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 \geq 0 .
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 _ { b } and \omega _ { 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 _ { eff } , and consider the impact of dark radiation perturbations and allowing \epsilon < 0 on the results .