Testing consistency of $\Omega_b h^2$ in the Planck data

We find that the cosmic microwave background temperature and polarization power spectra measurements from Planck constrain the parameter $\Omega_bh^2$ mostly through: A) the amplitude of Thomson scattering and B) a factor that ensures Thomson scattering does not violate momentum conservation of the baryon-photon fluid. This allows us to obtain two distinct but comparably strong constraints on $\Omega_b h^2$ from the Planck data alone. They are consistent, showing robustness of the Planck $\Omega_b h^2$ constraint. We can alternatively rephrase these constraints as A) the change of the Thomson scattering cross section since recombination is less than $\sim 2\%$ and B) momentum during recombination is conserved to better than $\sim 2\%$ by Thomson scattering. Decoupling the eight various ways in which $\Omega_b h^2$ affects the Planck data leads to $H_0$ only slightly higher than in the standard analysis, $(68.3 \pm 1.6)\,\mathrm{km/s/Mpc}$. The overall consistency of all $\Omega_b h^2$ constraints does not suggest any problem with the standard cosmological model.