Résumé
We revisit the prospects for detecting the Sunyaev Zel'dovich (SZ) effect induced by dark matter (DM) annihilation or decay. We show that with standard (or even extreme) assumptions for DM properties, the optical depth associated with relativistic electrons injected from DM annihilation or decay is much smaller than that associated with thermal electrons, when averaged over the angular resolution of current and future experiments. For example, we find: tau(DM) similar to 10(-9) - 10(-5) (depending on the assumptions) for m(chi) = 1 GeV and a density profile rho proportional to r(-1) for a template cluster located at 50 Mpc and observed within an angular resolution of 10 '', compared to tau(th) similar to 10(-3) - 10(-2). This, together with a full spectral analysis, enables us to demonstrate that, for a template cluster with generic properties, the SZ effect due to DM annihilation or decay is far below the sensitivity of the Planck satellite. This is at variance with previous claims regarding heavier annihilating DM particles. Should DM be made of lighter particles, the current constraints from 511 keV observations on the annihilation cross section or decay rate still prevent a detectable SZ effect. Finally, we show that spatial diffusion sets a core of a few kpc in the electron distribution, even for very cuspy DM profiles, such that improving the angular resolution of the instrument, e. g. with ALMA, does not necessarily improve the detection potential. We provide useful analytical formulae parameterized in terms of the DM mass, decay rate or annihilation cross section and DM halo features, that allow quick estimates of the SZ effect induced by any given candidate and any DM halo profile.