Could Hyper-Massive Black Holes be due to Dark Matter in the Early Universe?

Question

An interesting discussion started here: Is there a limit as to how fast a black hole can grow?

I am curious if Thompson Scattering and Eddington Luminosity have the same effect on Dark Matter (or alternatively, weakly-interacting massive particles) as they do with purely ionized hydrogen and other (known) particles that radiate (or can radiate due to decay, energy, friction, etc..).

Additionally, given dark matter's W.I.M.P.-like nature, would it not be possible that in the early universe where dark matter is believed to have materialized parallel to normal and anti-matter, and given its gravitational properties, could dark matter have contributed to the universe's earliest (and largest) black holes, and in turn explain black holes being larger than previously thought possible in a 13.7~ Billion Year history?

The properties of Dark Matter would render it virtually unaffected by the velocity and motion of the accretion disk. Since we are not yet sure if dark matter has a charge, it may also be unaffected by the black hole's electromagnetic field. It just pours into the event horizon virtually unhindered.

I do not expect an authoritative answer, as the only person who could provide such an answer would have earned a nobel prize already for dark matter's discovery (and thus be far too important to answer such a question), but a hypothetical answer based on what we do know about dark matter would be sufficient.

Answer 1

This is a good idea...

Dark matter by definition doesn't interact electromagnetically (i.e. it has no charge). Therefore its cross section $\sigma$ for absorbing radiation and being pushed away from an accreting object is $0$, at least to first order. You could look at higher-order effects, like its neutrino-absorption cross section, to calculate some effective absorption that is slightly greater than $0$. The end result is an Eddington luminosity that is extremely large, since it scales as $1/\sigma$.

This means dark matter can freely accrete onto a black hole without being stopped by outward going radiation.

...But dark matter is extremely diffuse

Even in our neighborhood of the galaxy, which one expects to have slightly more dark matter than some random place in the universe, the dark matter density amounts to less than a proton's worth of mass per cubic centimeter, as calculated for example in Bovy & Tremaine. Compare this to a giant molecular cloud, where the normal matter can be upwards of $10^6$ proton masses per cubic centimeter.

Since dark matter interacts essentially not at all except via gravity, dark matter particles fly on ballistic trajectories, unslowed by pressure gradients or friction. A cloud of dark matter not only cannot shed its angular momentum, it cannot even cool down enough to contract to a reasonable density.

Essentially, the only way for a dark matter particle to be captured by a black hole is for the particle to be flying directly at the hole. Now a black hole of mass $M$ has Schwarzschild radius $R = 2GM/c^2$. The cross sectional area is $A = \pi R^2$. Embedded in a gas of particles with density $\rho$ and typical velocity $v$, you would expect collisions (captures) amounting to a mass flux of $$ \dot{M} = \rho A v = \frac{4\pi G^2}{c^4} \rho v M^2. $$ Plugging in some rough estimates, let's consider a black hole like the one in the center of the Milky Way. Right now $M = 8.4\times10^{39}\ \mathrm{g}$, and let's use the local value of $\rho = 5.3\times10^{-25}\ \mathrm{g/cm^3}$. Suppose we simply want to accrete the present mass $M$ over $10^9$ years, giving our black hole the benefit of rounding and assuming it was always as good as it is now at accreting (it isn't). Then we are looking for an average accretion rate $\dot{M} = 2.6\times10^{23}\ \mathrm{g/s}$. This would require a velocity $$ v = 10^{23}\ \mathrm{cm/s}, $$ which is clearly much greater than the speed of light. Cold dark matter is nonrelativistic for much of the universe's history, and it certainly isn't superluminal. That is, our own supermassive black hole cannot even grow to its current size by running through diffuse dark matter.