Multi-frequency constraints on the non-thermal pressure in galaxy
  clusters
  The origin of radio halos in galaxy clusters is still unknown and is the
subject of a vibrant debate both from the observational and theoretical point
of view. In particular the amount and the nature of non-thermal plasma and of
the magnetic field energy density in clusters hosting radio halos is still
unclear. The aim of this paper is to derive an estimate of the pressure ratio X
between the non-thermal and thermal plasma in radio halo clusters that have
combined radio, X-ray and SZ effect observations. From the simultaneous
P_{1.4}-L_X and P_{1.4}-Y_{SZ} correlations for a sample of clusters observed
with Planck, we derive a correlation between Y_{SZ} and L_X that we use to
derive a value for X. This is possible since the Compton parameter Y_{SZ} is
proportional to the total plasma pressure in the cluster (that we characterize
as the sum of the thermal and non-thermal pressure) while the X-ray luminosity
L_X is proportional only to the thermal pressure of the intracluster plasma.
Our results indicate that the average (best fit) value of the pressure ratio in
a self-similar cluster formation model is X =0.55 \pm 0.05 in the case of an
isothermal beta-model with beta=2/3 and a core radius r_c = 0.3 R_{500} holding
on average for the cluster sample. We also show that the theoretical prediction
for the Y_{SZ}-L_X correlation in this model has a slope that is steeper than
the best fit value for the available data. The agreement with the data can be
recovered if the pressure ratio X decreases with increasing X-ray luminosity as
L_X^{-0.96}. We conclude that the available data on radio halo clusters
indicate a substantial amount of non-thermal pressure in cluster atmospheres
whose value must decrease with increasing X-ray luminosity, or increasing
cluster mass (temperature). (abridged)
