Invariant probability fluctuations

PBH formation is bound up with the primordial density fluctuations spectrum. Inflation produces a scale invariant spectrum the root mean square fluctuation e for an inhomogeneity just inside the horizon is M independent. When the propitious time (42) comes, a PBH can form only if the density contrast 5 = 5p/p at horizon scale is large enough the rule is that for equation of state p = wp, 5 must exceed w. Evidently, the fraction (5 of horizon-size volumes which actually collapse (the probability that the inhomogeneity makes a PBH), is M independent. 

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