A $f(R)$-gravity model of the Sunyaev-Zeldovich profile of the Coma   cluster compatible with {\it Planck} data
  In the weak field limit, analytic $f(R)$ models of gravity introduce a Yukawa-like correction to the Newtonian gravitational potential. These models have been widely tested at galactic scales and provide an alternative explanation to the dynamics of galaxies without Dark Matter. We study if the temperature anisotropies due to the thermal Sunyaev-Zeldovich effect are compatible with these Extended Theories of Gravity. We assume that the gas is in hydrostatic equilibrium within the modified Newtonian potential and it is well described by a polytropic equation of state. We particularize the model for the Coma cluster and the predicted anisotropies are compared with those measured in the foreground cleaned maps obtained using the Planck Nominal maps released in 2013. We show that the computed $f(R)$ pressure profile fits the data giving rise to competitive constraints of the Yukawa scale length $L=(2.19\pm1.02) \rm{\, Mpc}$, and of the deviation parameter $ \delta=-0.48\pm0.22$. Those are currently the tightest constraints at galaxy cluster scale, and support the idea that Extended Theories of Gravity provide an alternative explanation to the dynamics of self-gravitating systems without requiring Dark Matter. 
I. de Martino
astro-ph.CO gr-qc