The $H_0$ and $σ_8$ tensions and the scale invariant spectrum
Micol Benetti, Leila L. Graef, Jailson S. Alcaniz
TL;DR
This paper investigates whether a scale-invariant primordial spectrum (HZP) with a free effective number of relativistic species ($N_{eff}$) can alleviate the $H_0$ and $\sigma_8$ tensions by performing a Bayesian comparison against ΛCDM using Planck CMB data, the HST $H_0$ prior, and two BAO analyses (2PCF and 2PACF), supplemented by weak lensing and Sunyaev–Zel’dovich cluster counts. The analysis shows a strong BAO-data dependence: BAO 2PCF disfavors HZP+$N_{eff}$ relative to ΛCDM, while BAO 2PACF favors the HZP+$N_{eff}$ extension, and adding WL/SZ data reinforces these trends. The inferred $H_0$ values for the HZP+$N_{eff}$ scenario tend to align with the local measurements (around $H_0 \sim 73$–$74$ km s$^{-1}$ Mpc$^{-1}$) and $\sigma_8$ values shift in the range $\sim0.85$–$0.89$, illustrating how data choices drive model viability. The work underscores persistent tensions between datasets and the need for more model-independent observations of large-scale structure and the local expansion rate to adjudicate possible new physics versus systematics.
Abstract
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 $Λ$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, $σ_8$, 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 $Λ$CDM scenario whereas the combination CMB+HST+BAO 2PACF favours the former model, even when an extended dataset with NC and WL is considered.