Pole inflation and primordial black holes formation in Starobinsky-like supergravity

We extend the Cecotti-Kallosh model of Starobinsky inflation in supergravity by adding a holomorphic function to the superpotential in order to generate a large peak in the power spectrum of scalar (curvature) perturbations. In our approach, the singular non-canonical kinetic terms are largely responsible for inflation (as an attractor solution), whereas the superpotential is engineered to generate a production of PBH. We study the cases with (a) a linear holomorphic function, (b) a quadratic holomorphic function, and (c) an exponential holomorphic function, as regards the dependence of inflation and PBH production upon parameters of those functions and initial conditions, as well as verify viability of inflation with our superpotentials. We find that an efficient production of PBH consistent with cosmic microwave background measurements is only possible in the second (b) case. We calculate the masses of the produced PBH and find that they are below the Hawking (black hole) evaporation limit, so that they cannot be part of the current dark matter in our Universe.