An empirical investigation into cosmological tensions

Abstract: The possibility that the $H_0$ tension is a sign of a physics beyond the $\Lambda$CDM model is one of the most exciting possibilities in modern cosmology. The challenge of solving this problem is complicated by several factors, including the worsening of the tension on $\sigma_8$ parameter when that on $H_0$ is raised. Furthermore, the perspective from which the problem is viewed should not be underestimated, since the tension on $H_0$ can also be interpreted as a tension on the size of the acoustic horizon, $r_s$, which deserves proper discussion. The common approach in the literature consists in proposing a new model that can resolve the tension and treat the new parameters of the theory as free in the analysis. However, allowing additional parameters to vary often results in larger uncertainties on the inferred cosmological parameters, causing an apparent relaxing in the tension due to the broaden in the posterior, instead of a genuine shift in the central value of $H_{0}$. To avoid this, we consider here an empirical approach that assumes specific non-standard values of the $\Lambda$CDM extensions and we analyze how the important parameters in the context of the tension vary accordingly. For our purposes, we study simple extensions of the standard cosmological model, such as a phantom DE component (with Equation of State $w < -1$) and extra relativistic species in the early Universe (so that $N_{eff} > 3.046$). We obtain relations between variation in the value of $w$ and $N_{eff}$ and changes in $H_0$, $r_s$ and $\sigma_8$. In this way an empirical relation between $H_0$ and $\sigma_8$ is provided, that is a first step in understanding which classes of theoretical models, and with which characteristics, could be able to break the correlation between the two tensions.