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

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