A fresh look at linear cosmological constraints on a decaying dark
  matter component
  We consider a cosmological model in which a fraction $f$ 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)
$\Gamma$ and $f$ 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 $f\Gamma <
6.3\times10^{-3}$ 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 $\Omega_{\rm
m}$ 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, 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 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.
