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

TL;DR

This paper derives slow-roll and rapid-roll conditions for space-like vector fields with minimal and non-minimal coupling, revealing that small-field models are preferable in minimal coupling and that certain non-minimal cases mirror scalar field conditions.

Contribution

It introduces a unified framework for analyzing slow-roll conditions in vector fields, comparing minimal and non-minimal couplings, and highlights the suitability of small-field models.

Findings

Small-field models are more suitable than large-field models in minimal coupling.

The F=0 non-minimal case shares slow-roll conditions with scalar fields.

Derived explicit conditions for slow-roll and rapid-roll in vector field scenarios.

Abstract

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.