Gravitational Positivity Bounds on Higgs-Portal Dark Matter

Gravitational positivity bounds are constraints on a renormalizable theory in the presence of a massless graviton, under the assumption that the gravitational theory is ultraviolet-completed by a perturbative string theory. We derive these bounds for the Higgs-portal scalar dark matter model using the forward scattering process $\phi \phi \to \phi \phi$. We find that, in the absence of a dark matter self-coupling, new physics beyond the Higgs-portal dark matter interaction must appear below an energy scale of ${10}^{10}$ GeV if the dark matter mass is smaller than the Higgs boson mass. We further find that, in the presence of both interactions, achieving a cutoff scale at the grand unified theory scale generally requires a dark matter mass of order ${10}^{10}$-${10}^{11}$ GeV (or above), with larger values favored when the four-point self-coupling plays a significant role. For such heavy Higgs-portal dark matter, the observed relic abundance of dark matter in the Universe can be successfully reproduced via the freeze-in mechanism with a tiny Higgs-portal coupling, ${\lambda }_{h\phi }\lesssim 3.5\times {10}^{-11}$. The reheating temperature is then constrained to be $T_{\mathrm{r}\mathrm{e}\mathrm{h}}\lesssim {10}^{14}$ GeV by the positivity bounds on the dark matter mass.