Novel null tests for the spatial curvature and homogeneity of the Universe and their machine learning reconstructions

Abstract: A plethora of observational data obtained over the last couple of decades has allowed cosmology to enter into a precision era and has led to the foundation of the standard cosmological constant and cold dark matter paradigm, known as the $\Lambda$CDM model. Given the many possible extensions of this concordance model, we present here several novel consistency tests which could be used to probe for deviations from $\Lambda$CDM. First, we derive a joint consistency test for the spatial curvature $\Omega_{k,0}$ and the matter density $\Omega_\textrm{m,0}$ parameters, constructed using only the Hubble rate $H(z)$, which can be determined directly from observations. Second, we present a new test of possible deviations from homogeneity using the combination of two datasets, either the baryon acoustic oscillation (BAO) and $H(z)$ data or the transversal and radial BAO data, while we also introduce two consistency tests for $\Lambda$CDM which could be reconstructed via the transversal and radial BAO data. We then reconstruct the aforementioned tests using the currently available data in a model independent manner using a particular machine learning approach, namely the Genetic Algorithms. Finally, we also report on a $\sim 4\sigma$ tension on the transition redshift as determined by the $H(z)$ and radial BAO data.