Enabling Thermal Dark Matter within the Vanilla $L_\mu$-$L_\tau$ Model

Thermal dark matter is a compelling setup that has been probed by a multitude of experiments, mostly in the GeV-TeV mass range. The thermal paradigm in the sub-GeV range is about to experience the same experimental test with the next generation of low-energy accelerators and light dark matter detectors. Motivated by this, we investigate thermal dark matter in the $L_\mu-L_\tau$ and assess how the introduction of a matter-dominated era impacts the parameter that yields the correct relic density. Interestingly, we show that the projected experiments, such as MuSIC, FCC-ee, and LDMX, will probe a large region of the viable parameter space that yields the correct relic density. In the GeV-TeV mass regime, the usual large-scale detectors push the sensitivity. Our work highlights the rich interplay between early-universe dynamics, dark matter phenomenology, and the discovery potential of next-generation experiments.