# Is there an upper limit on the mass of black hole mergers we can detect?

From the LIGO website, black hole mergers have been observed between black holes with a mass up to roughly 50 $$M_\odot$$.

Are there no black holes with a mass above 100 $$M_\odot$$ or is this an observational bias? Why haven't we observed any mergers between black holes with a mass in the 100 - 1000 $$M_\odot$$ range?

It is quite likely there is an astrophysical upper limit to the mass of a black hole that can be produced during the core collapse of a massive star, caused by the pair instability supernova phenomenon. There isn't an observational bias against detecting more massive black holes in the range of 100 to a few hundred $$M_{\odot}$$.

Details:

The frequency of the gravitational waves is twice the orbital frequency of the binary system. The general scenario is that early in their evolution, a mrging binary system will be orbiting at relatively long periods (seconds !), but as gravitational waves take energy out of the orbit, the system becomes more compact, the orbital period gets smaller and the frequency of the emitted gravitational waves goes up. This continues until the black holes get so close together that their event horizons merge.

Very roughly, we can derive (from Kepler's third law, not going into detail), using Keplerian orbits $$f_{\rm max} \sim \left( \frac{GM}{\pi^2 a_{\rm merge}^3} \right)^{1/2}\ ,$$ where $$f_{\rm max}$$ is the peak frequency at merger (when the gravitational wave signal is also maximised), $$a_{\rm merge}$$ is the separation of the masses at merger and $$M$$ is the total mass of both black holes.

If we let $$a_{\rm merge} \sim 2GM/c^2$$, the sum of the two Schwarzschild radii of the black holes, then $$f_{\rm max} \sim \frac{c^3}{GM} \left( \frac{1}{8\pi^2}\right)^{1/2} \sim 2\times 10^4 \left(\frac{M}{M_{\odot}}\right)^{-1}\ {\rm Hz}$$

Now, LIGO is limited to observing frequencies above about 20 Hz. The sensitivity drops off rapidly below that because of seismic noise and other factors. If the mass of the merging black holes exceeds some critical value then the frequencies of the gravitational waves they produce never get into the sensitivity wndow of LIGO. Using the expression above, we can estimate that this happens only if the total mass exceeds $$1000 M_{\odot}$$. Observing the mergers of more massive black holes would require a detector that is sensitive to lower frequencies, probably beyond the surface of the Earth (e.g. LISA).

This calculation is only good to a factor of 2 or so, but we can check it. GW150914 had a total mass of around $$65 M_{\odot}$$ and merged at frequencies of about 120 Hz. Since $$f_{\rm max}$$ scales as $$M^{-1}$$ this suggests 360 solar mass mergers should be just about detectable, but clearly demonstrates that LIGO could detect black holes of 100-200 solar masses. What's more, at a given distance and frequency, the signals from such mergers would be more powerful than for less massive black holes -- something like $$h \propto M^{5/3}$$, which means the volume in which the mergers would be visible goes as $$M^5$$. Thus more massive black hole binaries would have to be extremely rare in order to have evaded detection.

The astrophyscal reason for an upper limit is the pehenomenon of pair instability supernovae (e.g. Farmer et al. 2019), which blows the star apart rather than leaving a black hole (or any other kind) of remnant. This likely happens for stars with initial masses of $$130+\ M_{\odot}$$, and means that leaving behind black holes with $$M > 50M_{\odot}$$ is very difficult, with even lower mass limits for stars with a metallicity more similar to the Sun, since they lose more mass in stellar winds during their lives.

For initial masses of $$250+ M_{\odot}$$ it is possible that the pair instability supernova mechanism ceases and direct collapse to a black hole becomes possible. In which case their might be a population of $$300+ M_{\odot}$$ mergers that are just below LIGO's sensitivity window. New Earth-based gravitational wave detectors like the Einstein Telescope and Cosmic Explorer aim to push their low frequency response down to a few Hz and might be capable of detecting mergers in the 300-1000$$M_{\odot}$$ range.

This means you could not get a merger pair between about $$100 M_{\odot}$$ and $$300 M_{\odot}$$ (unless they themselves were the products of a merger).

• I have no idea whether there is any theoretical work or observational (from mergers so far) data on it, but: Shouldn't the merged black hole shortly after the merge still be very asymmetric, still rotating like a handle? And hence: Shouldn't it still be emitting gravitational waves? Mar 24 '20 at 19:37
• @Peter-ReinstateMonica that would be a good question to post in its own thread. Mar 25 '20 at 5:23
• @Ryan_L A simple google search shows that it is indeed the case. There may even be two mechanisms. A simulation predicts my idea, "a cusp in the apparent horizon of the remnant black hole that repeatedly points to each of the observers as it fades away, resembling a sort of fading gravitational-wave lighthouse" (in the PDF of arxiv.org/abs/1906.01153). Observations show a stronger signal from the "ringdown", an oscillation of the freshly merged black hole, complete with overtones (journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.111102). Mar 25 '20 at 6:54
• @PM2Ring That is not true. It is perfectly possible for a BH binary to form dynamicaly. This typically requires a dense stellar environment (like a globular or nuclear star cluster), but is a perfectly plausible formation channel that could account for a portion of all observed mergers. Mar 25 '20 at 16:29
• @PM2Ring The black hole wouldn't even have to start in the global/nuclear cluster. It could have been captured by it after it was formed. I think I'm slightly confused about what point you were trying to make. Mar 25 '20 at 20:11

Adding to Rob Jeffries good answer - Observing stellar population and mass distribution shows a similar pattern ..

• many tiny / light objects
• medium number of medium sized / weighed objects
• rather few large / massive or even super massive objects

Many of those massive / super massive black holes are active galactic cores - that rarily will have said mergers. Many Astronomers assume those had their mergers in the early phases of their galaxies - since stars that produced them lived a rather short time.

This leaves way bigger chances for mergers of rather lighter black holes or neutron stars than for the massive or super massive ones.

• Also, we have only been observing these mergers for a short time. Larger-mass mergers will be rarer. Mar 24 '20 at 15:20
• This is partially countered by heavier mergers leading to stronger gravitational wave signals that can be observed at greater distances. Anyway, the current observations are increasingly incompatible with the black hole mass function being a standard power law. If this were the case we should have seen heavier mergers already. Mar 24 '20 at 15:39
• Just explained to ya that there are few heavier mergers if ANY at all Mar 24 '20 at 15:46
• This isn't correct. The reason that the statistics are dominated by black hole mergers with masses far larger than the ~10 solar mass stellar black holes inferred to be present in a number of X-ray binary systems, is that more massive black hole mergers are visible in a far greater volume. Signal strength at a given frequency/distance goes as $M^{5/3}$, which means if you double the mass you can see them to 3.2 times the distance and 32 times the volume. Mar 24 '20 at 16:02
• Thus black holes of double the mass would have to be more than 32 times rarer to avoid being seen. Stellar IMF goes as $M^{-2.3}$, so in fact they would only be about 5 times rarer. Supermassive black hole mergers are not seen because we cannot possibly see them at present. Mar 24 '20 at 16:03