A multi-frequency study of the SZE in giant radio galaxies

Context. Radio galaxy (RG) lobes contain relativistic electrons embedded in a tangled magnetic field that produce, in addition to low-frequency synchrotron radio emission, inverse-Compton scattering (ICS) of the cosmic microwave background (CMB) photons. This produces a relativistic, non-thermal Sunyaev-Zel'dovich effect (SZE). Aims. We study the spectral and spatial properties of the non-thermal SZE in a sample of radio galaxies and make predictions for their detectability in both the negative and the positive part of the SZE, with space experiments like Planck, OLIMPO, and Herschel-SPIRE. These cover a wide range of frequencies, from radio to sub-mm. Methods. We model the SZE in a general formalism that is equivalent to the relativistic covariant one and describe the electron population contained in the lobes of the radio galaxies with parameters derived from their radio observations, namely, flux, spectral index, and spatial extension. We further constrain the electron spectrum and the magnetic field of the RG lobes using X-ray, gamma-ray, and microwave archival observations. Results. We determine the main spectral features of the SZE in RG lobes, namely, the minimum, the crossover, and the maximum of the SZE. We show that these typical spectral features fall in the frequency ranges probed by the available space experiments. We provide the most reliable predictions for the amplitude and spectral shape of the SZE in a sample of selected RGs with extended lobes. In three of these objects, we also derive an estimate of the magnetic field in the lobe at the similar to mu G level by combining radio (synchrotron) observations and X-ray (ICS) observations. These data, together with the WMAP upper limits, set constraints on the minimum momentum of the electrons residing in the RG lobes and allow realistic predictions for the visibility of their SZE to be derived with Planck, OLIMPO, and Herschel-SPIRE. Conclusions. We show that the SZE from several RG lobes can be observed with mm and sub-mm experiments like Planck, OLIMPO, and Herschel-SPIRE, as well as with ground-based telescopes that have less than or similar to mJy sensitivity and sub-arcmin spatial resolution. These measurements will be crucial to disentangle the relativistic electron distribution from that of the magnetic field in RG lobes and to constrain the properties of their ICS emission, which is also visible at very high X-ray and gamma-ray energies.