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 Ansätze 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 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.

Figure 1

Fractional change in CMB temperature (left column) and $E$-mode polarization (right column) lensed angular power spectra, due to DM-baryon interactions, computed with class [29, 30]. Solid lines show the effect of a DM-baryon cross section saturating the CMB-anisotropy 95% upper limits of Ref. [17], and dashed lines show half the effect of twice that cross section. The two lines should overlap when CMB power spectra are linear in the cross section. We see that this is very nearly the case when all the DM is interacting with baryons (i.e., $f_{\chi }=100\%$, in the upper panels), but that CMB power spectra are significantly nonlinear in the cross section when only one percent of the DM is interacting ($f_{\chi }=1\%$, in the lower panels). These power spectra are computed specifically for a DM mass $m_{\chi }=1\mathrm{MeV}$ interacting with all baryons with a Coulomb-like cross section ${\sigma }_{\chi b}(v)\propto v^{-4}$, but we have explicitly checked that the same qualitative conclusions apply to any DM mass $\mathrm{MeV}\le m_{\chi }\le \mathrm{GeV}$, and also for ${\sigma }_{\chi b}(v)\propto v^{-2}$. Specifically, we used ${\sigma }_{\chi b}(v)={\sigma }_{95}{v}^{-4}$, with ${\sigma }_{95}=1.7\times {10}^{-41}{\mathrm{cm}}^2$ and $5.5\times {10}^{-39}{\mathrm{cm}}^2$ for $f_{\chi }=100\%$ and 1%, respectively.

Figure 2

Validity of the linear approximation for ${\sigma }_{\chi b}(v)\propto v^{-4}$ within the interacting-DM mass range, $\mathrm{MeV}\le m_{\chi }\le \mathrm{GeV}$, and for different fractions of interacting DM, $f_{\chi }=\{1\%,10\%,100\%\}$. Left: The shaded regions depict the range of cross-sections ${\sigma }_{\text{NL}}$ for which the $\chi$-$b$ scattering efficiency around recombination ($z_{\text{rec}}\approx 1100$) becomes nonlinear in ${\sigma }_{\chi b}$. Specifically, ${\overline{\rho }}_{\mathrm{DM}}{\mathrm{\Gamma }}_V/H$ becomes $\ge 0.1$ (light orange, \-hatched) and $\ge 1$ (dark orange, $\times$-hatched) for cross-sections ${\sigma }_{\text{NL}}$, where ${\overline{\rho }}_{\text{DM}}$ is the total average DM density, and $H$ is the Hubble expansion rate. The black solid curves show ${\sigma }_{95}$, the 95% C.L. CMB limits on cross-section [$\mathrm{BG}+18$]. Right: The ratio $\mathrm{\Delta }{C}_{\ell }^{TT}(2{\sigma }_{95})/(2\mathrm{\Delta }{C}_{\ell }^{TT}({\sigma }_{95}))$ (at fixed $\ell =1500$) as a function of $m_{\chi }$, for three values of $f_{\chi }$. The closer the solid curves are to 1 (dotted horizontal line), the more linear is the dependence of $\mathrm{\Delta }{C}_{\ell }^{TT}$ on ${\sigma }_{\chi b}$. We checked that the behavior of the curves does not change significantly for the $EE$ power spectrum, nor for any $\ell \gtrsim 1000$, and that the $f_{\chi }=100\%$ curve for ${\sigma }_{\chi b}(v)\propto v^{-2}$ nearly overlaps with the corresponding curve for ${\sigma }_{\chi b}(v)\propto v^{-4}$ shown here.