Constraining the Spatial Curvature of the Universe thanks to its Large-Scale Structure

In this work, we study the detectability of potential spatial curvature of the universe with the Euclid satellite. We begin by establishing the theoretical framework, presenting the general expression for the correlation function in both flat and curved space, while also accounting for wide-angle effects. Notably, we show that, in the flat regime, the ratio dipole-to-quadrupole simplifies to the expression $\frac{3}{5}\frac{r}{r_1}$ under the assumption that the comoving distance between surveyed objects $r$ is much smaller than their radial comoving distance $r_1$. Therefore, the ratio is used to survey the Euclid mock catalogues as they don’t include curvature. The comparison and the good matching between the empirical ratio and the theoretical one across different values of $r$ allows first to retrieve the flatness of the catalogues, validating the theoretical model, and then to set the superior limit of this model to $r$ approximately equal to 70 $h^{-1}Mpc$. Moreover, by analysing the errors of these measurements, it appears that at least 2 observations are needed to ensure the detectability of the ratio, meaning that the Euclid satellite will need to survey a volume twice larger than the one actually surveyed to properly detect the ratio and eventually look for deviations from its expected evolution in flatness regime to gain insights into potential curvature. Finally, this ratio is exploited to test cosmologies with different values of $\Omega_{m, 0}$. Among the ones tested, the $\Omega_{m, 0}=0.32$ cosmology is the best fit to the theoretical prediction.