A fresh look at linear cosmological constraints on a decaying dark matter component

International audienceWe consider a cosmological model in which a fraction $f$ of the Dark Matter(DM) is allowed to decay in an invisible relativistic component, and computethe resulting constraints on both the decay width (or inverse lifetime)$\Gamma$ and $f$ from purely gravitational arguments. We report a fullderivation of the Boltzmann hierarchy, correcting a mistake in previousliterature, and compute the impact of the decay --as a function of thelifetime-- on the CMB and matter power spectra. From CMB only, we obtain thatno more than 3.8 % of the DM could have decayed in the time betweenrecombination and today (all bounds quoted at 95 % CL). We also comment on theimportant application of this bound to the case where primordial black holesconstitute DM, a scenario notoriously difficult to constrain. For lifetimeslonger than the age of the Universe, the bounds can be cast as $f\Gamma <6.3\times10^{-3}$ Gyr$^{-1}$. For the first time, we also checked thatdegeneracies with massive neutrinos are broken when information from the largescale structure is used. Even secondary effects like CMB lensing suffice tothis purpose. Decaying DM models have been invoked to solve a possible tensionbetween low redshift astronomical measurements of $\sigma_8$ and $\Omega_{\rmm}$ and the ones inferred by Planck. We reassess this claim finding that withthe most recent BAO, HST and $\sigma_8$ data extracted from the CFHT survey,the tension is only slightly reduced despite the two additional freeparameters, loosening the bound to $f\Gamma < 15.9\times10^{-3}$ Gyr$^{-1}$.The bound however improves to $f\Gamma < 5.9\times10^{-3}$ Gyr$^{-1}$ if onlydata consistent with the CMB are included. This highlights the importance ofestablishing whether the tension is due to real physical effects or unaccountedsystematics, for settling the reach of achievable constraints on decaying DM