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(abridged) We explore in this paper the ability of spatially resolvedspectroscopic measurements of the SZ effect (SZE) to determine the temperatureprofile of galaxy clusters. We derive a general formalism for the thermal SZEin galaxy clusters with a non-uniform temperature profile that can be appliedto both cool-core clusters and non-cool core cluster with an isothermal ornon-isothermal temperature structure. We derive an inversion technique throughwhich the electron distribution function can be extracted from spectroscopicSZE observations over a wide frequency range. We study the fitting procedure toextract the cluster temperature from a set of simulated spatially resolvedspectroscopic SZE observations in different bands of the spectrum, from 100 to450 GHz. The results of our analysis for three different cluster prototypes(A2199 with a low-temperature cool core, Perseus with a relativelyhigh-temperature cool core, Ophiuchus with an isothermal temperaturedistribution) provide both the required precision of the SZE observations andthe optimal frequency bands for a determination of the cluster temperaturesimilar or better than that obtainable from X-ray observations. The precisionof SZE-derived temperature is also discussed for the outer regions of clusters.We also study the possibility to extract, from our method, the parameterscharacterizing the non-thermal SZE spectrum of the relativistic plasmacontained in the lobes of radio galaxies as well as the spectrum ofrelativistic electrons co-spatially distributed with the thermal plasma inclusters with non-thermal phenomena. We find that the next generation SZEexperiments with spectroscopic capabilities can provide precise temperaturedistribution measurements (...)

On the ability of spectroscopic SZ effect measurements to determine the temperature structure of galaxy clusters