Dynamics of Tachyon and Phantom Field beyond the Inverse Square Potentials

We investigate the cosmological evolution of the tachyon and phantom-tachyon scalar field by considering the potential parameter \(\Gamma\)(\(=\frac{V V"}{V'^2}\)) as a function of another potential parameter \(\lambda\)(\(=\frac{V'}{\kappa V^{3/2}}\)), which correspondingly extends the analysis of the evolution of our universe from two-dimensional autonomous dynamical system to the three-dimension. It allows us to investigate the more general situation where the potential is not restricted to inverse square potential and .One result is that, apart from the inverse square potential, there are a large number of potentials which can give the scaling and dominant solution when the function \(\Gamma(\lambda)\) equals \(3/2\) for one or some values of \(\lambda_{*}\) as well as the parameter \(\lambda_{*}\) satisfies condition Eq.(18) or Eq.(19). We also find that for a class of different potentials the dynamics evolution of the universe are actually the same and therefore undistinguishable.