d'Alfonso del Sordo, Antonio;
(2025)
Special Functions and Dynamical Systems in Cosmology.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
From a mathematical perspective, questions in cosmology often reduce to solving ordinary differential equations. The theories of special functions and dynamical systems provide powerful tools for analysing these equations, enabling us to draw physical conclusions. We investigate two distinct problems: (i) computing the angular distance travelled by a test particle along an azimuthal geodesic in closed Friedmann--Lemaître--Robertson--Walker (FLRW) universes; (ii) studying the dynamical evolution of a cosmological model with a boundary-term coupling. After reviewing exact solutions to the cosmological field equations and deriving the full set of FLRW geodesics, we compute an array of explicit formulae for the path of a photon over one cycle of expansion and recollapse in several scenarios. We begin with the simplest closed FLRW model with an arbitrary linear equation of state parameter and extend our analysis to include two-fluid models, a cosmological constant, and a massive particle. Special functions, particularly elliptic functions, naturally emerge in this context. Next, we apply dynamical systems techniques to cosmological models incorporating a perfect fluid, a scalar field, and boundary terms. After reviewing the use of dynamical systems in dark energy models, we adopt Brown’s formulation of the variational principle for relativistic fluids to introduce novel couplings. We focus on one specific coupling, which is particularly relevant to cosmology, as it behaves similarly to models where dark matter decays into dark energy, particularly in the case of a constant coupling. When the coupling is non-constant, however, it reveals a previously unseen, rich dynamical structure, featuring both early and late-time accelerated expansion. Using well-known variables, we work in a two-dimensional phase space, allowing for a clear physical interpretation and draw analogies with existing models. Whilst the first problem yields self-contained, complete results, the second opens several promising directions for future research.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Special Functions and Dynamical Systems in Cosmology |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10214939 |
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