Section 2.1

How dense does a region need to be to collapse?

Getting the density right

The other section in this chapter's related material describes the basic picture: in the first fraction of a second after the Big Bang, tiny random differences in density meant some patches of the universe were denser than their surroundings, and the densest of these could collapse under their own gravity into black holes. But not just any lump of extra density will do. The overdensity has to be large enough, and concentrated tightly enough, that gravity wins decisively over the enormous outward pressure of the hot, radiation-filled early universe pushing back against collapse.

How large is 'large enough'? This is one of the trickiest numbers in the whole theory, because it requires solving Einstein's equations for a violently collapsing, highly curved region rather than just estimating with rough physics. Recent large-scale computer simulations that solve the full equations of general relativity for a collapsing patch during the radiation-dominated era have pinned down a fairly narrow answer. They find that a patch collapses into a black hole only once its density contrast — how much denser it is than the average density around it — climbs to somewhere between about 0.77 and 0.83 or beyond. Below that range, gravity temporarily pulls the overdense region together, but the pressure of the surrounding radiation eventually wins and the patch simply spreads back out, leaving no black hole behind.

Horizon mass and the timing of collapse: A collapsing patch cannot simply be dense; it also has to be small enough, physically, to fit inside its own cosmological horizon — roughly, the largest distance light could have travelled since the Big Bang up to that moment. As the universe expands, that horizon grows, so patches of larger and larger size, and correspondingly larger mass, become able to collapse as time goes on. Because of this, the mass of the resulting black hole is set almost entirely by how early it formed: a patch that collapses a tiny fraction of a second earlier ends up as a much lighter black hole than one that collapses slightly later, simply because less mass had come within reach of gravity's grip by then. This is why calculations of primordial black hole formation are as much calculations about cosmic timing as they are about density.

Different proposed routes to the same required condition: a region collapsing inside its own horizon
	What has to happen	When it could occur
Enhanced fluctuations from unusual inflation	An inflaton field with an inflection point, bump or plateau amplifies density fluctuations at one particular size scale by a huge factor, since ordinary slow-roll inflation alone never makes fluctuations strong enough to collapse.	During inflation itself, imprinted on whichever scale re-enters the horizon later
The quark-hadron (QCD) transition	As the universe cools through this transition the pressure resisting collapse briefly dips, lowering the density contrast needed and making solar-mass-scale collapse easier without any special tuning of the fluctuations themselves.	A specific, brief moment, roughly a hundred-thousandth of a second after the Big Bang
Bubble collisions in a phase transition	Colliding bubbles of a new physical phase can trap a pocket of the old phase; if that pocket shrinks below its own gravitational radius it collapses into a black hole.	Whenever such a first-order phase transition occurs in the early universe
Collapsing topological defects	A closed loop of a domain wall or similar defect left over from a broken symmetry can concentrate enough energy inside its own gravitational radius to collapse.	Tied to whenever the relevant symmetry-breaking and later defect formation occurred

Which era of the early universe matters most

Almost all of this collapsing is thought to happen once the universe has settled into being dominated by radiation, rather than during inflation itself. Any overdensities that existed while inflation was still stretching space apart would have been diluted to nothing before they had a chance to collapse. Some versions of the theory also allow for a short era, before the radiation-dominated era properly begins, when the universe was instead dominated by slow-moving matter; if that happened, the conditions for collapse would have been somewhat less strict, potentially letting smaller, gentler irregularities collapse that would not have made it under ordinary radiation-era conditions.

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