The modified supergravity approach is applied to describe a formation of Primordial Black Holes (PBHs) after Starobinsky inflation. Our approach naturally leads to the two-(scalar)-field attractor-type double inflation, whose first stage is driven by scalaron and whose second stage is driven by another scalar field which belongs to a supergravity multiplet. The scalar potential and the kinetic terms are derived, the vacua are studied, and the inflationary dynamics of those two scalars is investigated. We numerically compute the power spectra and we find the ultra-slow-roll regime leading to an enhancement (peak) in the scalar power spectrum. This leads to an efficient formation of PBHs. We estimate the masses of PBHs and we find their density fraction (as part of Dark Matter). We show that our modified supergravity models are in agreement with inflationary observables, while they predict the PBH masses in a range between $$10^{16}$$10 16 g and $$10^{20}$$10 20 g. In this sense, modified supergravity provides a natural top-down approach for explaining and unifying the origin of inflation and the PBHs Dark Matter.

We propose a limited class of models, describing interacting chiral multiplets with a non-minimal coupling to a vector multiplet, in curved superspace of N=1 supergravity. Those models are suitable for the inflationary model building in supergravity with inflaton assigned to a massive vector multiplet and spontaneous SUSY breaking in Minkowski vacuum after inflation, for any values of the inflationary parameters ns and r, and any scale of SUSY breaking

We propose a novel model of the modified (Starobinsky-like) old-minimal-type supergravity coupled to a chiral matter superfield, that can simultaneously describe multi-field inflation, primordial black hole (PBH) formation, dark matter (DM), and spontaneous supersymmetry (SUSY) breaking after inflation in a Minkowski vacuum. The PBH masses in our supergravity model of double slow-roll inflation, with a short phase of “ultra-slow-roll” between two slow-roll phases, are close to $$10^{18}$$10 18 g. We find that a significant PBH fraction in the allowed mass window can be supplemented by spontaneous SUSY breaking in the vacuum with the gravitino mass close to the scalaron (inflaton) mass M of the order $$10^{13}$$10 13 GeV. Our supergravity model favors the composite nature of DM as a mixture of PBH and heavy gravitinos as the lightest SUSY particles. The composite DM significantly relaxes fine-tuning needed for the whole PBH-DM. The PBH-DM fraction is derived, and the second-order gravitational wave background induced by the enhanced scalar perturbations is calculated. Those gravitational waves may be accessible by the future space-based gravitational interferometers.

A supergravity extension of the $(R+R^2)$ gravity with the additional (Born-Infeld) structure of a massive vector multiplet gives rise to the specific $F(R)$ gravity, whose structure is investigated in detail. The massive vector multiplet has an inflaton (scalaron), goldstino, and massive vector field as its field components. The model describes Starobinsky inflation and allows us to extrapolate the $F(R)$ function beyond the inflationary scale (up to Planck scale). We observe some differences versus the $(R+R^2)$ gravity and several breaking patterns of the well-known correspondence between the $F(R)$ gravity and the scalar-tensor gravity.