Planck and the local Universe: quantifying the tension

TL;DR

This work formulates a Bayesian tension statistic $\\mathcal{T}$ to quantify the agreement between local, cosmology-independent measurements of $H_0$ and $t_U$ and Planck-derived high-redshift constraints within the $\\Lambda$CDM framework. Applying this to Planck+WP data and local measurements yields strong tension, driven primarily by $H_0$, with $t_U$ remaining broadly consistent. Exploring simple extensions to $\\Lambda$CDM shows some options (notably varying $N_{\\rm eff}$ or the dark-energy EOS $w$) can reduce tension, while others (e.g., nonzero $M_\\nu$) can worsen it; Bayesian model selection generally does not find strong evidence favoring these extensions. The results highlight potential avenues for alleviating tension—through either revised local systematics, new physics beyond $\\Lambda$CDM, or local cosmic variance—while setting the stage for forthcoming polarization data and GAIA-era stellar ages to clarify the discrepancy. The Appendix quantifies the information gain Planck provides relative to WMAP using KL divergence, showing Planck adds meaningful information, especially for the effective number of neutrino species $N_{\\rm eff}$.

Abstract

We use the latest Planck constraints, and in particular constraints on the derived parameters (Hubble constant and age of the Universe) for the local universe and compare them with local measurements of the same quantities. We propose a way to quantify whether cosmological parameters constraints from two different experiments are in tension or not. Our statistic, T, is an evidence ratio and therefore can be interpreted with the widely used Jeffrey's scale. We find that in the framework of the LCDM model, the Planck inferred two dimensional, joint, posterior distribution for the Hubble constant and age of the Universe is in "strong" tension with the local measurements; the odds being ~ 1:50. We explore several possibilities for explaining this tension and examine the consequences both in terms of unknown errors and deviations from the LCDM model. In some one-parameter LCDM model extensions, tension is reduced whereas in other extensions, tension is instead increased. In particular, small total neutrino masses are favored and a total neutrino mass above 0.15 eV makes the tension "highly significant" (odds ~ 1:150). A consequence of accepting this interpretation of the tension is that the degenerate neutrino hierarchy is highly disfavoured by cosmological data and the direct hierarchy is slightly favored over the inverse.

Figures (6)

• Figure 1: Constraints (1 and 2 $\sigma$ joint) in the $t_U$--$H_0$ plane from local measurements (black solid contours, the dashed contours corresponds to the single parameter, marginalized constraint) and CMB data (blue). The transparent set of contours correspond to WMAP and the filled contours to Planck.
• Figure 2: Left panel: blue: curvature extension to the $\Lambda$CDM model, magenta: equation of state parameter for the dark energy $w$ extension to the $\Lambda$CDM model. Right panel: non standard neutrino properties. Green: neutrino mass and primordial helium content extension and Orange: number of effective neutrino species and primordial helium content extension. The plot range and color scheme have been chosen so these figures can be compared directly, at a glance, with Fig. 3 of Ref. local for a direct comparison with WMAP.
• Figure 3: Posterior distributions in the $t_U$--$H_0$ plane for local and CMB measurements. A random sub-sample of the CMB MCMC points has been shown color-coded by the value of the effective number of neutrino species $N_{\rm eff}$ on the top panels. In the left panel the primordial helium fraction is kept fixed at the nucleosynthesis value while on the right it is left as a parameter which is then marginalized. A larger $N_{\rm eff}$ value brings in better agreement the $H_0$ determinations (but the agreement worsens for $t_U$). On the bottom panels we show the $M_{\nu}$ (left) and $w$ (right) extension to the $\Lambda$CDM model.
• Figure 4: $\ln\mathcal{T}$ for Planck and the local measurements as a function of the factor downweighting the measurement of $H_0$ or as a function of a shift in $H_0$.
• Figure 5: $\ln\mathcal{T}$ as a function of $M_{\nu}$ (top panel) and as a function of $N_{\rm eff}$ (bottom panel). The black solid line corresponds to the $\Lambda$CDM value, odds $\sim1\,:\,50$. Note that for values of $M_{\nu}$ higher than $0.15$ eV, the tension between local measurements and Planck derived values increases to highly significant (odds $\sim 1:150$). This indicates that the degenerate hierarchy for the neutrino mass spectrum is highly disfavored and that normal hierarchy is preferred over the inverted one. However no value of $M_{\nu}$ yields non-significant' or even substantial tension