  We consider the positivity bounds for WIMP scalar dark matter with effective
Higgs-portal couplings up to dimension-8 operators. Taking the superposed
states for Standard Model Higgs and scalar dark matter, we show that the part
of the parameter space for the effective couplings, otherwise unconstrained by
phenomenological bounds, is ruled out by the positivity bounds on the
dimension-8 derivative operators. We find that dark matter relic density,
direct and indirect detection and LHC constraints are complementary to the
positivity bounds in constraining the effective Higgs-portal couplings. In the
effective theory obtained from massive graviton or radion, there appears a
correlation between dimension-8 operators and other effective Higgs-portal
couplings for which the strong constraint from direct detection can be evaded.
Nailing down the parameter space mainly by relic density, direct detection and
positivity bounds, we find that there are observable cosmic ray signals coming
from the dark matter annihilations into a pair of Higgs bosons, $WW$ or $ZZ$.
