Modulated Inflation Models and Primordial Black Holes
  For the first time, the running of the running of the spectral index
$\beta_s$ has been measured by Planck. This parameter is crucial for dark
matter Primordial Black Holes (PBHs) formation. We study the PBHs formation in
inflation potentials with superimposed periodic oscillations. We show that the
value of $\beta_s$ suggested by the recent cosmic microwave background data is
easily explained in these models. As a by-product, we also show that the
nonproduction of (long--lived) PBHs puts a stronger upper bound on $\beta_s$.
