| Summary: | International audience We analyze the cooling and feedback properties of 48 galaxy clusters at redshifts $0.4 < z < 1.3$ selected from the South Pole Telescope (SPT) catalogs to evolve like the progenitors of massive and well-studied systems at $z{\sim}0$. We estimate the radio power at the brightest cluster galaxy (BCG) location of each cluster from an analysis of Australia Telescope Compact Array (ATCA) data. Assuming that the scaling relation between radio power and active galactic nucleus (AGN) cavity power $P_{\mathrm{cav}}$ observed at low redshift does not evolve with redshift, we use these measurements in order to estimate the expected AGN cavity power in the core of each system. We estimate the X-ray luminosity within the cooling radius $L_{\mathrm{cool}}$ of each cluster from a joint analysis of the available $Chandra$ X-ray and SPT Sunyaev-Zel'dovich (SZ) data. This allows us to characterize the redshift evolution of the $P_{\mathrm{cav}} / L_{\mathrm{cool}}$ ratio. When combined with low-redshift results, these constraints enable investigations of the properties of the feedback/cooling cycle across 9~Gyr of cluster growth. We model the redshift evolution of this ratio measured for cool core clusters by a log-normal distribution $\mathrm{Log}$-$\mathcal{N}(\alpha + \beta z, \sigma^2)$ and constrain the slope of the mean evolution $\beta = -0.05\pm 0.47$. This analysis improves the constraints on the slope of this relation by a factor of two. We find no evidence of redshift evolution of the feedback/cooling equilibrium in these clusters which suggests that the onset of radio-mode feedback took place at an early stage of cluster formation. High values of $P_{\mathrm{cav}} / L_{\mathrm{cool}}$ are found at the BCG location of non-cool core clusters which might suggest that the timescales of the AGN feedback cycle and the cool core / non-cool core transition are different.
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