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Cosmic rays in galaxy clusters are unique probes of energetic processesoperating with large-scale structures in the Universe. Precise measurements ofcosmic rays in galaxy clusters are essential for improving our understanding ofnon-thermal components in the intracluster-medium (ICM) as well as the accuracyof cluster mass estimates in cosmological analyses. In this paper, we perform across-correlation analysis with the extragalactic gamma-ray background and thethermal Sunyaev-Zel'dovich (tSZ) effect in the cosmic microwave background. Theexpected cross-correlation signal would contain rich information about thecosmic-ray-induced gamma-ray emission in the most massive galaxy clusters at$z\sim0.1-0.2$. We analyze the gamma-ray background map with 8 years of datataken by the Large Area Telescope onboard Fermi satellite and the publiclyavailable tSZ map by Planck. We confirm that the measured cross-correlation isconsistent with a null detection, and thus it enables us to put the tightestconstraint on the acceleration efficiency of cosmic ray protons at shocks inand around galaxy clusters. We find the acceleration efficiency must be below5\% with a $2\sigma$ confidence level when the hydrostatic mass bias ofclusters is assumed to be 30\%, and our result is not significantly affected bythe assumed value of the hydrostatic mass bias. Our constraint implies that thenon-thermal cosmic-ray pressure in the ICM can introduce only a $\le 3\%$ levelof the hydrostatic mass bias, highlighting that cosmic rays alone do notaccount for the mass bias inferred by the Planck analyses. Finally, we discussfuture detectability prospects of cosmic-ray-induced gamma rays from thePerseus cluster for the Cherenkov Telescope Array.

Cross Correlation of the Extragalactic Gamma-ray Background with Thermal Sunyaev-Zel'dovich Effect in the Cosmic Microwave Background