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Recently, gas entropy at the outskirts of galaxy clusters attracts muchattention. We propose that the entropy profiles could be used to studycosmic-ray (CR) acceleration around the clusters. If the CRs are effectivelyaccelerated at the formation of clusters, the kinetic energy of infalling gasis consumed by the acceleration and the gas entropy should decrease. As aresult, the entropy profiles become flat at the outskirts. If the accelerationis not efficient, the entropy should continue to increase outwards. Bycomparing model predictions with X-ray observations with Suzaku, which showflat entropy profiles, we find that the CRs have carried ~<7% of the kineticenergy of the gas away from the clusters. Moreover, the CR pressure at theoutskirts can be ~<40% of the total pressure. On the other hand, if the entropyprofiles are not flat at the outskirts as indicated by combined Plank and ROSATobservations, the carried energy and the CR pressure should be much smallerthan the above estimations.

Entropy at the Outskirts of Galaxy Clusters as Implications for Cosmological Cosmic-Ray Acceleration