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Comment on "The Mass-Radius relation for Neutron Stars in $f(R)$ gravity" by S. Capozziello, M. De Laurentis, R. Farinelli and S. Odintsov

In the present comment we discuss the qualitative and the quantitative differences between the neutron stars models in $f(R)$ gravity presented in the recent paper \cite{Capozziello2015} (Capozziello et al. (2015)) and in our paper \cite{Yazadjiev2014} (Yazadjiev et al. (2014)). The authors of \cite{Capozziello2015} claim that these differences come from the fact that they are using the physical Jordan frame while we are working in the Einstein frame. The results in \cite{Yazadjiev2014} are very well tested by two numerical codes using different formulations of the field equations that give us the confidence that they are correct. Thus we decided to perform an additional test. Namely, we derive the relevant set of field equations in the Jordan frame and solve them numerically in order to disprove the claim in \cite{Capozziello2015} that the difference comes from the use of different frame. The obtained results are in perfect agreement with our previous work \cite{Yazadjiev2014} and therefore the numerical results in \cite{Capozziello2015} seem to be incorrect. At the end we point out several possible sources of error in \cite{Capozziello2015} that can lead to the observed large deviations.