Anisotropic power-law k-inflation

It is known that power-law k-inflation can be realized for the Lagrangian P = Xg(Y), where X = -(partial derivative phi)(2)/2 is the kinetic energy of a scalar field phi and g is an arbitrary function in terms of Y = Xe-lambda phi/M-pl (lambda is a constant and M-pl is the reduced Planck mass). In the presence of a vector field coupled to the inflaton with an exponential coupling f(phi) alpha e(mu phi)/(Mpl), we show that the models with the Lagrangian P = Xg(Y) generally give rise to anisotropic inflationary solutions with Sigma/H constant, where Sigma is an anisotropic shear and H is an isotropic expansion rate. Provided these anisotropic solutions exist in the regime where the ratio Sigma/H is much smaller than 1, they are stable attractors irrespective of the forms of g(Y). We apply our results to concrete models of k-inflation such as the generalized dilatonic ghost condensate and the Dirac-Born-Infeld model and we numerically show that the solutions with different initial conditions converge to the anisotropic power-law inflationary attractors. Even in the de Sitter limit (lambda -> 0) such solutions can exist, but in this case the null energy condition is generally violated. The latter property is consistent with the Wald's cosmic conjecture stating that the anisotropic hair does not survive on the de Sitter background in the presence of matter respecting the dominant/strong energy conditions.