Mapping Poincar\'{e} gauge cosmology to Horndeski theory for emergent dark energy

The ten-parameter, quadratic Poincar\'e gauge theory of gravity is a plausible alternative to general relativity. We show that the rich background cosmology of the gauge theory is described by a non-canonical bi-scalar-tensor theory in the Jordan frame: the `metrical analogue'. This provides a unified framework for future investigation by the broader community. For many parameter choices, the non-canonical term reduces to a Cuscuton field of the form $\smash{\sqrt{|X^{\phi\phi}|}}$. The Einstein-Cartan-Kibble-Sciama theory maps to a pure quadratic Cuscuton, whereas the teleparallel theory maps to the Einstein-Hilbert Lagrangian. We apply the metrical analogue to novel unitary and power-counting-renormalisable cases of Poincar\'e gauge theory. These theories support the concordance $\Lambda$CDM background cosmology up to an optional, effective dark radiation component, we explain this behaviour in terms of a stalled Cuscuton. We also obtain two dark energy solutions from one of these cases: accelerated expansion from a negative bare cosmological constant whose magnitude is screened, and emergent dark energy to replace vanishing bare cosmological constant in $\Lambda$CDM.