Phantom k-essence cosmologies

We devise a method to obtain a phantom version of FRW k-essence cosmologies with homogeneous k-fields by applying form-invariance transformations. It can be seen that the transformation performs the maps $H\text{toH}$ and $\rho +p\to -(\rho +p)$, which in turn give $\gamma \to \gamma$ and $a\to a^{-1}$. The discussion is presented in a general setup, valid for FRW k-essence cosmologies, and then we discuss power-law solutions for illustration purposes. First, we deal with models such that the gradient of the k-field is not constant, which include standard and generalized tachyon cosmologies. We concentrate on the usual tachyon and show the phantom symmetry involves a change in the potential and that it generates an extended super-accelerated tachyon field. Then, we turn our attention to models for which the time derivative of the k-field is not constant, and we show the transformation can be implemented without changing the potential at all.