Testing distance duality with CMB anisotropies

We constrain deviations of the form $T\propto (1+z)^{1+\epsilon}$ from the standard redshift-temperature relation, corresponding to modifying distance duality as $D_L=(1+z)^{2(1+\epsilon)} D_A$. We consider a consistent model, in which both the background and perturbation equations are changed. For this purpose, we introduce a species of dark radiation particles to which photon energy density is transferred, and assume $\epsilon\ge0$. The Planck 2015 release high multipole temperature plus low multipole data give the limit $\epsilon<4.5\times 10^{-3}$ at 95% C.L. The main obstacle to improving this CMB-only result is strong degeneracy between $\epsilon$ and the physical matter densities $\omega_{\rm b}$ and $\omega_{\rm c}$. A constraint on deuterium abundance improves the limit to $\epsilon<1.8\times 10^{-3}$. Adding the Planck high-multipole CMB polarisation and BAO data leads to a small improvement; with this maximal dataset we obtain $\epsilon<1.3\times 10^{-3}$. This dataset constrains the present dark radiation energy density to at most 12% of the total photon plus dark radiation density. Finally, we discuss the degeneracy between dark radiation and the effective number of relativistic species $N_{\rm eff}$, and consider the impact of dark radiation perturbations and allowing $\epsilon<0$ on the results.