  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.
