When a dense patch of the early universe becomes a black hole
Sandbox Physics ·
A patch collapses when it re-enters the horizon with a density contrast above a threshold of about 0.4 to 0.6, and the black hole takes a fixed fraction of the horizon mass at that moment. Because the horizon mass grows in proportion to time, the formation time alone fixes the mass.
The horizon mass grows in proportion to time
Work in the radiation era, when the expansion rate is H=1/(2t). The distance over which regions can influence one another is the Hubble radius c/H. The Friedmann equation gives the mean density ρ=3H2/(8πG), where ρ is energy density divided by c2. The mass inside a sphere of radius c/H is the horizon mass.
The result grows linearly with time. Its coefficient c3/G is about 2.03×105 solar masses per second, which the Particle Data Group rounds to 2×105. A second line of the same calculation shows that the Schwarzschild radius of the horizon mass equals the Hubble radius. A patch that collapses as soon as it enters the horizon therefore makes a black hole comparable in size to the horizon itself.
MH=34πρ(Hc)3=2GHc3=Gc3t
Substitute ρ=3H2/(8πG) and then H=1/(2t).
rs(MH)=c22GMH=Hc
The horizon mass sits exactly at its own Schwarzschild radius.
Describe a patch by its density contrast δ=(ρ−ρˉ)/ρˉ, smoothed over the horizon scale at the moment it re-enters. While the patch is larger than the horizon it simply expands with everything else. After re-entry, pressure resists collapse. Only patches with δ above a threshold δc collapse before pressure wins.
The value of δc depends on the shape of the density profile and on exactly how δ is defined. Harada, Yoo and Kohri give an analytic estimate of about 0.41 for radiation in the measure used by numerical simulations. The 2026 review uses 0.55 for one of its examples. The Lab lets you move δc between 0.35 and 0.65 and starts at 0.45.
The black hole keeps a fraction of the horizon mass
The black hole mass is written M=γMH. The Particle Data Group quotes γ≃(1/3)3≈0.19 during radiation domination, and Sasaki and collaborators use γ=0.2. This series uses 0.2. In detailed collapse simulations the mass also depends on how far above threshold the patch was, which spreads the masses formed at one time. The Lab keeps one mass per formation time, so its spread of masses comes only from the spread of scales in the early density peak.
M=γMH≈γGc3t≈4×104M⊙(1st)(γ=0.2)
The formation time fixes the mass. Heavier black holes formed later.
Worked examples: an asteroid-mass and a stellar-mass black hole
Take the Lab’s default peak mass, M=3.16×10−11M⊙, about 6×1019 kilograms. The horizon mass is M/γ=1.58×10−10M⊙. Dividing by 2.03×105M⊙s−1 gives a formation time of 7.8×10−16 seconds. The Schwarzschild radius of the black hole is 2.95km×3.16×10−11≈93 nanometres, one fifth of the Hubble radius 2ct≈470 nanometres at that time.
Now take M=30M⊙, a typical mass in the gravitational-wave catalogues. The horizon mass is 150M⊙ and the formation time 7.4×10−4 seconds. The two examples differ by about 1012 in mass and in formation time; a black hole mass function spanning many decades corresponds to a long stretch of the first second.
Black holes lose mass by Hawking radiation, faster the lighter they are. According to the 2026 review, a non-spinning black hole formed with about 5×1014 grams, or 2.5×10−19M⊙, would be completing its evaporation today. The Lab’s mass ruler starts at 10−18M⊙, above this edge, and ignores the small mass loss of survivors.
Predict: if you move the black hole mass from 10−11 to 10−5 solar masses, how much later did the patch form?
Drag the mass grip along the ruler and read the most common black hole mass. Check your prediction with t=M/(γc3/G).
Drag the threshold plane up a little and watch the fraction of horizon patches that collapse fall.
Exercises with explanations
At what time after the beginning did a black hole of one solar mass form, with γ=0.2?
Read the explanation
The horizon mass is M/γ=5M⊙. Dividing by 2.03×105M⊙s−1 gives t≈2.5×10−5 seconds.
Show that rs(MH)=c/H using only MH=c3/(2GH).
Read the explanation
rs=2GMH/c2=2Gc3/(2GHc2)=c/H. The result does not depend on the radiation era; it holds for any flat Friedmann universe, where MH=c3/(2GH) follows from the critical density.
If the true efficiency were γ=0.1, when would a 3.16×10−11M⊙ black hole have formed, and what would change in the Lab’s mapping from spectrum peak to mass?
Read the explanation
The horizon mass doubles to 3.16×10−10M⊙, so the formation time doubles to 1.6×10−15 seconds. For the same early density peak every black hole would be half as heavy, and the present abundance per collapsed patch would change too, because it scales as γMH−1/2; that scaling is derived in the next chapter.
Review; arXiv lists the journal reference Riv. Nuovo Cim. (2026). Read in full: Sec. II.A.2 on non-Gaussianity, Sec. IV.C (gravitational waves), IV.E (accretion), V (asteroid-mass window), VI (Eqs. 25–29) and the captions of Figs. 2–6; parts of Sec. II and III.