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My question is a more detailed version of the one found here, which elicited some good information but the question was never really answered.

From table 4 in a 2010 paper we see the estimated rate of BH-BH mergers ranged from $10^{-4}$ to $0.3 \rm \,Mpc^{-3} Myr^{-1}$. This corresponds to $10^{-10}$ to $3\times10^{-7} \rm\,Mpc^{-3} yr^{-1}$. They write:

For BH–BH inspirals, horizon distances of ...2187 Mpc are assumed. These distances correspond to a choice of ... 10M for BH mass.

In the left panel of figure 4 found in this 2016 paper by LIGO there is a plot of BH mass vs horizon distance. For 2015-2016 sensitivity levels and 10M mass, the horizon distance is 300 Mpc. So we need to multiply the above numbers by $300^3=2.7\times10^{7}$. This gives a range of $2.7\times10^{-3}$ to $8.1$ detectable mergers per year, given the prior assumptions that a typical signal would come from black holes with ~10 solar masses. Also, in the discovery paper we find:

We present the analysis of 16 days of coincident observations between the two LIGO detectors from September 12 to October 20, 2015.

Therefore the duration of the experiment was $16/365$ years. This gives an expected rate ranging from $1.18\times10^{-4}$ to $3.55\times10^{-1}$ mergers during the experiment. As far as I can tell, this can be taken as an upper bound, since not all of these events will really be detected (due to non-optimal location, coincident noise, etc).

Are there other factors to take into account?

Community
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Livid
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1 Answers1

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I think there are some problems.

The figure 4 you refer to has total system mass on the lower x-axis. So the horizon distance for the merger of two 10 solar mass BHs appears to be 800 Mpc. So that increases your naive rate by a factor of 20.

In the right hand plot of the same Fig 4, you see that the authors say that the effective volume for a merger of two 10 solar mass black holes is 0.1 Gpc$^3$.

So we can compare this with the naive volume of $(4\pi/3)\times 0.8^3 = 2.14$ Gpc$^3$.

So they are saying they detect about 20 times fewer mergers than your way of calculating it would suggest, ironically cancelling out with the earlier error, and so suggesting that your headline figures are roughly correct.

Two interesting points arise. First, the naive estimate of the volume in which a detection can be made is way too large. The reason for this is explained in the appendix of the paper. The left hand panel curves are for an equal mass binary, overhead with a face-on orbit. All other geometries will result in a weaker signal by factors of a few. As signal amplitude scales as 1/distance this means a shorter horizon distance by factors of a few. The volume then goes as the cube of this. As the appendix explains, the curves in the right hand plot were produced by integrating over random directions and orientations.

Second, the headline figure in your calculation turns out to be right (because of two cancelling errors), and suggests that the detection of a merger of 10 solar mass BHs was unlikely in 16 days of data. However, the mass dependence is large - GW amplitude goes as $M^2$. The detected source had a total mass of 60 solar masses and could have been seen (RH panel of Fig. 4) in an effective volume that was 20 times bigger than for a 20 solar mass system.

ProfRob
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