A viable alternative to cosmological inflation is provided by the combined theory of quiescent cosmology and the Weyl curvature hypothesis. We augment the conformal framework of this theory by incorporating the spacetime property of self-similarity. A generalisation of the conformal Killing equation is developed as a definition of asymptotic self-similarity for use in the framework; we derive several propositions and theorems that facilitate the application of this definition, and demonstrate asymptotic self-similarity for FLRW and other models. We also detail the conditions under which self-similarity is preserved by conformal transformations, and investigate its relationship to other symmetry properties in the framework
