  A certain vector-tensor (VT) theory is revisited. It was proposed and
analyzed as a theory of electromagnetism without the standard gauge invariance.
Our attention is first focused on a detailed variational formulation of the
theory, which leads to both a modified Lorentz force and the true energy
momentum tensor of the vector field. The theory is then applied to cosmology. A
complete gauge invariant treatment of the scalar perturbations is presented.
For appropriate gauge invariant variables describing the scalar modes of the
vector field (A-modes), it is proved that the evolution equations of these
modes do not involve the scalar modes appearing in General Relativity
(GR-modes), which are associated to the metric and the energy momentum tensor
of the cosmological fluids. However, the A-modes modify the standard gauge
invariant equations describing the GR-modes. By using the new formalism, the
evolution equations of the A-perturbations are derived and separately solved
and, then, the correction terms --due to the A-perturbations-- appearing in the
evolution equations of the GR-modes are estimated. The evolution of these
correction terms is studied for an appropriate scale. The relevance of these
terms depends on both the spectra and the values of the normalization constants
involved in extended electromagnetism. Further applications of the new
formalism will be presented elsewhere.
