# Probability of a primordial black hole absorbing background radiation?

P.S. It occurred to that the early plasma phase of the universe is described as opaque, because photons that were constantly being emitted could not travel very far without being reabsorbed. A PBH with a radius of about a 0.25 millimeters is small, but relatively huge compared to the nuclei that form the form the bulk of the plasma and these nuclei have no problem absorbing the radiation.

#### Due to how rapidly the radiation density drops, a primordial black hole very quickly accretes essentially all of the radiation that it ever will.

Here is an estimate. Due to gravitational focusing, photons that would otherwise miss the black hole's center by a distance $$b<\sqrt{27/4}\, r_\mathrm{s}$$, where $$r_\mathrm{s}=2GM/c^2$$ is the Schwarzschild radius, are deflected enough to fall into it (see e.g. this question). So the cross section for accretion is $$\sigma = 27\pi (GM)^2/c^4.$$ For an energy density $$\rho$$ of photons, this implies an energy accretion rate $$\sigma\rho c$$, assuming isotropic accretion (so no net momentum transfer). That is, the black hole would grow at the rate $$\frac{\mathrm{d}M}{\mathrm{d}t}=\sigma \rho/c = 27\pi (GM)^2 \rho/c^5.$$

The right-hand side depends on time $$t$$ through both $$M$$ and $$\rho$$. The latter is trivial: $$\rho/c^2=3H^2/(8\pi G)$$ by the Friedmann equation, and $$H\simeq 1/(2t)$$ during radiation domination (this is exactly true if there are no phase changes in the radiation and no epoch prior to radiation domination). Here $$H$$ is the Hubble rate. Thus $$\frac{\mathrm{d}M}{\mathrm{d}t}= \frac{81}{32} \frac{GM^2}{c^3 t^2}.$$ The solution is $$M = \frac{M_0}{1-\frac{81G}{32c^3} M_0 (t_0^{-1}-t^{-1})}$$ for initial mass $$M_0$$ at time $$t_0$$.

But a primordial black hole's initial mass $$M_0$$ is some fraction $$f$$ of the horizon mass at formation, i.e., $$M_0= f \frac{4\pi}{3}(\rho/c^2)(c/H)^{3}$$, where the right-hand side is evaluated at formation. From the Friedmann equation and the expression for $$H$$, above, $$M_0 = f c^3 t_0/G$$, implying $$t_0\simeq G M_0/(c^3 f)$$. The mass thus grows as $$M = \frac{M_0}{1-\frac{81}{32}f\,(1-t_0/t)}.$$ This means that if $$f>32/81$$, then the black hole's mass grows without bound in finite time, although this is not exactly physical because accretion would be limited by the availability of radiation. On the other hand, if $$f<32/81$$, then $$M$$ asymptotes toward a finite multiple of $$M_0$$, $$M \to \frac{M_0}{1-\frac{81}{32}f},$$ over a time scale of order $$t_0$$.

Note that $$f$$ here can't directly be compared to a primordial black hole's fraction of horizon mass that one might find in the literature, because black hole formation depletes the radiation in its immediate surroundings (since it forms out of that radiation), so my assumption of uniform $$\rho$$ is not valid. Instead we should imagine that $$M_0$$ is the black hole's mass at time $$t_0$$ sufficiently long after formation that the radiation can be approximated as uniform. It is easy to believe that the black hole is a tiny fraction of the horizon mass ($$f\ll 1$$) by this time, and so over infinite time, black holes would grow by some fixed factor of order 1.

Should I think of the background radiation as literally a hail of 'bullet like particles' that have to score a direct hit on the PBH to be intercepted and absorbed, or should I think of the background radiation as wavelike and more likely to be absorbed due to being spread out?

The former. If photons are spread out due to their wavelike nature, then they are simply bullets whose trajectories are distributed probabilistically. Since we already assume a uniform and isotropic photon distribution, convolving it with any further distribution has no effect.

Additionally, does the gravitational field around the PBH effectively increase its intercept cross section due to otherwise near miss paths being curved inwards towards the PBH. Has anyone done some ball park calculations or estimates on this?

As noted at the beginning of the calculation above, gravitational focusing increases the black hole's cross section by a factor of $$27/4 = 6.75$$ compared to the cross section of the event horizon.

A PBH with a radius of about a 0.25 millimeters is small, but relatively huge compared to the nuclei that form the form the bulk of the plasma and these nuclei have no problem absorbing the radiation.

PBHs are astronomically less abundant than charged particles, though, by a factor of order $$m/M$$ after $$e^+e^-$$ annihilation, where $$m$$ is the proton mass and $$M$$ is the black hole mass, and by an even smaller factor beforehand. That's assuming they are all of the dark matter, which is the weakest upper limit on their abundance.

• Wow, what a great detailed and informed answer. Thanks!
– KDP
Commented May 11 at 11:04

No need for much maths. The Schwarzschild radius of your $$1.7\times 10^{11}$$ kg black hole is in fact $${\bf 2.5\times 10^{-16}}$$ m and therefore has a cross-section to photons that is similar to that of an electron.

But of course, if primordial black holes of that mass were dark matter, there would many, many orders of magnitude fewer of them than electrons (and protons).

i.e. The effect you are thinking of is negligible.