Pitfalls of a power-law parametrization of the primordial power spectrum for primordial black hole formation

Primordial black holes (PBHs) can form in the radiation dominated early Universe from the collapse of large density perturbations produced by inflation. A power-law parametrization of the primordial power spectrum is often used to extrapolate from cosmological scales, where the amplitude of the perturbations is well measured by the cosmic microwave background and large scale structure observations, down to the small scales on which PBHs may form. We show that this typically leads to large errors in the amplitude of the fluctuations on small scales, and hence extremely inaccurate calculations of the abundance of PBHs are formed.