1 Introduction

Binaries composed of neutron stars (NSs) and black holes (BHs) have long been of interest to astrophysicists. They provide many important constraints for models of massive star evolution and compact object formation, and are among the leading potential sources for detection by gravitational-wave (GW) observatories. While it remains uncertain whether mergers of compact binaries are an important contributor to the production of r-process elements, they are now thought to be the leading candidate to explain short-duration, hard-spectrum gamma-ray bursts (often abbreviated to “short-hard” GRBs, or merely SGRBs).

The first neutron star–neutron star (NS-NS) binary to be observed was PSR B1913+16, in which a radio pulsar was found to be in close orbit around another NS [135]. In the decades since its discovery, the decay of the orbit of PSR B1913+16 at exactly the rate predicted by Einstein’s general theory of relativity (see, e.g., [306, 325]) has provided strong indirect evidence that gravitational radiation exists and is indeed correctly described by general relativity (GR). This measurement led to the 1993 Nobel Prize in physics for Hulse and Taylor.

According to the lowest-order dissipative contribution from GR, which arises at the 2.5PN level (post-Newtonian; where the digit indicates the expansion order in [v∕c]2 in the Taylor expansion term), and assuming that both NSs may be approximated as point masses, a circular binary orbit decays at a rate da ∕dt = − a ∕τGW where a is the binary separation and the gravitational radiation merger timescale τGW is given by

4 4 τ = -5--a--- = -5-----a------- GW 64 μM 2 64 q(1 + q)M 31 ( )4 ( ) −3 = 2.2 × 108q−1(1 + q)−1 -a-- -M1---- yr, (1 ) R ⊙ 1.4M ⊙
where M1, M2, and M ≡ M1 + M2 are the individual NS masses and the total mass of the binary, μ = M1M2 ∕M is the reduced mass, q = M2 ∕M1 is the binary mass ratio, and we assume geometrized units where G = c = 1 (as we do throughout this paper, unless otherwise noted). The timescale for an elliptical orbit is shorter, and it can be shown that eccentricity is reduced over time by GW emission, leading to a circularization of orbits as they decay. A quick integration shows that the time until merger is given by τmerge = τGW ∕4.

The luminosity of such systems in gravitational radiation is

dEGW-- 32μ2M--3 32-m21m22(m1-+-m2-) LGW = − dt = 5 a5 = 5 a5 ( )5 ( ) −5 = 5.34 × 1032q2(1 + q) --M1--- -a-- erg∕s 1.4M ⊙ R⊙ ( )5 ( )−5 = 8.73 × 1051q2(1 + q) --M1--- ---a---- erg∕s, (2 ) 1.4M ⊙ 100 km
which, at the end of a binary’s lifetime, when the components have approached to within a few NS radii of each other, is comparable to the luminosity of all the visible matter in the universe (∼ 1053 erg/s). The resulting strain amplitude observed at a distance D from the source (assumed to be oriented face-on) is given approximately by
( )2 ( ) −1 4M1M2--- −23 --M1--- (---a----)−1 ----D---- h = aD = 5.53 × 10 q 1.4M ⊙ 100 km 100 Mpc , (3 )
at a characteristic frequency
∘ --- 1 M ( M )1 ∕2 ( a )−3∕2 fGW = 2forb = -- -3-= 194 ------- -------- Hz. (4 ) π a 2.8M ⊙ 100 km

The first measurement that will likely be made with direct GW observations is the orbital decay rate, with the period evolving (for the circular case) according to the relation

dT 192 π --- = − -----(ℳc ω)5∕3, (5 ) dt 5
where T is the orbital period and ω the angular frequency, and thus the “chirp mass,”
ℳc ≡ μ3∕5M 2∕5 = M 3∕5M 3∕5(M1 + M2 )−1∕5, (6 ) 1 2
is likely to be the easiest parameter to determine from GW observations.

Several NS-NS systems are now known, including PSR J0737–3039 [59], a binary consisting of two observed pulsars, which allows for the prospect of even more stringent tests of GR [149]. Even with the handful of observed sources to date, one may use this sample to place empirical limits on the expected rate of NS-NS mergers [143Jump To The Next Citation Point] and to constrain the many parameters that enter into population synthesis calculations [217]. With regard to the former, the very short merger timescale for J0737, τmerge = 85 Myr, makes it especially important for estimating the overall rate of NS-NS mergers since it is a priori very unlikely to detect a system with such a short lifetime.

Although black hole–neutron star (BH-NS) binaries are expected to form through the same processes as NS-NS binaries, none has been detected to date. This is generally thought to reflect their lower probability of detection in current surveys, in addition to intrinsically smaller numbers compared to NS-NS systems [39]. BH-NS systems are an expected byproduct of binary stellar evolution, and properties of the population may be inferred from population synthesis studies calibrated to the observed NS-NS sample (see, e.g., [37Jump To The Next Citation Point]).

In this review, we will summarize the current state of research on relativistic mergers, beginning in Section 2 with a description of the astrophysical processes that produce merging binaries and the expected parameters of these systems. The phases of the merger are briefly described in Section 3. In Section 4, we discuss the numerical techniques used to generate quasi-equilibrium (QE) sequences of NS-NS configurations, and we summarize the QE calculations that have been performed. These sequences yield a lot of information about NS physics, particularly with regard to the nuclear matter equation of state (EOS). They also serve as initial data for dynamical merger calculations, which we discuss next, focusing in turn on the numerical hydrodynamics techniques used to compute mergers and the large body of results that has been generated, in Sections 5 and 6, respectively. We pay particular attention to how numerical studies have taken steps toward answering a number of questions about the expected GW and electromagnetic (EM) emission from merging binaries, and we discuss briefly the possibility that they may be the progenitors of SGRBs and a source of r-process elements. We close with conclusions and a look to the future in Section 7.

While most of this review focuses on NS-NS mergers, many of the methods used to study NS-NS binaries are also used to evolve BH-NS binaries, and it has become clear that both merger types may produce similar observational signatures as well. For a review focusing on BH-NS merger calculations, we encourage the reader to consult the recent work by Shibata and Taniguchi [284Jump To The Next Citation Point].

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