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Title: How many black holes does a given primordial spectrum produce?
Plain: The number of primordial black holes depends on the rarest, most extreme ripples, so small changes in the assumed statistics change the prediction by huge factors. A reliable calculation is still missing.
Precise: For a curvature spectrum $P_{\zeta}(k)$ peaked at $k_p$, compute the mass fraction $\beta(M)$ collapsing into PBHs at horizon entry, including the profile-dependent collapse threshold $\delta_c$ (about 0.4 to 0.67), critical-collapse scaling M = K (delta - delta_c)^0.36 $M_H$ with M_H the horizon mass, the nonlinear relation between $\zeta$ and the density contrast, and non-Gaussian tails (including exponential tails from stochastic inflation). An answer is $\beta(M)$ with theoretical uncertainty below a factor of 10 for a benchmark spectrum, agreed between independent methods (peak theory, threshold statistics, numerical relativity).
Settled by: A benchmark comparison in which peak-theory, threshold-statistics and numerical-relativity calculations of $\beta(M)$ for the same spectrum agree within a factor of 10.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: (empty)
Posed since: (empty)