Tensions in cosmology: a discussion of statistical tools to determine inconsistencies

We present a comprehensive analysis of statistical tools for evaluating tensions in cosmological parameter estimates arising from distinct datasets. Focusing on the unresolved Hubble constant ($H_0$) tension, we explore the Pantheon Plus + SH0ES (PPS) compilation, which includes low-redshift Cepheid data from the SH0ES collaboration, and the Cosmic Chronometers (CC) dataset. Employing various tension metrics, we quantitatively assess the inconsistencies in parameter estimates, emphasizing the importance of capturing multidimensional tensions. Our results reveal substantial tension between PPS and Planck 2018 datasets. We highlight the importance of the adoption of these metrics to enhance the precision of future cosmological analyses and facilitate the resolution of existing tensions.