The Warm Dark Matter Doorframe for Light Dark Matter Direct Detection Experiments

If dark matter has even been in sufficient thermal contact with the visible sector and sufficiently light ($m_\chi\lesssim\mathcal{O}(10)~\text{keV}$), the thermal motion inherited from the visible sector will cause significant free streaming effect which is subject to the structure formation constraint, similar to the benchmark thermal warm dark matter model. Here we identify the interaction responsible for such thermal contact to be the interaction probed by the deep underground dark matter direct detection experiments. With the kinetic decoupling technique on the $m_\chi$ vs. $\sigma$ plot we determine the bound shape in detail, and find that recasting the current Lyman-$\alpha$ bound gives a constraint of $m_\chi\gtrsim73~\text{keV}$, and it gets relaxed to $m_\chi\gtrsim35~\text{keV}$ for a smaller cross section of $\sigma<10^{-46}~\text{cm}^2$ with some model dependence. That can be taken as a generic ``no go'' constraint for light dark matter direct detection experiments, and the known caveats are if dark matter is axion-like with an early Bose-Einstein condensation form, or if there is Brownian motion protection of the free streaming.