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We discuss a scenario where the DAMA modulation effect is explained by aWeakly Interacting Massive Particle (WIMP) which upscatters inelastically to aheavier state and predominantly couples to the spin of protons. In thisscenario constraints from xenon and germanium targets are evaded dynamically,due to the suppression of the WIMP coupling to neutrons, while those fromfluorine targets are evaded kinematically, because the minimal WIMP incomingspeed required to trigger upscatters off fluorine exceeds the maximal WIMPvelocity in the Galaxy, or is very close to it. In this scenario WIMPscatterings off sodium are usually sensitive to the large-speed tail of theWIMP velocity distribution and modulated fractions of the signal close to unityarise in a natural way. On the other hand, a halo-independent analysis withmore conservative assumptions about the WIMP velocity distribution allows toextend the viable parameter space to configurations where large modulatedfractions are not strictly necessary. We discuss large modulated fractions inthe Maxwellian case showing that they imply a departure from the usual cosinetime dependence of the expected signal in DAMA. However we explicitly show thatthe DAMA data is not sensitive to this distortion, both in time and frequencyspace, even in the extreme case of a 100 % modulated fraction. Moreover thesame scenario provides an explanation of the maximum in the energy spectrum ofthe modulation amplitude detected by DAMA in terms of WIMPs whose minimalincoming speed matches the kinematic threshold for inelastic upscatters. Forthe elastic case the detection of such maximum suggests an inversion of themodulation phase below the present DAMA energy threshold, while this is notexpected for inelastic scattering. This may allow to discriminate between thetwo scenarios in a future low-threshold analysis of the DAMA data.

Inelastic dark matter with spin-dependent couplings to protons and large modulation fractions in DAMA