Restrictions on Purely Kinetic K-Essence

We restrict purely kinetic k-essence. Assuming that the equation of state is a power law of the kinetic energy, i.e. w=w₀X^α, we find that α must be positive to obtain accelerated phases, constrained from the conditions for stability and causality. In this case the k-essence behaves like a phantom. We also study the evolutions of the equation of state and the speed of sound with numerical simulations.