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In the weak field limit, analytic $f(R)$ models of gravity introduce aYukawa-like correction to the Newtonian gravitational potential. These modelshave been widely tested at galactic scales and provide an alternativeexplanation to the dynamics of galaxies without Dark Matter. We study if thetemperature anisotropies due to the thermal Sunyaev-Zeldovich effect arecompatible with these Extended Theories of Gravity. We assume that the gas isin hydrostatic equilibrium within the modified Newtonian potential and it iswell described by a polytropic equation of state. We particularize the modelfor the Coma cluster and the predicted anisotropies are compared with thosemeasured in the foreground cleaned maps obtained using the Planck Nominal mapsreleased in 2013. We show that the computed $f(R)$ pressure profile fits thedata 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 galaxycluster scale, and support the idea that Extended Theories of Gravity providean alternative explanation to the dynamics of self-gravitating systems withoutrequiring Dark Matter.

A $f(R)$-gravity model of the Sunyaev-Zeldovich profile of the Coma cluster compatible with {\it Planck} data