Cosmological constraints on $R^2$-corrected Appleby-Battye model

Nowadays, efforts are being devoted to the study of alternative cosmological scenarios in which modifications of General Relativity have been proposed to explain the late cosmic acceleration without assuming the existence of dark energy. In this scenario, we investigate the $R^2$-AB model, which consists of an $f(R)$ model with only one extra free parameter, $b$, in addition to the 6 of the flat-$\Lambda$CDM. Regarding this model, it was already shown that a positive value for $b$ is required for the model to be consistent with Solar System tests, moreover, the condition for the existence of a de~Sitter state requires $b \ge 1.6$. To impose observational constraints on the $R^2$-AB model we consider three datasets: 31 $H(z)$ measurements from Cosmic Chronometers (CC), 20 $[{f\sigma}_{8}](z)$ measurements from Redshift-Space Distortion (RSD), and the most recent type Ia Supernovae (SNe Ia) sample from Pantheon+. Next, we perform two different analyses: we have considered only SNe Ia data and the combined likelihood SNe+CC+RSD. The first one has provided $b=2.28^{+6.52}_{-0.55}$, while the second one $b=2.18^{+5.41}_{-0.55}$. In the first case it was necessary to set the absolute magnitude $M_B = -19.253$ from SH0ES collaboration, while in the second we did a marginalization over the Hubble constant $H_0$ in the normalized growth function. We have also observed that the $H_0-M_B$ degeneracy was broken by adding CC data to the SNe data. Additionally, we perform illustrative analyses that compare this $f(R)$ model with the flat-$\Lambda$CDM model, considering several values of the parameter $b$, for diverse cosmological functions like the Hubble function $H(z)$, the equation of state $w_{\rm eff}(z)$, the parametrized growth rate of cosmic structures $[f\sigma_8](z)$, and $\sigma_8(z)$. We conclude that the model fits well the data, but the parameter $b$ was not unambiguously constrained.