1 Introduction

Close binaries consisting of two compact stellar remnants – white dwarfs (WDs), neutron stars (NSs) or black holes (BHs) are considered primary targets of the forthcoming field of gravitational wave (GW) astronomy (see, for a review, [758*, 252*, 673*, 672]), since their orbital evolution is entirely controlled by the emission of gravitational waves and leads to ultimate coalescence (merger) and possible explosive disruption of the components. Emission of gravitational waves accompanies the latest stages of evolution of stars and manifests instabilities in relativistic objects [10, 13]. Close compact binaries can thus serve as testbeds for theories of gravity [221]. The NS(BH) binary mergers that release  10⁵² erg as GWs [113, 114] should be the brightest GW events in the 10 – 1000 Hz frequency band of the existing or future ground-based GW detectors like LIGO [23], VIRGO [6], GEO600 [635], KAGRA(LCGT) [728] (see also [642] for a review of the current state of existing and 2nd- and 3rd-generation ground-based detectors). Mergers of NS(BH) binaries can be accompanied by the release of a huge amount of electromagnetic energy in a burst and manifest themselves as short gamma-ray bursts (GRBs). A lot of observational support for NS-NS/NS-BH mergers as sources of short GRBs have been obtained (see, e.g., studies of short GRB locations in the host galaxies [204*, 751*] and references therein). As well, relativistic jets, associated with GRB of any nature may be sources of GW in the ground-based detectors range [50].

WD binaries, especially those observed as AM CVn-stars and ultracompact X-ray binaries (UCXB), are potential GW sources within the frequency band (10^–4 – 1) Hz of the space GW interferometers like (the currently cancelled) LISA [187*],¹ NGO (eLISA) [10, 11], DEGIGO [857] and other proposed or planned low-frequency GW detectors [122, 40]. At the moment, eLISA is selected for the third large-class mission in ESA’s Cosmic Vision science program (L3). Its first step should be the launch of ESA’s LISA Pathfinder (LPF) mission in 2015; the launch of eLISA itself is currently planned for 2034.²

WD binary mergers are among the primary candidate mechanisms for type Ia supernovae (SNe Ia) explosions. The NIR magnitudes of the latter are considered as the best “standard candles” [26] and, in this guise, are crucial in modern cosmological studies [638, 578]. Further improvements in the precision of standardization of SNe Ia fluxes is possible, e.g., by account of their environmental dependence [640, 641, 820]. On the other hand, usually, as an event beneath the “standard candle”, an explosion of a non-rotating WD with the Chandrasekhar mass () is considered. Rotation of progenitors may increase the critical mass and make SNe Ia less reliable for cosmological use [154, 155].

SNe Ia are suggested to be responsible for the production of about 50% of all heavy elements in the Universe [761]. Mergers of WD binaries are, probably, one of the main mechanisms of the formation of massive () WDs and WDs with strong magnetic fields (see, e.g., [322*, 392] for recent studies and review of earlier work).

A comparison of SN Ia rates (for the different models of their progenitors) with observations may, in principle, shed light on both the star formation history and on the nature of the progenitors (see, e.g., [663, 873*, 453, 874, 871, 250, 457, 249, 458*]).

Compact binaries are the end products of the evolution of stellar binaries, and the main purpose of the present review is to describe the astrophysical knowledge of their formation and evolution. We shall discuss the present situation with the main parameters determining their evolution and the rates of coalescence of NS/BH binaries and WDs.

The problem is to evaluate as accurately as possible (i) the physical parameters of the coalescing binaries (masses of the components and, if possible, their spins, magnetic fields, etc.) and (ii) the occurrence rate of mergers in the Galaxy and in the local Universe. Masses of NSs in binaries are known with a rather good accuracy of 10% or better from, e.g., pulsar studies [760, 367]; see also [410, 550*] for recent summaries of NS mass measurements.

The case is not so good with the rate of coalescence of relativistic stellar binaries. Unfortunately, there is no way to derive it from first principles – neither the formation rate of the progenitors for compact binaries nor stellar evolution are known well enough. However, the situation is not completely hopeless, especially in the case of NS binary systems. The natural appearance of rotating NSs with magnetic fields as radio pulsars allows one to search for binary pulsars with a secondary compact companion using powerful methods of modern radio astronomy (for example, in dedicated pulsar surveys, such as the Parkes multi-beam pulsar survey [455, 192]).

Based on observational statistics of Galactic pulsar binaries with NS companions, one can evaluate the Galactic rate of NS binary formation and merging [586*, 505*]. On the other hand, a direct simulation of the evolution of population of binaries in the Galaxy (the population synthesis method) can also predict the formation and merger rates of close compact binaries as a function of (numerous) parameters of stellar formation and evolution. Both kinds of estimates are plagued by badly constrained parameters or selection effects, but it is, nevertheless, encouraging that most likely Galactic rates of events obtained in two ways currently differ by a factor of  3 only: 80/Myr from observations and 30/Myr from population synthesis; see [456] for a recent review of observational and theoretical estimates, also Section 6.

No BH or NS + BH binary systems have been found so far, so merger rates of compact binaries with BHs have been evaluated as yet only from population synthesis studies.

1.1 Formation of stars and end products of their evolution

Let us briefly remind the key facts about star formation and evolution. Approximately 6% of the baryonic matter in the Universe is confined to stars [217]. Recent observational data suggests that, first, long thin filaments form inside molecular clouds and, next, these filaments fragment into protostellar cores due to gravitational instability, if their mass-per-unit length exceeds a certain threshold [14]. An object may be called a “star” if it is able to generate energy by nuclear fusion at a level sufficient to halt the contraction [394]. For solar chemical composition, and increases if stellar metallicity is lower than solar [81]. Currently, among observed stars, the lowest dynamically determined mass has component C of a triple system AB Dor: [115].

The maximum mass of the star is set by the proximity of the luminosity to the Eddington limit and pulsational instability. For solar chemical composition, this limit is close to  [38, 880].³ But conditions of stellar formation, apparently, define a much lower mass limit. Currently, components of a eclipsing binary NGC3603-A1 have maximum dynamically-measured masses: and [683]. Since both stars are slightly-evolved main-sequence objects (WN6h), subject to severe–stellar-wind mass loss, their inferred initial masses could be higher: and [125]. Indirect evidence, based on photometry and spectral analysis suggests the possible existence of stars; see [813] for references and discussion.

For the metal-free stars that formed first in the Universe (the so-called Population III stars), the upper mass limit is rather uncertain. For example, it can be below [4, 745], because of the competition of accretion onto first formed compact core and nuclear burning and influence of UV-irradiation from nearby protostars. On the other hand, masses of Population III stars could be much higher due to the absence of effective coolants in the primordial gas. Then, their masses could be limited by pulsations, though, the upper mass limit remains undefined [703] and masses up to are often inferred in theoretical studies.

The initial mass function (IMF) of main-sequence stars can be approximated by a power-law [669] and most simply may be presented taking for and for [830]. Note, there are also claims that for the most massive stars () the IMF is much steeper – up to , see [403] and references therein. Estimates of the current star-formation rate (SFR) in the Galaxy differ depending on the method, but most at present converge to [108, 353, 537]. This implies that in the past the Galactic SFR was much higher. For the most recent review of observations and models pertinent to the star-formation process, the origin of the initial mass function, and the clustering of stars, see [389].

The evolution of a single star and the nature of its compact remnant are determined by the main-sequence mass and chemical composition. If is lower than the minimum mass of stars that ignite carbon in the core, , after exhausting the hydrogen and helium in its core, a carbon-oxygen WD forms. If exceeds a certain limiting , the star proceeds to form an iron core, which collapses into a NS or a BH. In the stars of intermediate range between and (called “super-AGB stars”⁴ ) carbon ignites in a partially-degenerate core and converts its matter (completely or partially) into an oxygen-neon mixture. As conjectured by Paczyński and Ziolkowski [557], the TP-AGB stage of the evolution of stars⁵ terminates when shells with positive specific binding energy appear in the envelope of the star. In the latter expression, the internal energy term accounts for ideal gas, radiation, ionisation, dissociation and electron degeneracy, as suggested by Han et al. [266]. The upper limit on masses of model precursors of CO white dwarfs depends primarily on stellar metallicity, on the treatment of mixing in stellar interiors, on accepted rates of mass loss during the AGB and on more subtle details of the models, see, e.g., [278, 279, 338]. Modern studies suggest that for solar metallicity models it does not exceed  [412].

As concerns more massive stars, if, due to He-burning in the shell, the mass of the core reaches , electron captures by ²⁴Mg and ²⁰Ne ensue and the core collapses to a neutron star producing an “electron-capture supernova” (ECSN) [488]). Otherwise, an ONe WD forms.⁶ Figure 1* shows, as an example, the endpoints of stellar evolution for intermediate mass stars, computed by Siess [707, 708].⁷ Note, results of computations very strongly depend on still uncertain rates of nuclear burning, the treatment of convection, the rate of assumed stellar-wind mass loss, which is poorly known from observations for the stars in this transition mass range, as well as on other subtle details of the stellar models (see discussion in [599, 325, 98, 749]). For instance, zero-age main sequence (ZAMS) masses of solar-composition progenitors of ECSN found by Poelarends et al. [599] are .⁸ In [295], the lower mass limit for core-collapse SN at solar metallicity is found to be equal to . According to Jones et al. [326], an star experiences ECSN, while a star evolves to Fe-core collapse. Takahashi et al. [749] find that a solar metallicity star explodes as ECSN, while an star experiences Fe-core collapse, consistent with Siess’ results.

Figure 1: Endpoints of evolution of moderate-mass nonrotating single stars depending on initial mass and metallicity. Image reproduced with permission from [709], copyright by IAU.

ECSN eject only a small amount of Ni (), and it was suggested that they may be identified with some subluminous type II-P supernovae [366*]. Tominaga et al. [764] were able to simulate ECSN starting from first principles and to reproduce their multicolor light curves. They have shown that observed features of SN 1054 (see [741]) share the predicted characteristics of ECSN.

We should enter a caveat that in [412] it is claimed that opacity-related instability at the base of the convective envelope of stars may result in ejection of the envelope and, consequently, prevent ECSN.

Electron-capture SNe, if they really happen, presumably produce NS with short spin periods and, if in binaries, systems with short orbital periods and low eccentricities. Observational evidence for ECSN is provided by Be/X-ray binaries, which harbor two subpopulations, typical for post-core-collapse objects and for post-ECSN ones [369]. Low kicks implied for NSs formed via ECSN [597*] may explain the dichotomous nature of Be/X-ray binaries. As well, thanks to low kicks, NS produced via ECSN may be more easily retained in globular clusters (e.g., [397]) or in regions of recent star formation.

The existence of the ONe-variety of WDs, predicted theoretically [24, 488, 301*] and later confirmed by observations [774], is important in the general context of compact-star binary evolution, since their accretion-induced collapse (AIC) may result in the formation of low-mass neutron stars (almost) without natal kicks and may thus be relevant to the formation of NS binaries and low-mass X-ray binaries (see [21, 146*, 798] and references therein). But since for the purpose of detection of gravitational waves they are not different from much more numerous CO WDs, we will, as a rule, not consider them below as a special class.

If (or ), thermonuclear evolution proceeds until iron-peak elements are produced in the core. Iron cores are subject to instabilities (neutronization, nuclei photo-disintegration), that lead to gravitational collapse. The core collapse of massive stars results in the formation of a neutron star or, for very massive stars, a black hole⁹ and is associated with the brightest astronomical phenomena such as supernova explosions (of type II, Ib, or Ib/c, according to the astronomical classification based on the spectra and light curves properties). If the pre-collapsing core retains significant rotation, powerful gamma-ray bursts lasting up to hundreds of seconds may be produced [850].

The boundaries between the masses of the progenitors of WDs or NSs and NSs or BHs are fairly uncertain (especially for BHs). Usually accepted typical masses of stellar remnants for single non-rotating solar–chemical-composition stars are summarized in Table 1.

Note that the above-described mass ranges for different outcomes of stellar evolution were obtained by computing 1D-models of non-rotating stars. However, all star are rotating and some of them possess magnetic fields, thus, the problem is 2D (or 3D, if magnetic-field effects are considered). For a physical description of effects of rotation on stellar evolution, see [406, 561]. Realistic models with rotation should account for deviation from spherical symmetry, modification of gravity due to centrifugal force, variation of radiative flux with local effective gravity, transfer of angular momentum and transport of chemical species. Up to now, as a rule, models with rotation are computed in a 1D-approximation which, typically, makes use of the fact that in rotating stars mass is constant within isobar surfaces [174]. Regretfully, for low and intermediate-mass stars, model sequences of rotating stars covering evolution to advanced phases of AGB-evolution are absent. However, as noted by Domínguez et al. [156], rotation must strongly influence evolution at core helium exhaustion and the formation of the CO-core stage, when the latter experiences very strong contraction. With an increase of angular velocity, lower pressure is necessary to balance the gravity, hence the temperature of He-burning shells of rotating stars should be lower than in non-rotating stars of the same mass. This extends AGB-lifetime and results in an increase of the mass of CO-cores, i.e., the C-ignition limit may be shifted to lower masses compared to non-rotating stars. Most recently, grids of 1-D models of rotating stars were published, e.g., in [69, 234, 236].

The algorithm and first results of self-consistent calculations of rapidly-rotating 2D stellar models of stars in the early stages of evolution are described, e.g., in [639, 178]. We may note that these models are very important for deriving physical parameters of the stars from astro-seismological data.

For a more detailed introduction to the physics and evolution of stars, the reader is referred to the classical fundamental textbook [121] and to several more modern ones [612, 51, 53, 171, 297, 298]. Formation and physics of compact objects is described in more detail in the monographs [691, 53]. For recent studies and reviews of the evolution of massive stars and the mechanisms of core-collapse supernovae we refer to [406, 79, 320, 881, 212, 721].

1.2 Binary stars

A fundamental property of stars is their multiplicity. Among stars that complete their nuclear evolution in the Hubble time, the estimated binary fraction varies from  (40 – 60)% for stars [168, 621] to almost 100% for more massive A/B and O-stars, e.g., [468, 382, 370, 469, 106, 671], (but, e.g., in [454, 670], a substantially lower binary fraction for massive stars is claimed).

Based on the summary of data on binary fraction provided in [381, 385, 671], van Haaften et al. [804] suggested an approximate formula

considering all multiple systems as binaries.

The most crucial parameters of binaries include the component separation and mass ratio , since for close binaries they define the outcome of the Roche lobe overflow (in fact, the fate of the system). The most recent estimates for M-dwarfs and solar-type stars confirm earlier findings that the -distribution does not strongly deviate from a flat one: , with [632]. This distribution is defined by the star-formation process and dynamical evolution in stellar clusters; see, e.g., [28, 569]. Distribution over is flat in log between contact and  [604*].

We note, cautionarily, that all estimates of the binary fraction, mass-ratios of components and distributions over separations of components are plagued by numerous selection effects (see [380] for a thorough simulation of observations and modeling observational bias). A detailed summary of studies of multiplicity rates, distributions over orbital periods and mass ratios of components for different groups of stars may be found in [167].

In binary stars with sufficiently-large orbital separations (“wide binaries”) the presence of the secondary component does not influence significantly the evolution of the components. In “close binaries” the evolutionary expansion of stars leads to the overflow of the critical (Roche) lobe and mass exchange between the components RLOF. Consequently, the formation of compact remnants in close binaries differs from single stars (see Section 3 for more details).

As discussed above, the lower mass limit of NS progenitors is uncertain by several . This limit is even more uncertain for the BH progenitors. For example, the presence of a magnetar (neutron star with an extremely large magnetic field) in the open cluster Westerlund 1 means that it descends from a star that is more massive than the currently-observed most-massive main-sequence cluster stars (because for a massive star the duration of the main-sequence stage is inversely proportional to its mass squared). The most massive main-sequence stars in this stellar cluster are found to have masses as high as , suggesting [643]. On the other hand, it was speculated, based on the properties of X-ray sources, that in the initial mass range the mass of pre-BHs may vary depending on such poorly-known parameters as rotation or magnetic fields [176].

Binaries with compact remnants are primary potentially-detectable GW sources (see Figure 2*). This figure plots the sensitivity of ground-based interferometers, LIGO, as well as the space laser interferometers LISA and eLISA, in the terms of dimensionless GW strain measured over one year.¹⁰ The strongest Galactic sources at all frequencies are the most compact NS binaries, WD binaries, and (still hypothetically) BHs. NS/BH binary systems are formed from initially massive binaries, while WD binaries descend from low-mass binaries.

Figure 2: Sensitivity limits of GW detectors and the regions of the – diagram occupied by some potential GW sources. (Courtesy G. Nelemans).

In this review we shall concentrate on the formation and evolution of compact star binaries most relevant to GW studies. The article is organized as follows. We start in Section 2 with a review of the main observational data on NS binaries, especially measurements of masses of NSs and BHs, which are most important for the estimate of the amplitude of the expected GW signal. We briefly discuss empirical methods to determine the NS binary coalescence rate. The basic principles of stellar binary evolution are discussed in Section 3. Then, in Section 4 we describe the evolution of massive stellar binaries. Next, we discuss the Galactic rate of formation of binaries with NSs and BHs in Section 5. Theoretical estimates of detection rates for mergers of relativistic stellar binaries are discussed in Section 6. Further, we proceed to the analysis of the formation of short-period binaries with WD components in Section 7, and consider observational data on WD binaries in Section 8. A model for the evolution of interacting double-degenerate systems is presented in Section 9. In Section 10 we describe gravitational waves from compact binaries with white dwarf components. Section 11 is devoted to the modeling of optical and X-ray emissions of AM CVn-stars. Our conclusions follow in Section 12.