Traversable wormholes with vanishing sound speed in $f(R)$ gravity

Abstract: We derive exact traversable wormhole solutions in the framework of $f(R)$ gravity with no exotic matter and with stable conditions over the geometric fluid entering the throat. For this purpose, we propose power-law $f(R)$ models and two possible approaches for the shape function $b(r)/r$. The first approach makes use of an inverse power law function, namely $b(r)/r\sim r^{-1-\beta}$. The second one adopts Pad\'e approximants, used to characterize the shape function in a model-independent way. We single out the $P(0,1)$ approximant where the fluid perturbations are negligible within the throat, if the sound speed vanishes at $r=r_0$. The former guarantees an overall stability of the geometrical fluid into the wormhole. Finally we get suitable bounds over the parameters of the model for the above discussed cases. In conclusion, we find that small deviations from General Relativity give stable solutions.