  The '$S_8$ tension' is a longstanding discrepancy between the cosmic
microwave background (CMB) and weak gravitational lensing determination of the
amplitude of matter fluctuations, parametrized as
$S_8\equiv\sigma_8(\Omega_m/0.3)^{0.5}$, where $\sigma_8$ is the root mean
square of matter fluctuations on a 8 $h^{-1}$Mpc scale, and $\Omega_m$ is the
total matter abundance. It was recently shown that dark matter (DM) decaying
into a massless (dark radiation) and a massive (warm DM) species, with a
lifetime $\Gamma^{-1} \simeq 55~(\varepsilon/0.007)^{1.4}$ Gyrs -- where
$\varepsilon$ represent the mass-energy fraction transferred to the massless
component -- can ease the tension. Thanks to a fast and accurate fluid
approximation scheme for the warm species, we perform a comprehensive study of
this 2-body decaying DM scenario, discussing in detail its dynamics and its
impact on the CMB and linear matter power spectra. We then investigate the
implications for the '$S_8$ tension' against a number of changes in the
analysis: different $S_8$ priors, marginalization over the lensing information
in Planck data, trading Planck high$-\ell$ polarization data for those from the
SPTpol and ACTPol surveys, and the inclusion of the recent results from the
Xenon1T collaboration. We conclude that the preference for decaying DM,
apparent only when the $S_8$ value determined from weak lensing data is added
to the analysis, does not sensibly degrade the fit to any of the cosmological
data-sets considered, and that the model could potentially explain the
anomalous electron recoil excess reported by the Xenon1T collaboration.
Furthermore, we explictly show that while current CMB data alone are not
sensitive enough to distinguish between standard CDM and decaying DM,
next-generation CMB observations (CMB-S4) can unambiguously detect its
signature.
