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.
