  We show that Einstein's field equations for spatially flat
Friedmann-Robertson-Walker (FRW) space times have a form invariance symmetry
(FIS) realized by the form invariance transformations (FIT) which are indeed
generated by an invertible function of the source energy density. These
transformations act on the Hubble expansion rate, the energy density, and
pressure of the cosmic fluid; likewise such transformations are endowed with a
Lie group structure. Each representation of this group is associated with a
particular fluid and consequently a determined cosmology, so that, the FIS
defines a set of equivalent cosmological models. We focus our seek in the FIT
generated by a linear function because it provides a natural framework to
express the duality and also produces a large sets of cosmologies, starting
from a seed one, in several contexts as for instance in the cases of a perfect
fluid source and a scalar field driven by a potential depending linearly on the
scalar field kinetic energy density.
