Self-interacting dark matter offers an interesting alternative to collisionless dark matter because of its ability to preserve the large-scale success of the cold dark matter model, while seemingly solving its challenges on small scales. We present here the first study of the expected dark matter detection signal taking into account different self-scattering models. We demonstrate that models with constant and velocity dependent cross sections, which are consistent with observational constraints, lead to distinct signatures in the velocity distribution, because non-thermalised features found in the cold dark matter distribution are thermalised through particle scattering. Depending on the model, self-interaction can lead to a 10% reduction of the recoil rates at high energies, corresponding to a minimum speed that can cause recoil larger than 300 km/s, compared to the cold dark matter case. At lower energies these differences are smaller than 5% for all models. The amplitude of the annual modulation signal can increase by up to 25%, and the day of maximum amplitude can shift by about two weeks with respect to the cold dark matter expectation. Furthermore, the exact day of phase reversal of the modulation signal can also differ by about a week between the different models. In general, models with velocity dependent cross sections peaking at the typical velocities of dwarf galaxies lead only to minor changes in the detection signals, whereas allowed constant cross section models lead to significant changes. We conclude that different self-interacting dark matter scenarios might be distinguished from each other through the details of direct detection signals. Furthermore, detailed constraints on the intrinsic properties of dark matter based on null detections, should take into account the possibility of self-scattering and the resulting effects on the detector signal.
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