Decaying dark matter (DDM) has emerged as an interesting framework to extend the $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) model, as many particle physics models predict that dark matter may not be stable over cosmic time and can impact structure formation. In particular, a model in which DM decays at a rate $\mathrm{\Gamma }$ and imprints a velocity kick $v$ onto its decay products leads to a low amplitude of fluctuations, as quantified by the parameter $S_8$, in better agreement with that measured by some past weak lensing surveys. Bayesian analyses have provided mixed conclusions regarding its viability, with a reconstructed clustering amplitude only slightly below the standard $\mathrm{\Lambda }\mathrm{CDM}$ value. In this paper, we contrast previous results with a frequentist analysis of Planck and SDSS baryon acoustic oscillation data. We find that the 68% confidence level region corresponds to a decay half-life of ${6.93}_{-2.85}^{+7.88}\mathrm{Gyr}$ and a velocity kick of ${1250}_{-1000}^{+1450}\mathrm{km}/\mathrm{s}$. These $1\sigma$ constraints strongly differ from their Bayesian counterparts, indicating the presence of volume effect in the Bayesian analysis. Moreover, we find that under the DDM model, the frequentist analysis predicts lower values of $S_8$, in agreement with those found by KiDS-1000 and DES-Y3 at $\sim 1.5\sigma$. We further show that previously derived KiDS-1000 constraints that appeared to exclude the best-fit model from Planck data were driven by priors on the primordial amplitude $A_s$ and spectral index $n_s$. When those are removed from the analysis, KiDS-1000 constraints on the DDM parameters are fully relaxed. It is only when applying Planck-informed priors on $A_s$ and $n_s$ to the KiDS-1000 analysis that one can constrain the model. We further highlight that in the absence of such priors, the region of scales best measured by KiDS-1000 does not exactly match the $S_8$ kernel (centered around $k\sim 0.1h/\mathrm{Mpc}$), but rather a slightly smaller range of scales centered around $k\sim 0.3h/\mathrm{Mpc}$. One must thus be careful in applying $S_8$ constraints to a model instead of the full data likelihood.

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• Cosmic microwave background
• Cosmological parameters
• Large scale structure of the Universe
• Particle dark matter