Relativistic effects in the power spectrum of the large scale structure
  The forthcoming Stage-IV experiments aim to map the large scale structure of
the Universe at high precision. The scales explored require a relativistic
description, in addition to statistical tools for their analysis. In this
thesis, we study the effects of adding relativistic and primordial
non-Gaussianity contributions to the power spectrum. We begin by reviewing the
standard cosmology, then we present the cosmological and Newtonian perturbation
theory, which are necessary mathematical tools in the computation of our main
results. Afterwards we present the main contributions to this thesis. First, we
present solutions to the Einstein equations in the long-wavelength
approximation, this allow us to obtain expressions for the relativistic density
power spectrum at second and third order, these expressions also include
contributions from primordial non-Gaussianity, in terms of the parameters
$f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$. These results are complemented with
the well known Newtonian solutions for the density contrast and are used in the
computation of the total (relativistic + Newtonian) one-loop power spectrum.
For completeness we also calculate the bispectrum at tree-level. We discuss the
possibility of these relativistic effects being detectable with the future
surveys considering different limiting values for $f_{\mathrm{NL}}$ and
$g_{\mathrm{NL}}$. Subsequently, we compute the real space galaxy power
spectrum, including relativistic and primordial non-Gaussianity effects. These
effects come from the relativistic one-loop power spectrum terms and from
factors of the non-linear bias parameter $b_{\mathrm{NL}}$. We use our
modelling to assess the ability of Stage-IV surveys to constrain primordial
non-Gaussianity. Finally, we show how this non-linear bias parameter can
effectively renormalize diverging relativistic contributions at large scales.
