The Slow-Roll and Rapid-Roll Conditions in The Space-like Vector Field Scenario

In this note we derive the slow-roll and rapid-roll conditions for the minimally and nonminimally coupled space-like vector fields. The function f(B) represents the non-minimal coupling effect between vector fields and gravity, the f = 0 case is the minimal coupling case. For a clear comparison with scalar field, we define a new function F = ±B2/12+f(B2) where B = AμA , Aμ is the “comoving” vector field. With reference to the slow-roll and rapid-roll conditions, we find the small-field model is more suitable than the large-field model in the minimally coupled vector field case. And as a non-minimal coupling example, the F = 0 case just has the same slow-roll conditions as the scalar fields. Email: [email protected]