Investigating $f(R)$-Inflation: background evolution and constraints

In this work, we investigate the possibility of generating an inflationary mechanism within the framework of a metric-$f(R)$ modified gravity theory, formulated in the Jordan frame. We explore whether the scalar field, non-minimally coupled to gravity and emerging in the Jordan frame, can play the role of the primordial inflaton. Particular attention is devoted to constructing a dynamical scenario in the Jordan frame that exhibits a slow-rolling phase for the scalar field and admits a quasi-de Sitter solution for cosmic evolution. To ensure consistency with the standard cosmological model, we impose a matching condition with the $\Lambda$CDM model at the end of the inflationary phase. Furthermore, to address the problem of the absence of matter after inflation, we consider a radiation-type particle creation process that maintains an approximately constant energy density. We test our theoretical model against background observational data, specifically Pantheon+ calibrated with SH0ES and DESI calibrated with BBN. We asses its viability and discuss implications for alleviating the Hubble constant tension.