From direct detection to relic abundance: the case of proton-philic
  spin-dependent inelastic Dark Matter
  We discuss strategies to make inferences on the thermal relic abundance of a
Weakly Interacting Massive Particle (WIMP) when the same effective
dimension-six operator that explains an experimental excess in direct detection
is assumed to drive decoupling at freeze-out, and apply them to the
proton-philic Spin-dependent Inelastic Dark Matter (pSIDM) scenario, a
phenomenological set-up containing two states $\chi_1$ and $\chi_2$ with
$m_{\chi_2}>m_{\chi_1}$ that we have shown in a previous paper to explain the
DAMA effect in compliance with the constraints from other detectors. We update
experimental constraints on pSIDM, extend the analysis to the most general
spin-dependent momentum-dependent interactions allowed by non-relativistic
Effective Field Theory (EFT), and consider for the WIMP velocity distribution
in our Galaxy both a halo-independent approach and a standard Maxwellian. The
problem of calculating the relic abundance by using direct detection data to
fix the model parameters is affected by a strong sensitivity on $f(v)$ and by
the degeneracy between the WIMP local density and the WIMP-nucleon scattering
cross section. As a consequence, a DM direct detection experiment is not
directly sensitive to the physical cut-off scale of the EFT, but on some
dimensional combination that does not depend on the actual value of the relic
abundance. However, such degeneracy can be used to develop a consistency test
on the possibility that the WIMP is a thermal relic in the first place. When we
apply it to the pSIDM scenario we find that only a WIMP with a standard
spin-dependent interaction ${\cal O}_{spin}$ with quarks can be a thermal
relic, for a galactic velocity distribution that departs from a Maxwellian.
However all the $\chi_2$ states must have already decayed today, and this
requires some additional mechanism besides that provided by the ${\cal
O}_{spin}$ operator.
