Slow-roll and rapid-roll conditions in the spacelike vector field scenario

In this note we derive the slow-roll and rapid-roll conditions for the minimally and nonminimally coupled spacelike vector fields. The function f(B2) represents the nonminimal coupling effect between vector fields and gravity, the f=0 case is the minimal coupling case. For a clear comparison with the scalar field, we define a new function F=±B2/12+f(B2), where B2=AμAμ, 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. As a nonminimal coupling example, the F=0 case just has the same slow-roll conditions as the scalar fields.