We show how two seemingly different theories with a scalar multiplicative coupling to electrodynamics are actually two equivalent parametrisations of the same theory: despite some differences in the interpretation of some phenemenological aspects of
the parametrisations, they lead to the same physical observables. This is illustrated on the interpretation of observations of the Cosmic Microwave Background.
This paper proposes a systematic study of cosmological signatures of modifications of gravity via the presence of a scalar field with a multiplicative coupling to the electromagnetic Lagrangian. We show that, in this framework, variations of the fine
structure constant, violations of the distance duality relation, evolution of the cosmic microwave background (CMB) temperature and CMB distortions are intimately and unequivocally linked. This enables one to put very stringent constraints on possible violations of the distance duality relation, on the evolution of the CMB temperature and on admissible CMB distortions using current constraints on the fine structure constant. Alternatively, this offers interesting possibilities to test a wide range of theories of gravity by analysing several datasets concurrently. We discuss results obtained using current data as well as some forecasts for future data sets such as those coming from EUCLID or the SKA.
