  We explore the potential of future cryogenic direct detection experiments to
determine the properties of the mediator that communicates the interactions
between dark matter and nuclei. Due to their low thresholds and large
exposures, experiments like CRESST-III, SuperCDMS SNOLAB and EDELWEISS-III will
have excellent capability to reconstruct mediator masses in the MeV range for a
large class of models. Combining the information from several experiments
further improves the parameter reconstruction, even when taking into account
additional nuisance parameters related to background uncertainties and the dark
matter velocity distribution. These observations may offer the intriguing
possibility of studying dark matter self-interactions with direct detection
experiments.
