  The annual modulation observed by DAMA/NaI and DAMA/Libra may be interpreted
in terms of elastic or inelastic scattering of dark matter particles. In this
paper we confront these two scenarios within the framework of a very simple
extension of the Standard Model, the Inert Doublet Model (IDM). In this model
the dark matter candidate is a scalar, the lightest component of an extra Higgs
doublet. We first revisit the case for the elastic scattering of a light scalar
WIMP, M_DM~10 GeV, a scenario which requires that a fraction of events in DAMA
are channelled. Second we consider the possibility of inelastic Dark Matter
(iDM). This option is technically natural in the IDM, in the sense that the
mass splitting between the lightest and next-to-lightest neutral scalars may be
protected by a Peccei-Quinn (PQ) symmetry. We show that candidates with a mass
M_DM between ~535 GeV and ~50 TeV may reproduce the DAMA data and have a cosmic
abundance in agreement with WMAP. This range may be extended to candidates as
light as ~50 GeV if we exploit the possibility that the approximate PQ symmetry
is effectively conserved and that a primordial asymmetry in the dark sector may
survive until freeze-out.
