affected by neutrino absorption. To capture these effects, lations, we turn to their findings concerning distinctly rela- numerical relativity simulations with energy-integrated M1 tivistic effects. The threshold mass Mth for prompt (i.e. on a neutrino transport have been carried out by Wanajo et al [138], dynamical timescale) collapse after merger to a black hole is Foucart et al [576, 577], and Sekiguchi et al [578]. found to be 30%–70% above MTOVmax, depending on the EoS. A number of studies have focused particularly on the Bauswein et al find that M =(2.43 3.38C )M th − TOVmax TOVmax dynamical ejecta [565, 577–579], which, like black hole–neu- to reasonable accuracy for all EoS studied, where CTOVmax is tron star ejecta, is potentially important for r-process nucle- the compaction of the TOV maximum mass configuration osynthesis and kilonovae, although there are two important [572]. For neutron star–neutron star binary systems with mass differences. Both come from the fact that dynamical ejecta below Mth, the massive (perhaps hypermassive) differentially comes not only from the tidal tail, but also from the collision rotating neutron star remnant is dynamically stable. interface. Much of this material is polar, leading to the first The remnant is formed with a strong quadrupolar dist major difference from black hole–neutron star ejecta: the dis- ortion, emitting strong—and, what’s even better, EoS- tribution of ejecta is much more isotropic [580, 582]. It has sensitive—gravitational waves [567, 580, 599, 600]. Recall even been suggested that the ejecta forms a cocoon around that the remnant is differentially rotating, so the overall rotat- the central object that can collimate the GRB outflow [583]. ing quadrupole in the matter profile is anm = 2 density mode, Additional ejecta is released in the following milliseconds as not solid body rotation. In fact, the mode angular frequency is the hot remnant neutron star settles. These extra sources of close to, but slightly higher than, the maximum angular fre- early-time ejecta lead to a second difference from the black quency in the star, for reasons that remain unclear [574, 576,

32 Rep. Prog. Phys. 82 (2019) 016902 Review 595]. This lack of a corotation radius rules out the possibility of a corotation shear instability in the m = 2 mode. The = 2, m = 1 mode has lower frequency and in fact does grow from the corotation instability [585, 601, 602]. As shown in figure 20, the post-merger gravitational wave spectra show a few distinct sharp peaks. The strongest peak, at frequency often called f2 in the literature, comes from the fundamental m = 2 mode described above, having a gravitational wave frequency between 2–3 kHz. For a wide range of EoS, f2 is an EoS-independent function of Rmax, the radius of a nonrotating neutron star of the mass MTOVmax [599], or to the radius at some other fiducial mass [580, 581]. Simulations also consist- ently find a weaker signal at a lower frequency f1, usually in the range 1.2–2.5 kHz, which has been explained in terms of oscillations in the distance between the two cores still distinct for a short time after merger [561], as a spiral density wave [603], or as a coupling between radial and quadrupolar modes [600]. For nonlinear perturbations of this sort, these inter- pretations may not be mutually exclusive. For a wide range of EoS, f1 seems to obey a universal relation to the average compaction of the premerger neutron stars [567]. Measuring these frequencies would significantly constrain the neutron Figure 20. Gravitational waveforms from a neutron star–neutron star binary merger for two possible nuclear equations of state. Top star EoS. Unfortunately, this would only be possible for close panel: evolution of h+ for representative binaries with the APR4 and mergers (<40 Mpc) with Advanced LIGO or with a next gen- GNH3 EoSs (dark-red and blue lines, respectively) for sources at a eration gravitational wave observatory. polar distance of 50 Mpc. Bottom panel: spectral density 2h˜( f ) f 1/2 One worry about the above studies is that they ignore windowed after the merger for the two EoSs and sensitivity curves magnetic field-related stresses. Simulations by Palenzuela of Advanced LIGO (green line) and ET (light-blue line); the dotted et al [604] with hot EoS, neutrino cooling, and MHD included lines show the power in the inspiral, while the circles mark the find that the large-scale magnetic fields (those resolved in contact frequency. Reprinted figure with permission from [568], Copyright (2014) by the American Physical Society. global merger simulations) are too weak to affect the post- merger waveform during the first 10 ms, even when Kelvin– Kelvin–Helmholtz instability currently frustrate numerical Helmholtz amplification is included in an approximate way convergence, to which we add a general observation that in via subgrid modeling. As we shall shortly see, the effects of any high Reynolds-number, turbulent system one must resolve subgrid-scale MHD turbulence may be a different story. a certain inertial range to accurately estimate mean stresses. Such difficulties are not distinctly relativistic; it is a triumph 6.3.4. Longer term evolution of remnants: subgrid scale mod of sorts that numerical relativity now stumbles against the eling. The subsequent evolution of the remnant depends on same challenges inherent in turbulence and dynamo modeling secular processes that drive the star from one equilibrium to that would confront us in Newtonian physics. another. During the first tens of milliseconds, hydrodynamic Several numerical relativity groups have attempted to torques redistribute angular momentum outward while gravita- capture unresolved transport processes using subgrid mod- tional radiation drains the star’s total angular momentum [580]. els, i.e. by evolving the fluid at large scales while adding 1 2 On longer timescales of ∼10 –10 ms, magnetic processes, contributions to Tµν meant to represent averaged Reynolds namely magnetic winding and turbulent motions triggered by and Maxwell stresses from unresolved velocity and magn the MRI, also redistribute angular momentum outward [367]. etic field fluctuations. One simple choice is to model these If the core depends on rotational support, any of these might transport processes as a viscosity. One then adds a viscous trigger collapse to a black hole-torus system. However, simula- stress term Tvisc = ησ , where σ is the shear tensor µν − µν µν tions with finite-temperature nuclear EoS tend to find thermally associated with the 4-velocity uα, and η(ρ, T) sets the strength αβ supported hypermassive remnants, so collapse of hypermassive of the viscosity. From T ;β = 0, one obtains the relativis- remnants may be delayed until the neutrino cooling timescale, tic Navier–Stokes equations. This was done in 2004 by Duez which would be of order seconds [131, 605]. Winds driven by et al [607], who tracked the secular evolution of a set of hyper- magnetic fields [606] or neutrinos [555] may carry off a small massive Γ=2 polytropes in 2D (axisymmetry) starting from −3 2 amount of mass (10 –10− M ) and some angular momentum an initial differential rotation with the angular velocity about during that time. Remnants that are merely supramassive can three times higher at the center than at the equator. For high survive beyond this time and collapse much later from angular mass cases, the core undergoes dynamical collapse when it momentum loss due to pulsar spindown. loses sufficient angular momentum, leaving a black hole sur- Direct modeling of this evolution is, for the time being, rounded by a massive torus. For certain cases, viscous heating out of reach, due to the multi-scale nature of the prob- provides enough support to avert collapse, or rather to delay it lem. We have seen how small-scale growth of the MRI and for the cooling timescale.

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A disadvantage of the Navier–Stokes equations is that it relativity [280, 313, 615, 616]. To date, there have been sev- results in a parabolic system that violates causality. A dec- eral published tests of general relativity using the observed ade after Duez et al’s work, Shibata et al [608] returned to waveforms [7–9, 614, 617]. One such test consists of compar- this problem, using a version of the Israel-Stewart formalism ing the observed phase evolution of the waveform with post- for relativistic viscosity [609], which introduces an evolution Newtonian predictions. Another test consists of comparing the visc equation for T µν and respects causality. Like in the earlier inferred parameters of the merged binary based on the inspiral study, they evolve hypermassive stars in 2D. A notable finding and merger parts of the waveform separately. A difference in 2 is that, for high viscosities (roughly α ηΩ/P > 10− , with the inferred parameters would then imply that the binary did Ω the angular frequency and P the pressure),∼ outflows driven not evolve according to the predictions of general relativity. In by viscous heating may be a major source of expelled mat- all cases, to the precision that current detectors can measure ter, with outflow masses comparable to that of the dynamical these effects, the data were consistent with the predictions of ejecta. general relativity. Both of the above simulations use artificial initial data. Recently, Radice [610] has performed neutron star neutron – 7.2. GW170817: The age of multimessenger astronomy be star binary merger simulations using a subgrid turbulence gins model very similar to a viscosity. Because the cores of neutron star–neutron star binary remnants are slowly rotating, trans- On August 17, 2017, LIGO-Virgo made the first gravitational port effects spin up the core but spin down the inner envelope, wave detection of a late-inspiral neutron star–neutron star so that collapse can be delayed or accelerated, depending on binary system, labeled GW170817 (see figure 21) [10] The the strength of the effective viscosity. Meanwhile, Shibata and identification as an neutron star–neutron star binary system Kiuchi [611] find that the effect of viscosity (with a reasona- could be made from the masses of the binary components, 2 ble α 10− ) on nonaxisymmetric deformations is dramatic, which from the waveform were estimated to be in the range ∼ with these and their corresponding gravitational wave signals 1.17–1.60M , with a total mass of Mtotal = 2.73–2.78M 19 damping on a viscous timescale of around 5 ms. and mass ratio in the range 0.7–1 . As with the first black Subscale effects can also affect the large-scale magnetic hole–black hole binary detection, nature was unexpectedly field (e.g. the ‘alpha effect’ in dynamo theory). Giacomazzo kind, supplying an neutron star–neutron star binary merger et al [612] have taken a step to incorporate these effects in at a quite close luminosity distance of 40Mpc. Tidal effects numerical relativity, adding a subgrid EMF to the induction were not seen in the waveform, leading to a maximum tidal equation. The added term is designed to grow the magnetic deformability of Λ < 800 at 90% confidence. A short gamma field to equipartition with the turbulent kinetic energy (mean- ray burst, GRB 170817A, was detected by the Fermi Gamma- ing at least the largest eddies should be present on the grid), as ray Burst Monitor at a time 1.7 s after the GW170817 merger predicted by local simulations of small-scale dynamo action time in a region of the sky consistent with LIGO-Virgo’s 31 [613]. In simulations with this added term, Giacomazzo et al deg2 localization. A bright optical/infrared/UV transient was find that the field can quickly be amplified to magnetar levels. identified about half a day later and labeled AT2017gfo, fol- Simulations with subgrid terms depend on the reliability lowed by x-ray and radio signals in the coming weeks [11]. of the subgrid model, an assumption that cannot easily be The GRB was unusually dim (isotropic) luminosity relaxed, but they do allow long-term evolutions including L 1047 erg s−1, perhaps due to some combination of being effects that would otherwise be inaccessible. seen∼ off-axis and various forms of interaction between the jet and enveloping matter ejected during merger. Comparisons to numerical relativity results have mostly focused on the optical 7. Comparison to observations and infrared AT2017gfo signal, which strongly resembles an anticipated kilonova signal. Recall that, for the v 0.1c out- 7.1. Gravitational wave astronomy ∼ flows expected from neutron star–neutron star binary merg- To date, the LIGO and Virgo collaborations have announced five ers, neutron-rich Ye < 0.25 outflow will synthesize lanthanide confirmed black hole binary merger observations, GW150914 elements, have high opacity, and is expected to peak after a [4], GW151226 [6], GW170104 [7], GW170608 [8], and week in the near infrared. Less neutron-rich outflow (perhaps GW170814 [9], as well as a potential sixth, LVT151012 [614]. made so by neutrino processing) will have lower opacity and The observed progenitor black-hole masses ranged from 7M is expected to peak after about a day in optical wavelengths. to 36M , the mass ratios ranged from 0.53 to 0.83, and the AT2017gfo showed signs of both signals, an early UV-blue final merged black hole masses ranged from18 M to 62M . signal with a near-IR tail [619], which can naturally be There has been a wealth of new information gleaned from explained if outflows of both kinds are present, and the high these events. Perhaps most importantly, a major prediction of 19 The exact allowed range of masses depends on whether one allows the general relativity in the strong-field regime was confirmed: possibility of large binary component spins. If spins are assumed to be not black holes, or at least extremely compact objects much larger than those observed Galactic binary neutron stars, the component masses are inferred to lie in the range 1.16–1.6M . Allowing for large spins, more massive than the maximum neutron star mass, exist, the range expands to 1.00–1.89M [618]. The data itself does not, at the form binaries, and merge through the emission of gravita- time of writing, exclude the possibility that one component was a black hole, tional waves [4, 5]. As mentioned above, the observed merger but known black hole formation scenarios would not produce black holes of waveforms are consistent with the predictions of numerical such low mass.

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Figure 21. A reproduction of figure 1 of [11] courtesy of the authors. The figure shows the localization of the gravitational-wave, gamma- ray, and optical signals from neutron star–neutron star binary merger GW170817. The left panel shows 90% credible regions in the sky from gravitational wave and gamma ray detections. On the right are optical images after (top) and before (bottom) the merger. Reproduced from [11]. © IOP Publishing Ltd. CC BY 3.0. opacity material does not completely occult the low opacity remnant must have suffered delayed collapse, meaning that material. Kilonova models can accommodate the observed it was above this mass. First, a long-lived magnetar would model with two components to the ejecta: a M = 0.01M , have released energy via dipole radiation at L 1050 erg −1 ∼ v = 0.3c lanthanide-poor ejecta for the blue component and (B/1015 G)2, which would accelerate the ejecta to v c and ‘ ’ ≈ a M = 0.04M , v = 0.1c lanthanide-rich ejecta for the ‘red’ produce bright emissions not matching observations [625]. component [620–622]. Second, the collapse to a black hole with disk may be needed These numbers can be rather directly compared to numer to explain the GRB. The supramassive mass limit is related to ical predictions. A first conclusion is that dynamical ejecta the TOV mass limit; the former is about 20% larger than the from tidal forces and the collision shock are inadequate. For latter. (Note, though, that thermal effects effectively alter the the range of realistic EoS, the total dynamical ejecta mass EoS and hence the supramassive limit, an additional source of does not exceed about 0.02M [623, 624]. More ejecta can be uncertainty.) Reasoning of this sort has been used in several produced by outflows from the stellar remnant or surrounding papers [625] to set an upper limit to the neutron star maximum accretion disk. Numerical relativity has already told us some- mass in the range 2.16–2.28M [625, 626]. thing interesting: the merged object must be of a kind to give Finally, there may be clues in the presence of the blue kilo- such outflows. nova. Shibata et al [623] compare merger simulations with Next, numerical relativity provides another crucial piece of the stiff DD2 EoS with those of the soft SFHo EoS, combin- information. If the remnant collapses promptly, it will leave a ing 3D merger simulations with 2D viscous simulations of the black hole and very low-mass disk, lower than the remaining subsequent ∼s of secular evolution, with neutrino transport mass that needs to be ejected. (A warning is in order here. If included in both. SFHo has a lower maximum mass and so the binary was very asymmetric, of the order q 0.7, it may predicts prompt collapse, while DD2 results in a long-lived be possible to get massive disks even in a high-mass,≈ prompt- stellar remnant. Viscosity drives ejecta from both the hyper- collapse scenario. The inferences below mostly assume that massive remnant (for DD2) and the disk. However, a long- this was not the case.) Therefore, it is surmised that the rem- lived remnant seems to be necessary to provide the neutrino irradiation needed for a high-Y outflow component that pro- nant did not promptly collapse. This means the binary’s mass, e which we know, is below the threshold mass for prompt col- duces the blue kilonova. Once again, the kilonova combined lapse, which is loosely connected to the neutron star maxi- with numerical relativity constrains the EoS generally in the mum mass. Alternatively, the need to avoid prompt collapse direction of excluding soft EoS. However, the EoS-related and small disk led Bauswein et al [646] to set a lower limit parameter to which the neutron star–neutron star binary out- on the radius of a 1.6M neutron star of 10.6 km and Radice come is most sensitive is the maximum mass. et al [624] to place a lower limit on the tidal deformability of Λ > 400, both based on numerical relativity neutron star– 8. Conclusion neutron star binary simulations with a variety of realistic EoS. Next, we may ask whether the remnant was above or below The birth of gravitational wave astronomy presents a remark- the supramassive–hypermassive cutoff mass. (See section 1.4 able story of long-term planning and investment, with numer above.) Arguments have been made to the effect that the ical relativity being only one of the fields built up largely in

35 Rep. Prog. Phys. 82 (2019) 016902 Review anticipation of discoveries known to be decades away. The allow distinct signatures if the neutron star undergoes spon- investment of time, money, and careers has now been vin- taneous scalarization [630], so these systems may be useful dicated. Although it is more broadly useful, the majority of gravity test probes as well. effort in numerical relativity has been devoted where it was When all else is done, numerical relativity results leave needed by LIGO-Virgo, to the three types of compact object us with a problem of their own, the problem of their inter- binaries. For each type, numerical relativity has established pretation. The asymptotic gravitational wave output is gauge some definite results that are beyond the reach of Newtonian invariant, but the dynamics of the interior is encoded in tensor physics or perturbation theory: the possibility of super-kicks functions on grids in an evolved coordinate system. It is actu- in black hole–black hole binary mergers and the threshold ally remarkable that the movies numerical relativists produce mass for prompt collapse in neutron star–neutron star binary of our mergers look so qualitatively reasonable, an artifact of merger, to name just two. The study of these systems is not gauges designed to minimize unnecessary coordinate dynam- yet finished. Of the three system types, the simulation of black ics. When one wants to know something concrete about the hole–black hole binaries is the most mature, but here also the strong-field region, the difficulty of disentangling coordinate binary parameter space is most intimidating, and the appli- effects becomes acute. Even to prove that the result of a black cation of numerical relativity to devising templates for the hole–black hole binary merger settles to a Kerr spacetime is a full 7D space is ongoing work. Neutron star–neutron star and surprisingly intricate affair [631]. Apparent horizons are foli- black hole–neutron star binary simulations are less accurate, ation-dependent. Event horizons are not, but even they will be and their ability to capture the multi-scale magnetic and neu- visualized on some arbitrary coordinate system. Ray tracing trino effects of the post-merger evolution might not even be can be used to reconstruct what a nearby observer would actu- qualitatively adequate. ally see from a black hole–black hole binary merger [632]. However, even where numerical relativity simulations of To try to give some intuition for the physics of the merger, black hole–neutron star binary and neutron star–neutron star Nichols et al [633] have proposed plotting (gauge-dependent) binary mergers are inadequate, there is no longer anything dis- field lines to illustrate the local tidal stretches and twists. The tinctively relativistic about the problems. Newtonian simula- challenge of making sense of a background-independent field tions of neutron star–neutron star binary mergers are no more theory extends more widely in theoretical physics, but it con- advanced; they face all the same difficulties. In fact, Newtonian fronts us in a particularly concrete form in numerical relativity. simulations of these systems are becoming less common. Since it is not much more difficult, why not just work in general relativity? With the advent of open-source, publicly availa- Acknowledgments ble numerical relativity codes such as the Einstein Toolkit [41, 145], the barrier to an interested astrophysicist doing numer We thank Andreas Bauswein, John Baker, Manuela Campanelli, ical relativity work has never been lower. This is for the best. Koutarou Kyutoku, Carlos Lousto, Luciano Rezzolla, Richard One could say that the goal of numerical relativity all along has O’Shaughnessy, Frans Pretorius, and Masaru Shibata for care- been to abolish itself as a distinct subfield, to make solving the ful reading of this manuscript. We thank the referees and the editorial board member for their many helpful suggestions. Einsteins equations as routine as solving Poisson’s equation, and thus to dissolve into computational astrophysics. MD gratefully acknowledge the NSF for financial support from And yet, numerical relativity will also remain as a tool Grant PHY-1806207. YZ gratefully acknowledge the NSF for for addressing questions in gravitational physics. Numerical financial support from Grants No. PHY-1607520, No. PHY- experiments are used to investigate features of black hole 1707946, No. PHY-1726215, No. OAC-1811228, No. ACI- physics such as cosmic censorship and the generic structure 1550436, No. AST-1516150, and No. ACI-1516125. of spacetime singularities [627]. Perhaps more important, as gravitational wave detections grow in number and accuracy, ORCID iDs numerical relativity will aid in testing alternative theories of gravity. Ultimately, to test general relativity (and—dare Matthew D Duez https://orcid.org/0000-0002-0050-1783 we hope?—supersede it), we will need to compare general Yosef Zlochower https://orcid.org/0000-0002-7541-6612 relativity predictions, say of black hole–black hole binary mergers, with more general possibilities, to identify the sig- References natures of new physics that cannot be reproduced in general relativity by tinkering with parameters in the vast 7D [1] Pretorius F 2005 Phys. Rev. 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Matthew D Duez, is an associate professor at Washington State University. He received his PhD from the University of Illinois at Urbana-Champaign in 2005. He then took a postdoctoral position at Cornell University, where he joined the SXS collaboration. His current position at WSU began in 2010. His research focuses on the application of numerical relativity to non-vacuum (hydrodynamic) systems such as compact binaries in which at least one of the components is a neutron star.

Yosef Zlochower, is an associate professor at the Rochester Institute of Technology. He received his PhD from the University of Pittsburgh in 2002 and held postdoctoral positions at the University of Pittsburgh, the University of Texas at Brownsville, and the Rochester Institute of Technology. He has spent the past decade studying numerical simulations of black-hole spacetimes and was one of the principle developers of the breakthrough “moving punctures” approach for evolving multiple black holes in full numerical relativity. His research focuses on binary dynamics and generating waveforms for LIGO data analysis.