Hyperbolic spaces that detect all strongly-contracting directions

Given a geodesic metric space X, we construct a corresponding hyperbolic space, which we call the contraction space, that detects all strongly contracting directions in the following sense; a geodesic in X is strongly contracting if and only if its parametrized image in the contraction space is a quasi-geodesic. If a finitely generated group G acts geometrically on X, then all strongly-contracting elements act as WPD elements on the contraction space. If the space X is CAT (0), or more generally Morse-dichotomous, that is if all Morse geodesics are strongly-contracting, then all generalized loxodromics act as WPD elements, implying that the action is what we call ``universally WPD''.