Alleviating the $H_0$ and $\sigma_8$ anomalies with a decaying dark matter model

The Hubble tension between the $\Lambda$CDM-model-dependent prediction of the current expansion rate $H_0$ using Planck data and direct, model-independent measurements in the local universe from the SH0ES collaboration disagree at $>3.5\sigma$. Moreover, there exists a milder $\sim 2\sigma$ tension between similar predictions for the amplitude $S_8$ of matter fluctuations and its measurement in the local universe. As explanations relying on unresolved systematics have not been found, theorists have been exploring explanations for these anomalies that modify the cosmological model, altering early-universe-based predictions for these parameters. However, new cosmological models that attempt to resolve one tension often worsen the other. In this paper, we investigate a decaying dark matter (DDM) model as a solution to both tensions simultaneously. Here, a fraction of dark matter density decays into dark radiation. The decay rate $\Gamma$ is proportional to the Hubble rate $H$ through the constant $\alpha_{\rm dr}$, the only additional parameter of this model. Then, this model deviates most from $\Lambda$CDM in the early universe, with $\alpha_{\rm dr}$ being positively correlated with $H_0$ and negatively with $S_8$. Hence, increasing $\alpha_{\rm dr}$ (and allowing dark matter to decay in this way) can then diminish both tensions simultaneously. When only considering Planck CMB data and the local SH0ES prior on $H_0$, $\sim 1$\% dark matter decays, decreasing the $S_8$ tension to $0.3\sigma$ and increasing the best-fit $H_0$ by $1.6$ km/s/Mpc. However, the addition of intermediate-redshift data (the JLA supernova dataset and baryon acoustic oscillation data) weakens the effectiveness of this model. Only $\sim 0.5$\% of the dark matter decays bringing the $S_8$ tension back up to $\sim 1.5 \sigma$ and the increase in the best-fit $H_0$ down to $0.4$ km/s/Mpc.