Disformal invariance of cosmological observables

We study the frame dependence/independence of cosmological observables under disformal transformations, extending the previous results regarding conformal transformations, and provide the correspondence between Jordan-frame and Einstein-frame variables. We consider quantities such as the gravitational constant in the Newtonian limit, redshift, luminosity and angular diameter distances, as well as the distance-duality relation. Also, the Boltzmann equation, the observed specific intensity, and the adiabaticity condition are discussed. Since the electromagnetic action changes under disformal transformations, photons in the Einstein frame no longer propagate along null geodesics. As a result, several quantities of cosmological interest are modified. Nevertheless, we show that the redshift is invariant and the distance-duality relation (the relation between the luminosity distance and the angular diameter distance) still holds in general spacetimes even though the reciprocity relation (the relation between two geometrical distances) is modified.