Résumé
We consider a cosmological model in which a fraction f(dcdm), of the Dark Matter (DM) is allowed to decay in an invisible relativistic component, and compute the resulting constraints on both the decay width (or inverse lifetime) Gamma(dcdm) and fdcdm from purely gravitational arguments. We report a full derivation of the Boltzmann hierarchy, correcting a mistake in previous literature, and compute the impact of the decay as a function of the lifetime on the CMB and matter power spectra. From CMB only, we obtain that no more than 3.8% of the DM could have decayed in the time between recombination and today (all bounds quoted at 95% CL). We also comment on the important application of this bound to the case where primordial black holes constitute DM, a scenario notoriously difficult to constrain. For lifetimes longer than the age of the Universe, the bounds can be cast as f(dcdm)Gamma(dcdm) < 6.3 x 10(-3) Gyr(-1). For the first time, we also checked that degeneracies with massive neutrinos are broken when information from the large scale structure is used. Even secondary effects like CMB lensing suffice to this purpose. Decaying DM models have been invoked to solve a possible tension between low redshift astronomical measurements of sigma(8) and Omega(m), and the ones inferred by Planck. We reassess this claim finding that with the most recent BAO, HST and sigma(8) data extracted from the CFHT survey, the tension is only slightly reduced despite the two additional free parameters. Nonetheless, the existing tension explains why the bound on f(dcdm)Gamma(dcdm) loosens to f(dcdm)Gamma(dcdm) < 15.9 x 10(-3) Gyr(-1) when including such additional data. The bound however improves to f(dcdm)Gamma(dcdm) < 5.9 x 10(-3) Gyr(-1) if only data consistent with the CMB are included. This highlights the importance of establishing whether the tension is due to real physical effects or unaccounted systematics, for settling the reach of achievable constraints on decaying DM.