# Are the masses of merging black holes correlated?

LIGO/VIRGO recently released data on observed "gravitational wave transients", mostly BH-BH mergers. The data includes a plot of the primary and secondary mass (the primary mass is defined as the larger one) of each observed event. There seems to be a clear correlation between these two quantities:

I'm wondering whether this comes from a bias in detection (i. e. mergers with similar masses are easier to detect, therefore we observe more) or if this is a correlation in actual compact binary mergers. In the latter case, is there any known explanation?

EDIT: A quick linear regression (disregarding the error bars) gives $$r \approx 0.97$$. This however also ignores that which mass is primary and which is secondary is arbitrary (so probably the diagram should be symmetrized with respect to the diagonal).

EDIT 2: The paper accompanying the data release (arXiv:1811.12907) has plots of the estimated masses (showing the correlation between the errors of primary and secondary masses missing from the plot above) on page 13 and the estimated ratios of the masses on page 14.

• The radiated power is proportional to $m_1^2m_2^2(m_1+m_2)$, so the main observational bias would simply seem to be toward higher masses, not equal masses. And I do think your plot shows a huge bias toward high masses, since in the general stellar population stars with such high masses would be extremely rare. – Ben Crowell Dec 5 '18 at 2:01
• Since you don't seem to be getting any answers here, you might want to move this to astronomy.SE. – Ben Crowell Dec 6 '18 at 23:41
• @BenCrowell if thought about it bout I doubt it would be more successful. AFAIK posting the same question on two sites is discouraged, so I think I'll keep trying my Luck here (I didn't expect to get any quick answers anyway). – 0x539 Dec 6 '18 at 23:45
• There is a mechanism for moving the question rather than cross-posting. – Ben Crowell Dec 7 '18 at 5:31

I believe this could largely be explained as a selection effect.

Some analytical approximations for estimating the signal-to-noise ratios (SNR) of detections are presented by Flanagan & Hughes (1997). They show that the SNR of the detection of an inspiral depends on the strain amplitude $$h$$, where $$h$$ (see equation 1 here) is given by $$h = \propto \mu M^{2/3} \nu^{2/3} r^{-1},$$ where $$\mu$$ is the reduced mass, $$M$$ is the total mass, $$\nu$$ is the source frequency and $$r$$ is the distance to the source.

We can express the reduced mass as $$M f(1-f)$$, where the mass of binary component 1 is $$fM$$ and the mass of the other component is $$(1-f)M$$.

Keeping $$M$$ and the other quantities constant (note that during inspiral, $$\nu$$ increases, so all binaries go through the same frequency band unless they merge), then we see that $$h$$ is maximised if $$f = 0.5$$ and we have an equal-mass binary system.

Flanagan & Hughes do a more careful job show that at a given distance the SNR during the inspiral phase is reduced by a factor of $$(4 \mu/M)^{1/2}$$ for unequal mass binaries and by a factor of just $$4 \mu/M$$ when considering the ringdown phase of a black hole merger. This means that for a given SNR threshold, the volumes of space sampled are decreased by the same factors cubed.

For example if we take a binary with $$M_2 = 0.5M_1$$, then $$\mu= 2M/9$$ and the volume of space sampled decreases by 0.83 (inspiral) or 0.7 (ringdown), so a modest reduction, but if $$M_2 = 0.1M_1$$ then $$\mu = 0.08M$$ and the reduction factors are 0.19 (inspiral) and 0.04 (ringdown).

There is therefore a big detection efficiency bias against finding very unequal mass binary systems, but only a modest reduction for things within a factor of two or so.

Now, having said all that, there could also be astrophysical reasons for correlated masses and generally quite high mass ratios.

The mechanisms for forming massive black hole binaries that will merge on timescales less than the age of the universe are summarised by Podsialowski (2017).

Production of these systems via dynamical interactions in dense clusters may produce binary black holes with strongly correlated masses. The pairing of black holes with similar masses is favoured (e.g. Banerjee 2017).

Another possibility is the massive overcontact binary model, where two massive stars form a contact binary whilst on the main sequence. They swap mass back and forth and evolve towards an equal mass configuration, before detaching in the He burning phase and later becoming a close-to-equal mass binary black hole system (e.g. Marchant et al. 2016).

Finally, it should be noted that because gravitational wave power losses are maximised for $$f=0.5$$, this must also mean that such binaries will merge on a shorter timescale from any given separation and so it's mechanism produces a spectrum of mass ratios, the binaries with $$f=0.5$$ will have a higher merger rate.

• AFAIK the frequency of such a merger depends on the masses of the BHs (and is actually lower for larger BHs). How does this influence the above calculation? – 0x539 Dec 7 '18 at 15:43
• I would have expected the SNR to be proportional to the power of the wave (so in particular $\propto r^{-2}$), as for EM waves. Why is this not the case here? – 0x539 Dec 7 '18 at 15:57
• It is wave amplitude that is detected. @0x539 Literally the amount by which space is stretched and compressed. – Rob Jeffries Dec 8 '18 at 9:00
• @0x539 The sensitive frequency band of a detector is fixed. Dealing with a fixed band is dealt with in the reference I quote. All "chirps" increase in frequency and either pass through the band or merge in it. – Rob Jeffries Dec 8 '18 at 9:02