Cosmology of a polynomial model for de Sitter gauge theory sourced by a fluid
In the de Sitter gauge theory (DGT), the fundamental variables are the de Sitter (dS) connection and the gravitational Higgs/Goldstone field $ξ^A$. Previously, a model for DGT was analyzed, which generalizes the MacDowell--Mansouri gravity to have a variable cosmological constant $Λ=3/l^2$, where $l$ is related to $ξ^A$ by $ξ^Aξ_A=l^2$. It was shown that the model sourced by a perfect fluid does not support a radiation epoch and the accelerated expansion of the parity invariant universe. In this work, I consider a similar model, namely, the Stelle--West gravity, and couple it to a modified perfect fluid, such that the total Lagrangian 4-form is polynomial in the gravitational variables. The Lagrangian of the modified fluid has a nontrivial variational derivative with respect to $l$, and as a result, the problems encountered in the previous work no longer appear. Moreover, to explore the elegance of the general theory, as well as to write down the basic framework, I perform the Lagrange--Noether analysis for DGT sourced by a matter field, yielding the field equations and the identities with respect to the symmetries of the system. The resulted formula are dS covariant and do not rely on the existence of the metric field.