AGN feedback and entropy injection in galaxy cluster cores
  The non-gravitational energy feedback is of crucial importance in
modeling/simulating clusters to be used as cosmological probes. AGNs are,
arguably, of primary importance in injecting energy in the cluster cores. We
make the first estimate of non-gravitational energy {\it profiles} in galaxy
cluster cores (and beyond) from observational data. Comparing the observed
entropy profiles within $r_{500}$, from the Representative {\it XMM-Newton}
Cluster Structure Survey (REXCESS), to simulated entropy profiles from both AMR
and SPH non-radiative simulations, we estimate the amount of non-gravitational
energy, $E_{\rm ICM}$, contained in the ICM. Adding the radiative losses we
estimate the total energy feedback, $E_{\rm Feedback}$, into the clusters. The
profiles for the energy deposition, $\Delta E_{\rm ICM}(x)$, in the inner
regions differ for Cool-Core (CC) and Non Cool-Core (NCC) clusters, decreasing
after accounting for the radiative cooling. The total feedback energy scales
with the mean spectroscopic temperature as $E_{\rm Feedback} \propto T_{\rm
sp}^{2.52 \pm0.08}$ and $E_{\rm Feedback} \propto T_{\rm sp}^{2.17 \pm 0.11}$,
when compared with the baseline SPH and AMR profiles respectively. The scatter
in the two cases is 15% and 23%, respectively. The mean non-gravitational
energy per particle within $r_{500}$, is $\epsilon_{\rm ICM} = {2.8} \pm {0.8}$
keV for the SPH theoretical relation and $\epsilon_{\rm ICM} = {1.7} \pm {0.9}$
keV for the AMR theoretical relation. We use the {\it NRAO/VLA Sky Survey}
(NVSS) source catalog to determine the radio luminosity, $L_R$, at 1.4 GHz of
the central source(s) of our sample. For $T_{\rm sp} > 3$ keV, the $E_{\rm
Feedback}$ correlates with $L_R$. We show that AGNs could provide a significant
component of the feedback. (Abridged)
