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

TL;DR

This work studies a viable $f(R)$ gravity model intended to unify early-universe inflation with late-time acceleration by solving the background dynamics via the dynamical variable $y_H(z)$ and the Hubble rate $H(z)$. The authors introduce a polynomial $f(R)$ with an $R^2$ term and extra components, compute cosmological evolutions, and demonstrate consistency with SN Ia data and Planck/ΛCDM expectations for several parameter choices. They employ a broad set of geometric and perturbation diagnostics—statefinder pairs $(r,s)$ and $(r,q)$, the statefinder hierarchy, growth-rate diagnostics, the $w_D-w'_D$ plane, and the $Om(z)$ statistic—to contrast the model with $Λ$CDM, finding distinguishable low-redshift signatures while noting convergence toward ΛCDM at late times. The results indicate that the model provides a viable, testable alternative to ΛCDM, with clear predictions for upcoming observations and a known issue of ill-oscillatory behavior to resolve in future work.

Abstract

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 $Λ$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 $Λ$CDM. We found that evolutionary images can be clearly distinguished this model from the $Λ$CDM.

Figures (9)

• Figure 1: When $\lambda$ takes different values ($\lambda_{1} =1.0\Lambda$, $\lambda_{2} = 1.24\Lambda$, $\lambda_{3} =1.5\Lambda$), $\eta=0.09$ and $\mu^{2}=1.37\times10^{-67}eV^{2}$, the density parameters $\Omega_{DE}$ and $\Omega_{m}$ (left plot) versus $z$ and $w_{DE}(z)$ (right plot) for the polynomial type $f(R)$ model of Eqs.(\ref{['17']})-(\ref{['18']}).
• Figure 2: When $\lambda$ takes different values ($\lambda_{1} =1.0\Lambda$, $\lambda_{2} = 1.24\Lambda$, $\lambda_{3} = 1.5\Lambda$), $\eta=0.09$ and $\mu^{2}=1.37\times10^{-67}eV^{2}$, $w(z)$ and $q$are plotted against $z$ within the polynomial type $f(R)$ model of Eqs.(\ref{['19']}) and (\ref{['20']}).
• Figure 3: When $\lambda$ takes different values ($\lambda_{1} =1.0\Lambda$, $\lambda_{2} = 1.24\Lambda$, $\lambda_{3} = 1.5\Lambda$), $\eta=0.09$ and $\mu^{2}=1.37\times10^{-67}eV^{2}$, $H(z)$ (left plot) and $\mu$ comparing with $1048$ SN Ia observation data points(gray bars) are plotted versus $z$ within for the polynomial type $f(R)$ model of Eqs.(\ref{['24']}) and (\ref{['27']}).
• Figure 4: When $\lambda$ takes different values ($\lambda_{1} =1.0\Lambda$, $\lambda_{2} = 1.24\Lambda$, $\lambda_{3} = 1.5\Lambda$), $\eta=0.09$ and $\mu^{2}=1.37\times10^{-67}eV^{2}$, the evolutions of statfinder parameters for the polynomial type $f(R)$ model.
• Figure 5: When $\lambda$ takes different values ($\lambda_{1} =1.0\Lambda$, $\lambda_{2} = 1.24\Lambda$, $\lambda_{3} = 1.5\Lambda$), $\eta=0.09$ and $\mu^{2}=1.37\times10^{-67}eV^{2}$, the evolutions of $S_{3}^{(1)}$ and $S_{4}^{(1)}$ versus $z$ for the polynomial type $f(R)$ model.