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.
