| Summary: | Observations of the local Universe indicate that we are currently in an accelerated phase of cosmic expansion. This behaviour is compatible with our best theory of gravity, General Relativity, with the addition of a non-luminous form of energy that exerts a negative pressure known as ‘dark energy’. Alternatively, the observed acceleration could be an indication of the presence of new degrees of freedom active on cosmological scales. These degrees of freedom could be in the form of a dynamical field called ‘quintessence’, which is typically isolated from the rest of energy-momentum; although models which allow for interactions with ‘dark matter’ have recently been explored. This thesis explores a more radical idea; that the acceleration is a result of additional gravitational degrees of freedom acting on the largest scales. Any modification to our description of gravity on cosmological scales must take into account that we have very accurate tests of General Relativity within the solar system. Therefore the theories we explore rely on non-linearities that allow them to evade solar system tests via a ‘screening mechanism’. In particular, we focus on the use of special, non-linear, derivative structures which dominate on scales shorter than a characteristic length; suppressing the coupling to energy-momentum and thus screening the effect of the additional force. On the other hand, these interactions are negligible over cosmic scales and we recover a linear theory that communicates a fifth force. A typical theory of this type and the first to be discovered is the ‘Galileon’. We introduce this theory and discuss its behaviour around a static a spherically symmetric source which provides an example of ‘Vainshtein screening’. We introduce Ostrogradsky’s construction which proves that non-degenerate theories with higher order time derivatives always have instabilities. However, by being degenerate, the special structure of Galileons ensures that they evade this result. The same structure is used to construct vector theories with non-linear derivative self-interactions. These theories, named ‘vector Galileons’, break gauge symmetries and have been shown to have interesting cosmological applications. We introduce a way to spontaneously break the gauge symmetry and construct these theories via a Higgs mechanism. In addition to the purely gauge field interactions, our method generates new ghost-free scalar-vector interactions between the Higgs field and the gauge boson. We show how these additional terms are found to reduce, in a suitable decoupling limit, to scalar bi-Galileon interactions between the Higgs field and Goldstone bosons. Our formalism is first developed in the context of abelian symmetry, which allows us to connect with earlier work on the extension of the Proca action. We then show how this formalism is straightforwardly generalised to generate theories with non-abelian symmetry. Using an Arnowitt-Deser-Misner approach, we carefully reconsider the coupling with gravity of vector Galileons, with the aim of studying the necessary conditions to avoid the propagation of ghosts. We develop arguments that put on a more solid footing the results previously obtained in the literature. Moreover, working in analogy with the scalar counterpart, we find indications for the existence of a ‘beyond Horndeski’ theory involving vector degrees of freedom. After identifying the decoupled longitudinal mode of the vector Galileon with the scalar Galileon, we investigate the number of degrees of freedom present in the theory. We discuss how to construct the theory from the extrinsic curvature of the constant scalar field hypersurface, and find a simple expression for the action which guarantees the existence of the primary constraint necessary to avoid the Ostrogradsky instability. We then return to the ‘Galileonic Higgs mechanism’ and consider the effect of interactions between the higher order operators and a dynamical metric. We find a consistent covariantisation through the use of gravitational counter-terms that serve to also restrict the parameter space of the theory. After a brief introduction to cosmological perturbation theory, we explore the cosmological applications of the Galileonic Higgs. We find self-accelerating background solutions, associated with a non-trivial profile of the vector. We then expand the action to quadratic order in linear perturbations, diagonalise and discover that one of the modes is a ghost. This is in contrast with the positive results of related scenarios where an instability on Minkowski space is removed by gravitational interactions.
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