Not as big as a barn: Upper bounds on dark matter-nucleus cross sections
Critical probes of dark matter come from tests of its elastic scattering with nuclei. The results are typically assumed to be model-independent, meaning that the form of the potential need not be specified and that the cross sections on different nuclear targets can be simply related to the cross section on nucleons. For point-like spin-independent scattering, the assumed scaling relation is $\sigma_{\chi A} \propto A^2 \mu_A^2 \sigma_{\chi N}\propto A^4 \sigma_{\chi N}$, where the $A^2$ comes from coherence and the $\mu_A^2\simeq A^2 m_N^2$ from kinematics for $m_\chi\gg m_A$. Here we calculate where model independence ends, i.e., where the cross section becomes so large that it violates its defining assumptions. We show that the assumed scaling relations generically fail for dark matter-nucleus cross sections $\sigma_{\chi A} \sim 10^{-32}-10^{-27}\;\text{cm}^2$, significantly below the geometric sizes of nuclei, and well within the regime probed by underground detectors. Last, we show on theoretical grounds, and in light of existing limits on light mediators, that point-like dark matter cannot have $\sigma_{\chi N}\gtrsim10^{-25}\;\text{cm}^2$, above which many claimed constraints originate from cosmology and astrophysics. The most viable way to have such large cross sections is composite dark matter, which introduces significant additional model dependence through the choice of form factor. All prior limits on dark matter with cross sections $\sigma_{\chi N}>10^{-32}\;\text{cm}^2$ with $m_\chi\gtrsim 1\;\text{GeV}$ must therefore be re-evaluated and reinterpreted.