Relativistic Redshift Space Distortions In Stage IV Galaxy Surveys

We map the Universe through galaxy redshift surveys. Statistically analysing these galaxy maps allows us access to the nature of dark matter and dark energy. To interpret these galaxy maps, however, the measured redshifts are converted into distances assuming a homogeneous and isotropic universe. That assumption ignores the rich structure of the Universe and thus contaminates the estimated distances through redshift space distortions (RSD). The leading order RSD is caused by the peculiar velocities of the sources which in turn are sourced by gravitational instability. These RSD change the observed clustering patterns of galaxies which allows us to probe the nature of gravity. However, the photons emitted by distant galaxies are also affected by the inhomogeneities they encounter on their way to the observer. These additional relativistic RSD include multiple effects that alter the radial and transversal positions as well as the observed fluxes of galaxies. While these effects are subdominant at small scales, they gain prominence at the larger scales that will be probed by Stage IV surveys such as Euclid and SKAO2, which are poised to explore unprecedentedly vast regions of the late Universe. This thesis investigates these relativistic RSD in-depth, addressing two central questions: Can they be detected in Stage IV galaxy surveys? and can they be used as novel probes to measure cosmological parameters? To these ends, we adopt the LIGER method which produces galactic mock catalogues tailored to match user-defined surveys after accounting for relativistic RSD at linear order in cosmological perturbations. To answer the first inquiry, we analyze the impact of relativistic RSD on the angular power spectrum, two-point correlation function, and power spectrum multipoles measured from mock catalogues that match the radial and angular distribution of the Euclid Wide Spectroscopic Survey. We quantitatively assess the significance of relativistic RSD, concentrating specifically on wide-angle effects, weak lensing and velocity of the observer effects. We use the latter's impact on the power spectrum multipoles, dubbed the 'Finger of the Observer' (FOTO) effect, as our key in answering the second inquiry. Using mock Euclid and SKAO2 catalogues, we find that the FOTO effect can be used to measure both the magnitude and direction of the peculiar velocity of the observer as well as constrain cosmological parameters in optimistic conditions.