Cosmological analysis of a viable $f(R)$ gravity model

Since viable $f(R)$ gravity models must reconcile early-universe inflation with late-time acceleration, we specifically study the dynamical behavior of such a theory during the matter-dominated to dark-energy-dominated transition epoch. By using $y_{H}(z)$ versus $z$ and the Hubble parameter, we solved the field equations. After appropriately choosing appropriate parameter values , we plotted a series of images. We mentioned that their current values are similar to latest observations data and $\Lambda$CDM-model values. Furthermore, we plotted the fitting of the distance modulus about this model using SN Ia observation data. Therefore we find that the $f(R)$ gravity model is consistent with the SN Ia data, meanwhile, explains the late-stage acceleration of the Universe. Finally, we used various diagnostic tools including $( r, s)$, $( r, q)$, $w_{D}-w'_{D}$ plane, growth rate analysis, statefinder hierarchy and $Om(z)$-diagnostic to evaluate the observational viability of our model, we perform a systematic comparison with the standard $\Lambda$CDM. We found that evolutionary images can be clearly distinguished this model from the $\Lambda$CDM.