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 non-minimally coupled space-like vector fields. The function $f(B^{2})$ 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 $F=\pm B^{2}/12+f(B^{2})$ where $B^{2}=A_{\mu}A^{\mu}$, $A_{\mu}$ 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.