Non-thermal neutrino-like hot dark matter in light of the $S_8$ tension
  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.
