Viable f(R) Scenarios Unifying Inflation with Realistic Dynamical Dark Energy

Two $F(R)$ gravity models are tested on the basis of their viability during all stages of cosmological evolution. It is shown that these models can describe both the early-time inflationary epoch and the dark energy epoch. The models are confronted with the latest observational data, including the Pantheon+ catalogue with Type Ia supernovae, the Dark Energy Spectroscopic Instrument measurements of baryon acoustic oscillations, the Hubble parameter estimations and data from cosmic microwave background radiation. Investigation of the viability conditions for these models, in particular, the condition $\frac{dF}{dR}>0$ required a deep analysis. Both models appeared to be viable during the early-time era, but for the late-time evolution the viability conditions are not fulfilled in definite domains in the parameter spaces of these models. However the best fitted parameters, determined in confrontation with the mentioned observational data, lie far from the forbidden domains for both models. These $F(R)$ gravity models describe the observations with the large advantage over the $Λ$-Cold-Dark-Matter model, not only in $χ^2$ statistics, but also with Akaike and Bayesian information criteria. This success of the two $F(R)$ gravity scenarios is connected with their capability to mimic dynamical dark energy, similarly to models with variable equation of state, that is necessary for describing the latest Pantheon+ and DESI observational data.