The \(H_0\) and \(\sigma_8\) tensions and the scale invariant spectrum

In a previous communication we showed that a joint analysis of Cosmic Microwave Background (CMB) data and the current measurement of the local expansion rate favours a model with a scale invariant spectrum (HZ) over the minimal \(\Lambda\)CDM scenario provided that the effective number of relativistic degrees of freedom, \(N_{eff}\), is taken as a free parameter. Such a result is basically obtained due to the Hubble Space Telescope (HST) value of the Hubble constant, \(H_0 = 73.24 \pm 1.74\) \(\rm{km.s^{-1}.Mpc^{-1}}\) (68\% C.L.), as the CMB data alone discard the HZ+\(N_{eff}\) model. Although such a model is not physically motivated by current scenarios of the early universe, observations pointing to a scale invariant spectrum may indicate that the origin of cosmic perturbations lies in an unknown physical process. Here, we extend the previous results performing a Bayesian analysis using joint CMB, HST, and Baryon Acoustic Oscillations (BAO) measurements. In order to take into account the well-known tension on the value of the fluctuation amplitude parameter, \(\sigma_8\), we also consider Cluster Number counts (CN) and Weak Lensing (WL) data. We use two different samples of BAO data, which are obtained using two-point spatial (BAO 2PCF) and angular (BAO 2PACF) correlation functions. Our results show that a joint CMB+HST+BAO 2PCF analysis discards the HZP\(+N_{eff}\) model with respect to the minimal \(\Lambda\)CDM scenario whereas the combination CMB+HST+BAO 2PACF favours the former model, even when an extended dataset with NC and WL is considered.