Continuing Dynamical Systems in Homogeneous Cosmologies Through The Big Bang Singularity

Hoffmann, Joshua and Sloan, David (2024) Continuing Dynamical Systems in Homogeneous Cosmologies Through The Big Bang Singularity. PhD thesis, Lancaster University.

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Abstract

General Relativity is the most successful theory of gravity developed so far. Among its many successes lie just as many open questions. This thesis is concerned primarily with the description of cosmological dynamical systems within General Relativity. The Hawking-Penrose theorems have shown General Relativity contains, quite generically, singular solutions for which it is not possible to uniquely and deterministically evolve dynamical systems describing the cosmology through these singular points. Possibly the most well known example of a such a singular point in the spacetime manifold is the Big Bang. Most conventional approaches to resolving this singularity impose some kind of quantization of the gravitational field, as one expects quantum gravity effects to be relevant under such conditions. In this work, we argue that if one takes the relationalist point of view, initially developed by Barbour and Bertotti, in which the measurable dynamical variables are dimensionless ratios, it is possible to find a description of cosmological dynamical systems defined on a contact manifold rather than the typical symplectic manifold of geometric classical mechanics. The framework of contact-reduction is applied to FLRW, Bianchi I and Bianchi IX cosmologies. We show that in the relational, shape space description there exist unique solutions to the equations of motion describing the cosmological dynamical system that pass smoothly through the Big Bang singularity. We also present work on addressing fundamental issues with several classes of cosmological inflation models, with a particular emphasis on the Quartic Hilltop model. These models are typically investigated under assumptions that make them physically unviable. We show that relaxing these assumptions and accounting for how they exit inflation significantly alters their predictions of the spectral index and tensor-to-scalar ratio.