  We consider the relic density and positivity bounds for freeze-in scalar dark
matter with general Higgs-portal interactions up to dimension-8 operators. When
dimension-4 and dimension-6 Higgs-portal interactions are proportional to mass
squares for Higgs or scalar dark matter in certain microscopic models such as
massive graviton, radion or general metric couplings with conformal and
disformal modes, we can take the dimension-8 derivative Higgs-portal
interactions to be dominant for determining the relic density via the 2-to-2
thermal scattering of the Higgs fields after reheating. We discuss the
implications of positivity bounds for microscopic models. First, massive
graviton or radion mediates attractive forces between Higgs and scalar dark
matter and the resultant dimension-8 operators respect the positivity bounds.
Second, the disformal couplings in the general metric allow for the subluminal
propagation of graviton but violate the positivity bounds. We show that there
is a wide parameter space for explaining the correct relic density from the
freeze-in mechanism and the positivity bounds can curb out the dimension-8
derivative Higgs-portal interactions nontrivially in the presence of the
similar dimension-8 self-interactions for Higgs and dark matter.
