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The galaxy cluster MS 0735.6+7421 hosts two large X-ray cavities, filled withradio emission, where a decrease of the Sunyaev-Zel'dovich (SZ) effect has beendetected, without establishing if its origin is thermal (from a gas with veryhigh temperature) or non-thermal. In this paper we study how thermal andnon-thermal contributions to the SZ effect in the cavities are related; infact, Coulomb interactions with the thermal gas modify the spectrum of lowenergy non-thermal electrons, which dominate the non-thermal SZ effect; as aconsequence, the intensity of the non-thermal SZ effect is stronger for lowerdensity of the thermal gas inside the cavity. We calculate the non-thermal SZeffect in the cavities as a function of the thermal density, and compare the SZeffects produced by thermal and non-thermal components, and with the one fromthe external Intra Cluster Medium (ICM), searching for the best frequency rangewhere it is possible to disentangle the different contributions. We find thatfor temperatures inside the cavities higher than $\sim1500$ keV the non-thermalSZ effect is expected to dominate on the thermal one, particularly at highfrequencies ($\nu>500$ GHz), where it can also be a non-negligible fraction ofthe SZ effect from the external ICM. We also discuss the possible sources ofastrophysical bias (as kinetic SZ effect and foreground emission from Galacticdust) and possible ways to address them, as well as necessary improvements inthe modeling of the properties of cavities and the ICM.

Thermal and non-thermal Sunyaev-Zel'dovich effect in the cavities of the galaxy cluster MS 0735.6+7421: the role of the thermal density in the cavity