Anisotropic $k$-essence

In this paper, we study the late time cosmology of a non-canonical scalar field ($k$-essence) coupled to a vector field in a Bianchi-I background. Specifically, we study three cases: canonical scalar field (quintessence) with exponential potential as a warm up, the dilatonic ghost condensate, and the Dirac Born Infeld field with exponentials throat and potential. By using a dynamical system approach, we show that anisotropic dark energy fixed points can be attractors for a suitable set of parameters of each model. We also numerically integrate the associated autonomous systems for particular initial conditions chosen in the deep radiation epoch. We find that the three models can give an account of an equation of state of dark energy close to $-1$ nowadays. However, a non-negligible spatial shear within the observational bounds nowadays is possible only for thequintessence and the Dirac Born Infeld field. We also find that the equation of state of dark energy and the shear oscillate around nowadays, whenever the coupling of the k-essence field to the vector field is strong enough. The reason of these oscillations is explained in the appendix.