Coupling metric-affine gravity to a Higgs-like scalar field

General Relativity (GR) exists in different formulations. They are equivalent pure gravity but generically lead to distinct predictions once matter is included. After a brief overview of various versions GR, we focus on metric-affine gravity, which avoids any assumption about the vanishing curvature, torsion or non-metricity. We use it construct an action scalar field coupled non-minimally gravity. It encompasses as special cases numerous previously studied models. Eliminating non-propagating degrees freedom, derive theory metric formulation GR. Finally, give outlook implications for Higgs inflation.