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.
