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1 Introduction

The estimate of the compact object binary coalescence rates has been investigated by many authors. The subject is mentioned in several fields of astrophysics: from the binary pulsar studies, through gamma ray-bursts, and finally in the gravitational wave astrophysics.

Coalescences of compact object binaries are the most promising sources of gravitational waves to be detected by the interferometric detectors. Currently LIGO and VIRGO are being reassembled to achieve the sensitivity roughly ten times better then in the previous configurations. KAGRA is in the construction phase, and LIGO India is in the early stage. Thus the coming years are bound to bring a number of observatories, which raises expectations for detection of gravitational wave sources.

It is therefore important to investigate and review our knowledge of properties of compact object binaries and the constraints that can be imposed on their coalescence rates. This paper does not aim at a comprehensive review of the subject. We must note that a comprehensive review of the rates has been recently compiled (Abadie et al. 2010).

In the literature there are several conventions for the units of the coalescence rates. Some papers quote the number per unit time in the Milky Way. This has been to found to be too Earth centered, and basically irrelevant for the gravitational wave searches since we do not expect to see a coalescence of compact binary in the Milky Way, neither in our lifetime nor probably even in the lifetime of our civilisation. Thus people started using the units of the coalescence rate per MWEG, which stands for Milky Way Equivalent Galaxy. However, the MWEG is an imaginary galaxy, a copy of the Milky Way distributed uniformly in the Universe with the density of 0. 02 Mpc−3. Thus it is a unit of volume albeit quite strange. In this paper I will there fore use the units that are quite common and easy to understand i.e. number of coalescences per year per megaparsec cube.

The outline of the paper is as follows. In Sect. 2 we describe and review the observation based estimates. Section 3 contains the outline of population synthesis results. Finally in Sect. 4 we present the summary and discussion of uncertainties.

2 Observation Based Estimates

2.1 Double Neutron Stars

Table 1 Pulsars in binary neutron star systems according to ATNF database (Manchester et al. 2005)
Full size table

There are currently ten double neutron star binaries known, out of which four have coalescence times smaller than the Hubble time. The list of known binary neutron stars is shown in Table 1. The calculation of the coalescence rate of double neutron star systems can be described quite easily. Let us take into focus the only known real double pulsar: J0737-3039, where both neutron stars have been detected as radio pulsars. Given its radio luminosity the pulsar J0737-3039 could be detected in about 10 % of the volume of the Milky Way. Moreover given the width of the beam we can calculate that only about 3 % of the sky is covered by the radio emission. Thus there are about 300 not detected binaries like J0737-3039 for each one that we can see. The lifetime and the age of the pulsar have been determined by Piran and Shaviv (2005) and Willems et al. (2006). The pulsar will merge in about 80 Myrs, and its current age is approximately 90 Myrs, giving a total lifetime as a binary pulsar of 170 Myrs. Thus, we expect a coalescence of a binary pulsar like J0737-3039 every:

$$\displaystyle{ \delta t \approx \frac{170\,\mathrm{Myrs}} {300} \approx 5.6 \times 10^{5}\,\mathrm{yrs} }$$
(1)

This means that the Galactic merger rate is 1. 8 × 10−6 yr−1. In order to present this number as the merger rate density we need to use the average galaxy density in the Universe ρ G  = 0. 0 Mpc−3:

$$\displaystyle{ \mathcal{R} =\rho _{G}\delta t^{-1} \approx 4 \times 10^{-8}\,\mathrm{Mpc}^{-3}\,\mathrm{yr}^{-1} }$$
(2)

A more detailed calculation includes detailed models of the pulsar luminosity function, neutron star binaries distribution in the Galaxy, detectability as a function of position in the sky by various surveys, see e.g. Kim et al. (2010).

Estimating the properties of the binaries containing BHs is not straightforward, as as neither BHNS nor BHBH binary has been detected directly. It must be mentioned however that there is a potential BHBH candidate detected by the OGLE microlensing experiment (Dong et al. 2007). The microlensing event OGLE-2005-SMC-001, can be interpreted as a binary BH with the likely masses of 3 and 7 M\(_{\odot }\), with an orbital separation of 4.7 AU. Such a system has merger time much longer than the Hubble time and there fore does not contribute to the rate of observable gravitational wave systems. Nevertheless it is interesting to see that there is evidence for existence of binary BHs.

2.2 Binary Black Holes

The observational estimates of the binary BH coalescence rates have been based on the analysis of X-ray binaries and their future evolution. The first object to found to be candidate progenitor of the coalescing BH binary is IC10 X-1 (Bulik and Belczynski 2010). The X-ray binary IC10 X-1 is a member of a small class of binaries which host a Wolf-Rayet donor transferring mass onto a BH. The orbit is quite tight—the orbital period is ≈ 30 h. The mass of the BH is 23–33 M\(_{\odot }\), and the mass of the WR star is 17–35 M\(_{\odot }\). The lifetime of the WR star is only about 1 − 2 × 105 years. After this time the mass of the WR star will decrease to primarily due the wind mass loss, and will explode as a supernova creating a BH with very little mass loss. This will lead to formation of the binary BH on a tight orbit. The potential natal kick of the BH is unlikely to disrupt the system as it is already very tight and the orbital velocity is ≈ 600 km s−1. The binary BH that will form will have the time to merger of about 2–3 Gyrs.

The estimate of the merger rate of binary BH based on the IC10 X-1 is pretty straightforward. Since the merger time is shorter than the Hubble time we can assume that the merger rate is equal to the birth rate of such systems. The birth rate can can be estimated by considering the observability of IC10 X-1. There are two key key points that determine this: the volume in which one obtain the spectroscopy required to find the masses, and the time that IC10 X-1 is detectable as an X-ray source. The distance out to which one can obtain spectroscopy of a WR star is about R = 2 Mpc, while the source is X-ray active for about the lifetime of the WR star, i.e. up to t D  = 200 kyr. Thus the formation, and merger rate is:

$$\displaystyle{ \mathcal{R}_{\mathit{BBH}} = \left (\frac{4\pi R^{3}} {3} \right )^{-1}t_{ D}^{-1} \approx 1.6 \times 10^{-7}\,\mathrm{Mpc}^{-3}\,\mathrm{yr}^{-1} }$$
(3)

This is a surprisingly high rate, comparing to the one obtained for the binary neutron stars above.

A more detailed calculation has been done with the inclusion of another WR binary—NGC300 X-1 (Bulik et al. 2011). This paper contains a much more detailed derivation of the probability distribution of the expected detection rate by the detectors like VIRGO and LIGO. In Fig. 1 we present the analysis of the probability distribution of the merger rate density of binary BHs. Recently another object, a Wolf Rayet binary in NGC 253 was added to the list of binaries that may lead to formation of the binary BH (Maccarone et al. 2014). However this object needs still a more detailed spectroscopic study before its contribution to the merger rate density can be estimated.

Fig. 1
figure 1

The probability distribution of the merger rate of binary BHs. We present the contributions of the IC10 X-1 and NGC300 X-1 separately as well as the combined value. The horizontal bars correspond to the regions that contain, 68, 90, and 98 % of probability

Full size image

2.3 Black Hole Neutron Star Binaries

A number of X-ray binaries have been analysed in search for progenitors of black hole neutron star binaries. It has been found that Cyg X-3 is a potential progenitor of such a merging binary (Belczynski et al. 2013). Cyg X-3 hosts a 2–4.5 M\(_{\odot }\) black hole and 7.5–14.2 M\(_{\odot }\) Wolf-Rayet star on 4.8hr orbit. The most likely fate of this system is it will be disrupted by a supernova. However, if the Wolf-Rayet star mass is large it may end up as a merging binary BH. In the case the Wolf-Rayet star mass is in the lower end of the possible range the system may end as a BHNS binary. The probability of such an outcome is quite low, and hence the expected merger rate density is of the order of

$$\displaystyle{ \mathcal{R}_{\mathit{BHNS}} \approx 10^{-8}\,\mathrm{Mpc}^{-3}\,\mathrm{yr}^{-1} }$$
(4)

This estimate is highly uncertain. The derivation of this number carried a number of assumptions on the binary evolution, especially the wind mass loss, and the natal kicks imparted on the newly formed compact object. It is based on a single object but the problem of small number statistics is a plague of the observational estimates of the compact object coalescence rates. It must be stressed that this is the lower limit on the BHNS merger rate density as the estimate has not taken into account the selection effects that lead to non detection of a some fraction of Cyg X-3 like binaries in the Milky Way

3 Population Synthesis Based Results

3.1 How to Make a Compact Object Binary?

Formation of compact object binaries has been known for a long time see, e.g. Lipunov et al. (1995). A short description of an evolutionary scenario leading to formation of a merging compact object binary is described below. The main problem is that we need form a merging binary, with the orbital separation much smaller that the size of the initial stars. This requires that there are processes that tighten the orbit.

The general scenario goes along the following steps:

  1. 1.

    We start we two massive stars at an orbit separation a few hundred solar radii. The masses must be in excess of 20 M\(_{\odot }\) so that each of the components can form a compact object.

  2. 2.

    The initially more massive star evolves first, fills its Roche lobe and initiates mass transfer. The system losses some mass the orbit is not changed a lot. The initially more massive star looses its envelope and becomes a Wolf Rayet star, the mass ratio is inverted.

  3. 3.

    The Wolf Rayet star undergoes a supernova explosion, and forms the first compact object. the system may be disrupted by the natal kick imparted on the newly formed compact object.

  4. 4.

    The initially less massive star starts to evolve, becomes a giant, and fills its Roche lobe. The mass transfer is unstable and the common envelope phase starts. The orbit shrinks and the giant losses its envelope. The system may end up in a merger at this time forming a Thorne-Zytkow object.

  5. 5.

    If the system survives the common envelope it consists of a Wolf-Rayet star and a compact object on a tight orbit. It can be seen as an X-ray binary accreting either from the wind or through the Roche lobe overflow.

  6. 6.

    The Wolf-Rayet star ends up its life as supernova and forms a second compact object. The system can again be disrupted by the natal kick.

  7. 7.

    If the system survives formation of the second compact object it is now a tight binary consisting of two compact objects.

  8. 8.

    The system tightens emitting gravitational waves and merges.

This scenario has been published in a many papers (Lipunov et al. 1997; Belczynski et al. 2002) and was a basis to the belief that binary black hole formation is quite well understood. It was also believed that the binary BH binaries are likely to be the most common sources of gravitational waves.

This belief ran into trouble about 8 years ago when it was found that making compact object binaries containing black holes showed some problems. In particular,surviving stage 4. above was difficult as most systems simulated would not survive the common envelope and merge forming a Thorne-Zytkow object. This lead us to a pessimistic conclusion that the number of binaries containing black holes will be very small, published under a title “On the Rarity of Double Black Hole Binaries: Consequences for Gravitational Wave Detection” (Belczynski et al. 2007).

This pessimistic view has changed with the discovery and interpretation of the two Wolf-Rayet binaries described in Sect. 2. These are the system that have survived the difficult stage 4. of the evolutionary scenario above and nothing can prevent them from becoming merging compact object binaries containing black holes. What made them special is that they originate in the low metallicity dwarf galaxies: the metallicity of IC10 is \(0.3\,\mathrm{Z}_{\odot }\), and that of NGC300 is \(0.6\,\mathrm{Z}_{\odot }\). This lead to realization that a crucial factor in the formation of compact object binaries containing black holes is played by metallicity.

Metallicity affects the stellar evolution quite significantly. It influences the stellar winds and mass loss. It also changes the sizes of stars because of strong dependence of the opacity on metallicity. These two factors play a major role in allowing formation of the compact object binaries. The stellar mass loss determines the stellar masses at the moment they collapse and form a compact object. Thus they affect the mass spectrum of the compact objects. In fact the maximum mass of a stellar black hole in a low metallicity environment (\(0.1\,\mathrm{Z}_{\odot }\)) rises to about 30 M\(_{\odot }\), and for the very low metallicity like the one in globular clusters it can reach even 100 M\(_{\odot }\) (Belczynski et al. 2010a). The second factor is connected with stellar structure. The low metallicity giants have smaller radii and therefore the survival of the common envelope phase may be easier. These factors cause a very strong dependence of the compact object formation rate on metallicity, first shown in Belczynski et al. (2010b).

3.2 Rate Density Estimates

The population synthesis of compact objects has been performed by several groups and several population synthesis codes are now in use in the astrophysical community (Portegies Zwart and Yungelson 1998; Bethe and Brown 1998; Hurley et al. 2002; Belczynski et al. 2002; Tutukov and Yungelson 2002; Voss and Tauris 2003).

The estimate of the rate density must take into account three inputs: the population synthesis results, the star formation history and the cosmology model. As a result one obtains the merger rate density as a function of the redshift and masses of the binaries. Let us assume that the population synthesis provides an estimate of the distribution of masses and merger times of compact object binaries \(\frac{d^{2}N} {dMdt}\), where t is the delay time and M is the mass (or masses) of the compact object binary. The star formation rate dependence on the redshift is SFR(z). In general the star formation can also depend on the metallicity. The differential coalescence rate is then

$$\displaystyle{ \frac{d^{2}R} {\mathit{dzdM}} =\int \mathit{dt}' \frac{d^{2}N} {\mathit{dM}\mathit{dt}'}\mathit{SFR}(z')\delta (z' - z(t(z) + t')) }$$
(5)

where z(t) is the dependence of redshift on cosmic time, and t(z) is the inverse relation. The cosmology enters through the z(t) relation.

The recent modelling of the compact binary formation has been done in a series of papers (Dominik et al. 2012,  2013). This work includes the metallicity dependence of the compact binary formation, as well as the metallicity evolution in the Universe. For each metallicity there is a different star formation rate history. The calculation tales into account various time delays between the formation of each binary and its coalescence. The rate density range found in this study is summarized in Table 2.

Table 2 The summary of local merger rate density of compact object binaries from Dominik et al. (2012)
Full size table

The uncertainty of the rate densities presented in Table 2 is quite high, i.e up to a factor of ten up or down.

3.3 Population III Binaries

The extreme case of the low metallicity evolution corresponds to the very first stars in the Universe—Population III stars. The properties of Population III stars are highly uncertain: as yet we have not detected even a single such star. We need to relay on simulations to find their properties. Thus the initial mass function and the binary fraction of Population III stars are only known very approximately. It is however believed that Population III stars can lead to formation of massive black holes—with masses reaching 500–1,000 M\(_{\odot }\), provided that the initial mass function reaches that high.

An early analysis of the detectability of Population III binaries (Belczynski et al. 2004) has shown that they can dominate the observed rate in the detectors. This work has used quite optimistic assumptions about the number of binaries, and their initial mass function. Since this paper was published a number of detailed studies investigated the process of collapse of metal free clouds. The possibility of formation of binaries has been confirmed by several studies (Stacy et al. 2010; Machida et al. 2008; Saigo et al. 2004). Recently a very detailed study has shown that the Population III binaries will dominate the rate above the total mass of 30 M\(_{\odot }\) (Kinugawa et al. 2014).

4 Summary

I have presented several approaches to find the merger rate density of the compact object binaries. The list and the discussion are meant to provide an outline of the problem and it is by no means to be considered as an exhaustive description of the subject.

A relatively clear picture emerges based on these various leads. First, there exists a population of potential coalescing compact object binaries that will become sources of gravitational waves. The evidence comes from double pulsars as well as Wolf Rayet binaries that will evolve to form binary black holes soon. Thus the existence of such sources can not be denied. The estimate of their coalescence rate is much tougher. Nevertheless one can obtain several estimates of the compact object binary merger rate density. At the top end of the estimates we obtain the values reaching ≈ 10−6 Mpc−3 yr−1, and at the low end they are two orders of magnitude lower ≈ 10−8 Mpc−3 yr−1. It is expected that the DNS and BBH merger rates densities are close to the upper value, while the BHNS merger rates are rather low and close to the lower bound. There are some models that quench formation of compact object binaries, however at this point they seem quite unlikely. This rates can be used to calculate the detector range require to obtain a single detection in 1 year. For the optimistic rates the required range is 62 Mpc, and for the very pessimistic ones it is 287 Mpc. Note that I have not specified the types of the binaries and their masses. The expected angle and orientation averaged range for detection of a binary neutron star with the Advanced detectors will reach ≈ 200 Mpc. Thus, even for the pessimistic rates we expect the advanced detectors to see merger of a coalescing compact object binary.

In the discussion above I have not touched on the subject of the dynamically formed coalescing binaries. This has been investigated in a number of papers (Grindlay et al. 2006; Hopman et al. 2006; Sadowski et al. 2008; Lee et al. 2010; Downing et al. 2010,  2011). The evolution in clusters may lead to formation of a significant number of merging compact object binaries, increasing it average merger density. However, the studies of these rates is more complicated than in the case of solitary evolution. Consideration of the dynamically formed binaries will only increase the expected merger rate density. A different question is whether the dynamically formed binaries can be distinguished from those in the filed by gravitational wave observations only.

In summary the merger rate density estimates are quite solid and based on multiple astrophysical arguments. The uncertainty in the rate is quite large, however it seems that the direct detection of gravitational waves from a compact object merger is possible in the coming years.