Résumé
Phys. Rev. D 105, 103503 (2022) The$\Lambda$ CDM prediction of$S_8\equiv\sigma_8(\Omega_m/0.3)^{0.5}$-- where$\sigma_8$is the root mean square of matter fluctuations on a 8$h^{-1}$ Mpc scale -- once calibrated on Planck CMB data is$2-3\sigma$lower than its direct estimate by a number of weak lensing surveys. In this paper, we explore the possibility that the ' $S_8$ -tension' is due to a non-thermal hot dark matter (HDM) fractional contribution to the universe energy density leading to a power suppression at small-scales in the matter power spectrum. Any HDM models can be characterized by its effective mass$ m_{sp}^{\rm eff}$and its contribution to the relativistic degrees of freedom at CMB decoupling$\Delta N_{\rm eff}$ . Taking the specific example of a sterile particle produced from the decay of the inflaton during a matter dominated era, we find that from Planck only the tension can be reduced below$2\sigma$ , but Planck does not favor a non-zero${m_{sp}^{\rm eff},\Delta N_{\rm eff}}$ . In combination with a measurement of$S_8$from KIDS1000+BOSS+2dfLenS, the$S_8$ -tension would hint at the existence of a particle of mass$ m_{sp}^{\rm eff} \simeq 0.67_{-0.48}^{+0.26}$${\rm eV}$with a contribution to$\Delta N_{\rm eff} \simeq0.06\pm0.05$ . However, Pantheon and BOSS BAO/ $f\sigma_8$data restricts the particle mass to$m_{sp}^{\rm eff} \simeq 0.48_{-0.36}^{+0.17}$and contribution to$\Delta N_{\rm eff} \simeq 0.046_{-0.031}^{+0.004}$ . We discuss implications of our results for other canonical non-thermal HDM models -- the Dodelson-Widrow model and a thermal sterile particle with a different temperature in the hidden sector. We report competitive results on such hidden sector temperature which might have interesting implications for particle physics model building, in particular connecting the$S_8$ -tension to the longstanding short baseline oscillation anomaly.