List of Footnotes

1		http://lisa.gsfc.nasa.gov/cosmic_vision_changes.html, http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48661
2		https://www.elisascience.org/news/top-news/gravitationaluniverseselectedasl3
3		The existence of such high-mass very short-living objects is deemed possible due to runaway stellar collisions in dense young compact stellar clusters [605 ]; for the prospects of observing them see, e.g., [564 ].
4		AGB – Asymptotic Giant Branch on the Hertzsprung–Russell diagram
5		TP-AGB – thermally pulsing AGB, the late stage of the AGB evolution, characterized by the development of strong pulsations leading ultimately to the ejection of the entire envelope of the red giant star.
6		Yet another possibility for the formation of neutron stars exists: if a sufficiently cold white dwarf close to critical mass accretes matter, the time scale of its gravitational collapse allows neutronization before the onset of pycnonuclear reactions [85 ].
7		See these papers for detailed references and discussion of earlier studies.
8		Results obtained by the same group for lower metallicities may be found in Poelarend’s PhD thesis at http://www.astro.uni-bonn.de/~nlanger/siu_web/pre.html.
9		Pair-creation instability in the oxygen cores of slowly rotating $M0 ≈ (140 –260)M ⊙$ and metallicity $Z ≲ Z ⊙∕3$ stars or rapidly-rotating stars with $M0 > 80M ⊙$ and $Z∕Z⊙ < 10−3$ may lead to their collapse, which results in the explosive ignition of oxygen, which leads to the disruption of the star [407 ].
10		To compare the detector’s sensitivity, characterized by the noise power spectral density $∘ ---- S (f)$ with the dimensionality $√--- 1 Hz$ , with the dimensionless GW strain amplitude $h$ , one conventionally considers the detector’s sensitivity in an integration time of $T$ years for a monochromatic signal, which is $∘ ------- S (f)∕T$ and corresponds to the detection at the signal-to-noise level equal one.
11		Unless the companion is directly observable and its mass can be estimated by other means.
12		We neglect here the highly unlikely case of a direct collision between a newly born BH and its companion due to the possible BH natal kick.
13		A signal with such an increasing frequency is reminiscent of the chirp of a bird. This explains the origin of the term “chirp mass” for the parameter $ℳ$ , which fully determines the GW frequency and amplitude behavior.
14		Actually, no CVs were known in the “gap” when this fact was first realized [647 ].
15		Actually, the Roche potential should also incorporate the effects of radiation pressure from the binary components. Then in some cases the resulting potentials do not exhibit the contact surfaces of the classical Roche potential [684 , 144 ].
16		Hydrogen burning time for solar-composition single stars with $M0 ≲ 0.95 M⊙$ exceeds the age of the universe.
17		In principle, it is possible to form a close binary system with a compact star (NS or BH) from a hierarchical triple system without an initial CE phase due to dynamical interactions (Kozai–Lidov mechanism), see [693 ].
18		Indeed, to put a gas element of unit mass with specific volume $v$ at a distance $R$ around mass $M$ from infinity, the work equal to the gravitational potential $ϕ$ should be done; but then one should heat up the gas element, which is equivalent to give it the specific internal energy $𝜖$ , and also do $Pv$ -work to empty space in the already present gas, hence $ϕ+ H$ is relevant in calculating the binding energy of an isentropic envelope, where $dH = Pdv$ .
19		Attempts to apply the $γ$ -formalism to both stages of mass exchange ( $γ,γ$ -formalism), which failed to reproduce evolution of close WD binary and criticized the $γ$ -formalism, are unjustified extrapolation of the limited application of the formalism suggested by [517 ].
20		Tutukov and Yungelson’s paper [782 ] was published in 1973 prior to the discovery of the first pulsar binary PSR 1913+16 in 1974 [757 , 292 *] and, though the evolutionary path was traced to a NS (or BH) binary, since no such objects were known at that time, it was suggested that all pairs of NSs are disrupted at the second NS formation.
21		For the first supernova explosion without kick this is always satisfied.
22		Instead of Monte Carlo simulations one may use a sufficiently dense grid in the 3D space of binary parameters and integrate over this grid (see, e.g., [791 *] and references therein).
23		Note that similar low values of $λ$ for $20$ to $50 M⊙$ stars were obtained also in [598 ]. If confirmed, these results may have major impact on the estimates of merger rates for relativistic binaries.
24		If a network of three detectors, such as two LIGOs and VIRGO, runs simultaneously, the $S∕N$ ratio in an individual detector should be $√ - > 7∕ 3 ≈ 4$ .
25		Flickering is a fast intrinsic brightness scintillation occurring on time scales from seconds to minutes with amplitudes of 0.01 – 1 mag and suggesting an ongoing mass transfer.
26		Remarkably, a clear-cut confirmation of the binary nature of AM CVn and the determination of its true period awaited for almost 25 years [513 ].
27		It is expected that recently launched astrometric space mission GAIA will detect about 200 eclipsing double degenerates [511 ].
28		The advent of new, fast and stable stellar evolutionary codes like MESA [574 , 575 ], http://mesa.sourceforge.net will, hopefully, facilitate achieving this aim in the near future.
29		Note that the existence of these winds has never been rigorously proven by radiative transfer theory.
30		The latter is characterized by the parameter $ψ = EF,c∕kTc$ , where $EF,c$ is the electron Fermi energy, $Tc$ – the central temperature of a WD.
31		http://www.astro.caltech.edu/ptf
32		The “optimistic” model of the population of IDD was taken [512 *], which assumes effective tidal interaction and inefficient destruction of hypothetical progenitors of AM CVn stars due to double-degenerate SN Ia. Thus, the numbers provided in the table are upper limits.
33		These numbers, as for models discussed below, represent one random realization of the model and are subject to Poissonian noise.
34		http://www.eso.org/public/teles-instr/e-elt/
35		http://www.tmt.org
36		http://www.jwst.nasa.gov/
37		This is true for long-period systems, but becomes an oversimplification at shorter periods and for $Porb ≲ 40 min$ , heating of the WD by accretion has to be taken into account [49 ].

	Abstract
1	Introduction
	1.1	Formation of stars and end products of their evolution
	1.2	Binary stars
2	Observations of Double Compact Stars
	2.1	Compact binaries with neutron stars
	2.2	How frequent are NS binary coalescences?
	2.3	Black holes in binary systems
	2.4	A model-independent upper limit on the BH-BH/BH-NS coalescence rate
3	Basic Principles of the Evolution of Binary Stars
	3.1	Keplerian binary system and radiation back reaction
	3.2	Mass exchange in close binaries
	3.3	Mass transfer modes and mass and angular momentum loss in binary systems
	3.4	Supernova explosion
	3.5	Kick velocity of neutron stars
	3.6	Common envelope stage
	3.7	Other notes on the CE problem
4	Evolutionary Scenario for Compact Binaries with Neutron Star or Black Hole Components
	4.1	Compact binaries with neutron stars
	4.2	Black-hole–formation parameters
5	Formation of Double Compact Binaries
	5.1	Analytical estimates
	5.2	Population synthesis results
6	Detection Rates
7	Short-Period Binaries with White-Dwarf Components
	7.1	Formation of compact binaries with white dwarfs
	7.2	White-dwarf binaries
	7.3	Type Ia supernovae
	7.4	Ultra-compact X-ray binaries
8	Observations of Double-Degenerate Systems
	8.1	Detached white dwarf and subdwarf binaries
9	Evolution of Interacting Double-Degenerate Systems
	9.1	“Double-degenerate family” of AM CVn stars
	9.2	“Helium-star family” of AM CVn stars
	9.3	Final stages of evolution of interacting double-degenerate systems
10	Gravitational Waves from Compact Binaries with White-Dwarf Components
11	AM CVn-Type Stars as Sources of Optical and X-Ray Emission
12	Conclusions
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables