The non-minimal 3-form cosmology and the rise of the cuscuton

A bstract We consider the 3-form theory with non-minimal coupling to gravity in an expanding Universe. First, we assume that the background is homogeneous and isotropic, and that the three-form is coupled to both the Ricci scalar and the Ricci tensor. We show that in this case, it propagates three degrees of freedom: a scalar mode and two tensor ones. Then, we consider an anisotropic background that corresponds to a Bianchi Type I Universe, and set the coupling with the Ricci tensor to zero. We show that, similarly to the Proca theory with non-minimal coupling to gravity, this case leads to two branches for the background solutions — depending on the values of the 3-form. However, in contrast to the Proca case, we show that no extra modes appear. We explore the no-ghost conditions and speed of propagation for all three modes in both branches. Finally, we show that one of the branches can be written as a theory of a constrained scalar, coupled to a cuscuton field.