  We combine and extend the analyses of effective scalar, vector, Majorana and
Dirac fermion Higgs portal models of dark matter (DM), in which DM couples to
the Standard Model (SM) Higgs boson via an operator of the form
$\mathcal{O}_{\textrm{DM}}\, H^\dagger H$. For the fermion models, we take an
admixture of scalar $\overline{\psi} \psi$ and pseudoscalar $\overline{\psi}
i\gamma_5 \psi$ interaction terms. For each model, we apply constraints on the
parameter space based on the Planck measured DM relic density and the LHC
limits on the Higgs invisible branching ratio. For the first time, we perform a
consistent study of the indirect detection prospects for these models based on
the WMAP7/Planck observations of the cosmic microwave background, a combined
analysis of 15 dwarf spheroidal galaxies by Fermi-LAT and the upcoming
Cherenkov Telescope Array (CTA). We also perform a correct treatment of the
momentum-dependent direct search cross section that arises from the
pseudoscalar interaction term in the fermionic DM theories. We find, in line
with previous studies, that current and future direct search experiments such
as LUX and XENON1T can exclude much of the parameter space, and we demonstrate
that a joint observation in both indirect and direct searches is possible for
high mass weakly interacting massive particles. In the case of a pure
pseudoscalar interaction of a fermionic DM candidate, future gamma-ray searches
are the only class of experiment capable of probing the high mass range of the
theory.
