  We consider a model of gravity and matter fields which is invariant only
under unimodular general coordinate transformations (GCT). The determinant of
the metric is treated as a separate field which transforms as a scalar under
unimodular GCT. Furthermore we also demand that the theory is invariant under a
new global symmetry which we call generalized conformal invariance. We study
the cosmological implications of the resulting theory. We show that this theory
gives a fit to the high-z supernova data which is identical to the standard Big
Bang model. Hence we require some other cosmological observations to test the
validity of this model. We also consider some models which do not obey the
generalized conformal invariance. In these models we can fit the supernova data
without introducing the standard cosmological constant term. Furthermore these
models introduce only one dark component and hence solve the coincidence
problem of dark matter and dark energy.
