Designing Anisotropic Inflation with Form Fields

We study inflation with anisotropic hair induced by form fields. In four dimensions, the relevant form fields are gauge (one-form) fields and two-form fields. Assuming the exponential form of potential and gauge kinetic functions, we find new exact power-law solutions endowed with anisotropic hair. We also explore the phase space of anisotropic inflation and find fixed points corresponding to the exact power-law solutions. Moreover, we perform the stability analysis around the fixed points to reveal the structure of the phase space. It turns out that one of the fixed points becomes an attractor and others (if any) are saddle points. In particular, the one corresponding to anisotropic inflation becomes an attractor when it exists. We also argue that various anisotropic inflation models can be designed by choosing coupling constants.