Exact treatment of weak dark matter-baryon scattering for linear-cosmology observables

Elastic scattering of dark matter (DM) particles with baryons induce cosmological signals that may be detectable with modern or future telescopes. For DM-baryon scattering cross sections scaling with negative powers of relative velocity, $\sigma_{\chi b}(v) \propto v^{-2}, v^{-4}$, such interactions introduce a momentum-exchange rate that is nonlinear in DM-baryon bulk relative velocities, thus not amenable for inclusion as-is into standard linear cosmological Boltzmann codes. Linear ansatzes have been adopted in past works, but their accuracy is unknown as they do not arise from first-principles derivations. In this work, for the first time, we construct a rigorous framework for computing linear-cosmology observables as a perturbative expansion in $\sigma_{\chi b}$. We argue that this approach is accurate for Cosmic Microwave Background (CMB) angular power spectra when most or all of the DM is scattering with baryons with cross section $\sigma_{\chi b}(v) \propto v^{-2}, v^{-4}$. We derive exact formal expressions for CMB power spectra at linear order in $\sigma_{\chi b}$, and show that they only depend on a specific velocity integral of the momentum-exchange rate. Consequently, we can obtain the exact power spectra at linear order in $\sigma_{\chi b}$ by substituting the original nonlinear momentum-exchange rate with a uniquely specified linear rate. Serendipitously, we find that the exact substitution we derive from first principles precisely coincides with the most widely used linear ansatz, thus placing previous CMB-anisotropy upper bounds on a more solid footing. In addition to finally providing an exact cosmological solution to the DM-baryon scattering problem in a well-defined region of parameter space, the framework we construct opens the way to computing higher-order correlation functions, beyond power spectra, which are promising yet unexplored probes of DM-baryon scattering.