Self-similarity in the conformal framework of quiescent cosmology and the Weyl curvature hypothesis

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The prevalent theory of cosmological inflation, which attempts to explain the high degree of isotropy observed from the Earth’s location in the Universe, has been criticised for being ad hoc and thermodynamically unsound. A viable alternative to inflation is the combined theory of quiescent cosmology and the Weyl curvature hypothesis, in which cosmological models are studied within a mathematical framework that features conformal transformations between physical and unphysical spacetimes. The focus of this thesis is to augment the conformal framework by incorporating a symmetry-related spacetime property known as self-similarity, or scale invariance. An initial obstacle to this purpose is the lack of a satisfactory definition in the literature for asymptotic self-similarity, i.e. approximate self-similarity at early or late times in a cosmological model’s evolution. In this thesis, we conduct an example-driven development of a working definition that is both suitable for use in the conformal framework and sufficiently concordant with existing notions of asymptotic self-similarity. The definition is an asymptotic generalisation of the homothetic equation (which formalises the property of exact self-similarity), and is modified appropriately to generate better agreement with various results in the dynamical systems approach to cosmology. One unavoidable difficulty with our working definition is that the asymptotic selfsimilarity of a specified cosmological model is generally not trivial to determine: the existence of a vector field satisfying given conditions is required under the definition, but no universal method of constructing said vector field is provided. We derive several propositions and theorems that seek to address this problem, although such results are limited in their applicability. After settling on an adequate working definition of asymptotic self-similarity, we employ it in the conformal framework of quiescent cosmology and the Weyl curvature hypothesis. Example spacetimes that have been studied within the framework are examined for self-similarity in this thesis; most significantly, we are able to demonstrate asymptotic self-similarity for the Friedmann–Lemaitre–Robertson–Walker models, i.e. the class of all isotropic and homogeneous cosmological models (with some exceptions). To better understand the characterisation of self-similarity in the conformal framework, we detail the conditions under which it is preserved by conformal transformations. We also investigate the relationships between self-similarity and other symmetryrelated spacetime properties in the framework: many of these properties are shown to be pairwise independent via relevant counterexamples, but whether self-similarity stands completely apart remains an open question. It is hoped that the definition and analysis of asymptotic self-similarity in this thesis will contribute an additional facet to the conformal framework, thereby facilitating further research on quiescent cosmology and the Weyl curvature hypothesis.