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Extended tachyon field using form invariance symmetry

In this work we illustrate how form-invariance transformations (FIT) can be used to construct phantom and complementary tachyon cosmologies from standard tachyon field universes. We show how these transformations act on the Hubble expansion rate, the energy density, and pressure of the tachyon field. The FIT generate new cosmologies from a known "seed" one, in particular from the ordinary tachyon field we obtain two types of tachyon species, denominated phantom and complementary tachyon. We see that the FIT allow us to pass from a non-stable cosmology to a stable one and vice-versa, as appeared in the literature. Finally, as an example, we apply the transformations to a cosmological fluid with an inverse square potential, $V \propto ϕ^{-2}$, and generate the extended tachyon field.