6.1 Enhancement of the detection rate for binary BH mergers

Coalescing binaries emit gravitational wave signals with a well known time-dependence (waveform) (see Section 3.1 above). This allows one to use the technique of matched filtering [399Jump To The Next Citation Point]. The signal-to-noise ratio S ∕N for a particular detector, which is characterized by 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 ℳ = (M1 + M2 ) (M1M2 ) and its distance r. Here, we will use the simplified version for S∕N ([399]; see also [106]):
S −1∕2 −2∕3G5∕6ℳ5 ∕6 f− 1∕6 ---= 3 π -3∕2-------------. (59 ) N c r hrms(f)
At a fixed level of S ∕N, the detection volume is proportional to r3 and therefore it is proportional to ℳ5 ∕2. 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 with the detector’s registration volume 5∕2 ∝ ℳ for these binaries.

It is seen from Table 4 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 can be 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 double NS and BH systems, the expected ratio of their detection rates by a given detector is

𝒟 ℛ (ℳ )5∕2 --BH-= -BH-- ---BH- , (60 ) 𝒟NS ℛNS ℳNS
where 𝒟 BH and 𝒟 NS refer to BH + BH and NS + NS pairs, respectively. Here, we discuss the ratio of the detection rates, rather than their absolute values. The derivation of absolute values requires detailed evolutionary calculations, as we discussed above. Taking ℳBH = 8.7 M ⊙ (for 10M ⊙ + 10 M ⊙) and ℳNS = 1.22 M ⊙ (for 1.4 M ⊙ + 1.4M ⊙), Equation (60View Equation) yields
𝒟 ℛ --BH-≈ 140 --BH-. (61 ) 𝒟NS ℛNS
As ℛBH- ℛNS is typically 0.1 – 0.01 (see Table 4), this relation suggests that the registration rate of BH mergers can be higher than that of NS mergers. This estimate is, of course, 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 [420228124Jump To The Next Citation Point].
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