Anisotropic inflation in non-minimal kinetic coupling model

Abstract: We study anisotropic inflation in non-minimal derivative coupling model where the scalar field non-minimally coupled to the $U(1)$ gauge fields and derivative of the scalar field non-minimally coupled to the Einstein tensor. Within the framework we find power-law anisotropic solutions in this model when both the inflaton potential and the gauge kinetic function are power-law type in the high friction regime. We show the ratio of anisotropy to the expansion rate is nearly constant, small and proportional to the slow-roll parameters of the theory. As a demonstration, we consider numerically calculation of the model to show that the behavior of anisotropy by changing the parameters of the model for quadratic inflationary potential. There is anisotropic attractor solution for a wide range of values of the model parameters. We show both numerically and analytically that there are two phases of inflation, similar to those an anisotropic inflation in minimal coupling model, isotropic and anisotropic phase. We can change the number of e-folds corresponding to each phase of slow roll inflation by changing the gauge coupling constant or non-minimal derivative coupling constant. There are the best agreement with the numerically solutions and analytically solutions in this investigation.