Observational constraints on Starobinsky $f(R)$ cosmology from cosmic expansion and structure growth data.

The unknown physical nature of the Dark Energy motivates in cosmology the study of modifications of the gravity theory at large distances. One of these types of modifications are the theories known as $f(R)$ gravity. In this paper we use observational data to both constraint and test the Starobinsky $f(R)$ model \cite{Starobinsky2007}, using updated measurements from the dynamics of the expansion of the universe, $H(z)$, and the growth rate of cosmic structures, $[f\sigma_8](z)$, where the distinction between the concordance $\Lambda $CDM model and modified gravity models, $f(R)$, become clearer. We use MCMC likelihood analyses to explore the parameters space of the $f(R)$ model using $H(z)$ and $[f\sigma_8](z)$ data, both individually and jointly, and further check which of the models better fit the joint data. To further test the Starobinsky model, we use a method proposed by Linder \cite{Linder2017}, where the data from the observables is jointly binned in redshift space. This allows to further explore the model's parameter that better fits the data in comparison to the $\Lambda$CDM model. Our analyses show that the result from the MCMC run using $[f\sigma_8](z)$ data alone fits the analyzed model better than using the joint data. At the end, we reaffirm that this joint analysis is able to break the degenerescence between modified gravity models through the $f\sigma_8$ observable, as the original author proposed; our results show that the $f(R)$ Starobinsky model although not favored by the available data, cannot be discarded and is a possible alternative to the $\Lambda$CDM model.