Reconstruction of potentials of the hybrid inflation in the light of primordial black hole formation

The large enhancement of the primordial power spectrum of the curvature perturbation can seed the formation of primordial black hole, that can play as a dark matter component in the Universe. In multi-filed inflation models, the curved trajectory of the scalar fields in the field space can generate a peak in the power spectrum on small scales due to the existence of the isocurvature perturbation. Here we show that a potential can be reconstructed from a given power spectrum, which is made of a scale-invariant one on large scales and the other function with a peak on small scales. In multi-field inflation models the reconstructed potential may not be unique and we can find different potentials from a given power spectrum.
Comment: 14 pages, 4 figures