Dark Matter Primordial Black Holes and Inflation Models
  A broad range of single field models of inflation are analyzed in light of
all relevant recent cosmological data, checking whether they can lead to the
formation of long--lived Primordial Black Holes (PBHs) as candidate for dark
matter. To that end we calculate the spectral index of the power spectrum of
primordial perturbations as well as its first and second derivatives. PBH
formation is possible only if the spectral index $n_S(k_0)$ increases
significantly at small scales. Since current data indicate that the first
derivative $\alpha_S$ of the spectral index is negative at the pivot scale, PBH
formation is only possible in the presence of a sizable and positive second
derivative ("running of the running") $\beta_S$. Among the three small-field
and five large-field models we analyze, only the "running-mass" model allows
PBH formation, for a narrow range of parameters.
