Recently, the formation of primordial black holes (PBHs) from the collapse of
primordial fluctuations has received much attention. The abundance of PBHs
formed during radiation domination is sensitive to the tail of the probability
distribution of primordial fluctuations. We quantify the level of fine-tuning
due to this sensitivity. For example, if the main source of dark matter is PBHs
with mass 10−12M⊙, then anthropic reasoning suggests that the dark
matter to baryon ratio should range between 1 and 300. For this to happen, the
root-mean-square amplitude of the curvature perturbation has to be fine-tuned
within a 7.1% range. As another example, if the recently detected
gravitational-wave events are to be explained by PBHs, the corresponding degree
of fine-tuning is 3.8%. We also find, however, that these fine-tunings can
be relaxed if the primordial fluctuations are highly non-Gaussian, or if the
PBHs are formed during an early-matter-dominated phase. We also note that no
fine-tuning is needed for the scenario of a reheating of the universe by
evaporated PBHs with Planck-mass relics left to serve as dark matter.Comment: 6 page
