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We study the parameter space of the effective (with two scalars) models ofcosmological inflation and primordial black hole (PBH) formation in themodified $(R+R^2)$ supergravity. Our models describe double inflation, whosefirst stage is driven by Starobinsky's scalaron coming from the $R^2$ gravity,and whose second stage is driven by another scalar belonging to thesupergravity multiplet. The ultra-slow-roll regime between the two stages leadsa large peak (enhancement) in the power spectrum of scalar perturbations, whichresults in efficient PBH formation. Both inflation and PBH formation aregeneric in our models, while those PBH can account for a significant part orthe whole of dark matter. Some of the earlier proposed models in the same classare in tension (over $3\sigma$) with the observed value of the scalar tilt$n_s$, so that we study more general models with more parameters, andinvestigate the dependence of the cosmological tilts $(n_s,r)$ and the scalarpower spectrum enhancement upon the parameters. The PBH masses and theirdensity fraction (as part of dark matter) are also calculated. A good agreement(between $2\sigma$ and $3\sigma$) with the observed value of $n_s$ requiresfine tuning of the parameters, and it is only realized in the so-called$\delta$-models. Our models offer the (super)gravitational origin of inflation,PBH and dark matter together, and may be confirmed or falsified by futureprecision measurements of the cosmic microwave background radiation andPBH-induced gravitational waves.

Exploring the parameter space of modified supergravity for double inflation and primordial black hole formation