"The Evolution of Compact Binary Star Systems"
Konstantin A. Postnov and Lev R. Yungelson 
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

6 Detection Rates

From the point of view of detection, a gravitational-wave signal from merging close binaries is characterized by the signal-to-noise ratio S∕N, which depends on the binary masses, the distance to the binary, the frequency, and the detector’s noise characteristics. A pedagogical derivation of the signal-to-noise ratio and its discussion for different detectors is given, for example, in Section 8 of the review [252*].

Coalescing binaries emit gravitational wave signals with a well known time-dependence (waveform) h (t) (see Section 3.1 above). This allows one to use the technique of matched filtering [758*]. The signal-to-noise ratio S ∕N for a particular detector, which is characterized by the noise spectral density −1∕2 Sn (f) [Hz ] or the dimensionless noise rms amplitude hrms at a given frequency f, depends mostly on the “chirp” mass of the binary system −1∕5 3∕5 3∕5 2∕5 ℳ = (M1 + M2 ) (M1M2 ) = μ M (here μ = M1M2 ∕M is the reduced mass and M = M1 + M2 is the total mass of the system) and its (luminosity) distance r: S ∕N ∝ ℳ5 ∕6∕r [758, 200]. For a given type of coalescing binary (NS + NS, NS + BH or BH + BH), the signal-to-noise ratio will also depend on the frequency of the innermost stable circular orbit fISCO ∼ 1∕M, as well as on the orientation of the binary with respect to the given detector and its angular sensitivity (see, e.g., Section 8 in [252*] and [673] for more detail). Therefore, from the point of view of detection of the specific type of coalescing binaries at a prerequisite signal-to-noise ratio level, it is useful to determine the detector’s maximum (or “horizon”) distance D hor, which is calculated for an optimally oriented (“ideal”) coalescing binary with a given chirp mass ℳ. For a secure detection, the S∕N ratio is usually raised up to 7 – 8 to avoid false alarms over a period of a year (assuming Gaussian noise).24 This requirement determines the maximum distance from which an event can be detected by a given interferometer [126, 2*]. The horizon distance Dhor of LIGO I/VIRGO (advanced LIGO, expected) interferometers for relativistic binary inspirals with account of the actual noise curve attained in the S5 LIGO scientific run are given in [2*]: 33(445) Mpc for NS + NS (1.4M ⊙ + 1.4 M ⊙, ℳNS = 1.22M ⊙), 70(927) Mpc for NS + BH (1.4M ⊙ + 10 M ⊙), and 161(2187) Mpc for BH + BH (10 M ⊙ + 10 M ⊙, ℳBH = 8.7M ⊙). The distances increase for a network of detectors.

It is worth noting that the dependence of the S ∕N for different types of coalescing binaries on fISCO changes rather slightly (to within 10%) as 5∕6 S ∕N ∝ ℳ, so, for example, the ratio of BH and NS detection horizons scales as S∕N, i.e., Dhor,BH∕Dhor,NS = (ℳBH ∕ℳNS )5∕6 ≈ 5.14. This allows us to estimate the relative detection ratio for different types of coalescing binaries by a given detector (or a network of detectors). Indeed, at a fixed level of S∕N, the detection volume is proportional to D3 hor and therefore it is proportional to 5∕2 ℳ. The detection rate 𝒟 for binaries of a given class (NS + NS, NS + BH or BH + BH) is the product of their coalescence rate ℛV and the detector’s horizon volume ∝ ℳ5 ∕2 for these binaries.

It is seen from Table 6 that the model Galactic rate ℛG of NS + NS coalescences is typically higher than the rate of NS + BH and BH + BH coalescences. However, the BH mass is significantly larger than the NS mass. So a binary involving one or two black holes, placed at the same distance as a NS + NS binary, produces a significantly larger amplitude of gravitational waves. With the given sensitivity of the detector (fixed S ∕N ratio), a BH + BH binary can be seen at a greater distance than a NS + NS binary. Hence, the registration volume for such bright binaries is significantly larger than the registration volume for relatively weak binaries. The detection rate of a given detector depends on the interplay between the coalescence rate and the detector’s response to the sources of one or another kind.

If we assign some characteristic (mean) chirp mass to different types of NS and BH binary systems, the expected ratio of their detection rates by a given detector is

𝒟BH ℛBH (ℳBH )5∕2 -----= ----- ------ , (65 ) 𝒟NS ℛNS ℳNS
where 𝒟BH and 𝒟NS refer to BH + BH and NS + NS pairs, respectively. Taking ℳBH = 8.7 M ⊙ (for 10 M + 10 M ⊙ ⊙) and ℳ = 1.22 M NS ⊙ (for 1.4 M + 1.4M ⊙ ⊙), Eq. (65*) yields
𝒟BH--≈ 140 ℛBH--. (66 ) 𝒟NS ℛNS
As ℛBH- ℛNS is typically 0.1 – 0.01 (see Table 6), this relation suggests that the registration rate of BH mergers can be higher than that of NS mergers. Of course, this estimate is very rough, but it can serve as an indication of what one can expect from detailed calculations. We stress that the effect of an enhanced detection rate of BH binaries is independent of the desired S∕N and other characteristics of the detector; it was discussed, for example, in [788, 429, 252*, 157*].

Unlike the ratio of the detection rates, the expected value of the detection rate of a specific type of compact coalescing binaries by a given detector (network of detectors) requires detailed evolutionary calculations and the knowledge of the actual detector’s noise curve, as discussed. To calculate a realistic detection rate of binary mergers the distribution of galaxies should be taken into account within the volume bounded by the detector’s horizon (see, for example, the earlier attempt to take into account only bright galaxies from Tully’s catalog of nearby galaxies in [425], and the use of the LEDA database of galaxies to estimate the detection rate of supernovae explosions [27]). In this context, a complete study of galaxies within 100 Mpc was done by Kopparapu et al. [376*]. Based on their results, Abadie et al. [2] derived the approximate formula for the number of the equivalent Milky-Way-type galaxies within large volumes, which is applicable for distances ≳ 30 Mpc:

( )3 NG (Dhor) = 4π- Dhor- (0.0116)(2.26)−3. (67 ) 3 Mpc
Here the factor 0.0116 is the local density of the equivalent Milky-Way-type galaxies derived in [376], and the factor 2.26 takes into account the reduction in the detector’s horizon value when averaging over all sky locations and orientations of the binaries. Then the expected detection rate becomes 𝒟 (Dhor) = ℛG × NG (Dhor ).

However, not only the mass and type of a given galaxy, but also the star formation rate and, better, the history of the star formation rate in that galaxy are needed to estimate the expected detection rate 𝒟 (since the coalescence rate of compact binaries in the galaxies strongly evolves with time [432, 479, 158]).

So to assess the merger rate from a large volume based on galactic values, the best one can do at present appears to be using formulas like Eq. (67*) or (5*), given in Section 2.2. However, this adds another factor two of uncertainty to the estimates. Clearly, a more accurate treatment of the transition from galactic rates to larger volumes with an account of the galaxy distribution is very desirable.

To conclude, we will briefly comment on the possible electromagnetic counterparts of compact binary coalescences. It is an important issue, since the localization error boxes of NS + NS coalescences by GW detectors network only are expected to be, in the best case, about several square degrees (see the detailed analysis in [649], as well as [1*] for a discussion of the likely evolution of sensitivity and sky localization of sources for the advanced detectors), which is still large for precise astronomical identification. Any associated electromagnetic signal can greatly help to pinpoint the source. NS binary mergings are the most likely progenitors of short gamma-ray bursts ([500*, 204] and references therein). Indeed, recently, a short-hard GRB 130603B was found to be followed by a rapidly fading IR afterglow [43*], which is most likely due to a ‘macronova’ or ‘kilonova’ produced by decaying radioactive heavy elements expelled during a NS binary merging [416*, 658, 483, 290, 253]. Detection of electromagnetic counterparts to GW signals from coalescing binaries is an essential part of the strategy of the forthcoming advanced LIGO/VIRGO observations [706]. Different aspects of this multi-messenger GW astronomy are further discussed in papers [562, 530, 343, 1, 466], etc.

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