Tests of the CMB temperature-redshift relation, CMB spectral distortions and why adiabatic photon production is hard

In the expanding Universe, the average temperature of the cosmic microwave background (CMB) is expected to depend like TCMB~(1+z) on redshift z. Adiabatic photon production (or destruction) or deviations from isotropy and homogeneity could modify this scaling and several observational tests have been carried out in response. Here, we explain why `adiabatic' conditions are extremely difficult to establish in the redshift range targeted by these tests. Thus, instead of leading to a simple rescaling of the CMB temperature, a spectral distortion should be produced, which can be constrained using COBE/FIRAS. For scenarios with late photon production, tests of the temperature-redshift relation (TRR) should therefore be reinterpreted as weak spectral distortion limits, directly probing the energy dependence of the photon production process. For inhomogeneous cosmologies, an average y-type distortion is produced, but this type of distortion can be created in several other ways. Here, we briefly discuss possible effects that may help disentangling different contributions to the distortion signal, finding this to be very challenging. We furthermore argue that tests of the TRR, using the SZ effect have limited applicability and that for non-gravitational changes to the TRR the CMB anisotropy spectrum should exhibit an additional y-type dependence.