Free-streaming neutrinos raise the threshold for primordial black hole formation

Primordial black holes heavier than about $10^5\,\Msun$ have been proposed as the seeds of the supermassive black holes in galaxies. They would form seconds to minutes after the big bang, when the neutrinos, which carry 41--49\% of the energy of the radiation, stop interacting and begin to stream freely. Yet calculations of their formation have so far treated the neutrinos as part of a single fluid. We simulate the collapse in general relativity through the epoch of electron--positron annihilation, following the neutrinos as they decouple from the plasma and from each other and stream out of the collapsing region. We model their collisions with energy-dependent weak rates in approximate collision terms. Free streaming makes collapse harder: for all the fluctuation shapes we studied, the minimum strength needed to form a black hole rises by 9--10\% in the linear compaction, the measure that sets their abundance. Because primordial black holes form only from the rarest, largest fluctuations, raising the bar by 10\% can remove most of them. For Gaussian fluctuations, at the abundances relevant for seeds (collapse probabilities of $10^{-12}$ to $10^{-15}$), black holes heavier than about $10^6\,\Msun$ become, in a Press--Schechter estimate, 140 to 750 times rarer than perfect-fluid calculations predict. This leaves a step in their mass distribution between $10^4$ and $10^6\,\Msun$, where the neutrinos begin to stream freely; electron--positron annihilation only partly offsets it. At these masses the cosmic microwave background already excludes Gaussian fluctuations this large; for strongly non-Gaussian examples that could still produce seeds, a one-point estimate makes the black holes 1.2 to 8 times rarer.