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.
