Derivation and analysis of gauge transformations for third-order cosmological perturbations.

Cosmological perturbation theory plays a central role in advancing our understanding of the Universe. While linear perturbation theory has long been well established, extending the formalism beyond the linear regime remains one of the most active areas of research in modern theoretical cosmology. In this thesis, we address the problem of gauge choice in the analysis of relativistic matter density perturbations up to third-order. After reviewing the formal framework of cosmological perturbation theory, we derive the gauge transformation rules for scalar, vector, and tensor perturbations of the metric and matter variables up to third-order. Our analysis makes use of several gauges that are widely employed in the literature: the synchronous-comoving gauge, the Poisson gauge, the total matter gauge, the C-gauge, and the comoving-orthogonal gauge. Among these, the synchronous-comoving gauge provides the unique relativistic Lagrangian frame of reference, while the others serve as convenient choices for Eulerian frames. We systematically examine the correspondence between these gauges, construct explicit maps among them, and identify gauge-invariant combinations of perturbations up to third-order. The last part concerns a discussion on how to extend this work in this continually growing field. Our results have in fact direct outcomes for relating theory with observations of the Large Scale Structure of the Universe due to the fact that measurements of the galaxy number density require the knowledge of both gauge and General Relativity effects.