$H_0$ from cosmic chronometers and Type Ia supernovae, with Gaussian Processes and the novel Weighted Polynomial Regression method

TL;DR

This work tackles the $H_0$ tension by employing two independent, model-free reconstruction techniques—Gaussian Processes (GPs) and a novel Weighted Polynomial Regression (WPR)—to infer the current expansion rate from cosmic chronometers and Type Ia supernovae data, while also examining the impact of the local $H_0$ measurement from HST. The authors extend previous GP analyses by incorporating the Pantheon+MCT SnIa data, propagate kernel-hyperparameter uncertainties, and assess SPS-systematics in the cosmic chronometer data, finding $H_0$ values around $67$–$68$ km s$^{-1}$ Mpc$^{-1}$ with reduced uncertainties when SnIa data are included. The WPR method complements GP by weighting an ensemble of cosmographic polynomials through information-criterion-based Bayes factors, yielding a consistent $H_0$ estimate of $H_0\approx 68.9\pm1.96$ km s$^{-1}$ Mpc$^{-1}$, and a constrained version further tightens uncertainties. Collectively, the results favor the Planck-era, lower-$H_0$ regime and show a persistent tension with the Riess et al. local HST value at about $2$–$3\sigma$, reinforcing the view that the HST result may be outlier-like and underscoring the value of model-independent, cross-validated approaches for precision cosmology.

Abstract

In this paper we present new constraints on the Hubble parameter $H_0$ using: (i) the available data on $H(z)$ obtained from cosmic chronometers (CCH); (ii) the Hubble rate data points extracted from the supernovae of Type Ia (SnIa) of the Pantheon compilation and the Hubble Space Telescope (HST) CANDELS and CLASH Multy-Cycle Treasury (MCT) programs; and (iii) the local HST measurement of $H_0$ provided by Riess et al. (2018), $H_0^{\rm HST}=(73.45\pm1.66)$ km/s/Mpc. Various determinations of $H_0$ using the Gaussian processes (GPs) method and the most updated list of CCH data have been recently provided by Yu, Ratra and Wang (2018). Using the Gaussian kernel they find $H_0=(67.42\pm 4.75)$ km/s/Mpc. Here we extend their analysis to also include the most released and complete set of SnIa data, which allows us to reduce the uncertainty by a factor $\sim 3$ with respect to the result found by only considering the CCH information. We obtain $H_0=(67.06\pm 1.68)$ km/s/Mpc, which favors again the lower range of values for $H_0$ and is in tension with $H_0^{\rm HST}$. The tension reaches the $2.71σ$ level. We round off the GPs determination too by taking also into account the error propagation of the kernel hyperparameters when the CCH with and without $H_0^{\rm HST}$ are used in the analysis. In addition, we present a novel method to reconstruct functions from data, which consists in a weighted sum of polynomial regressions (WPR). We apply it from a cosmographic perspective to reconstruct $H(z)$ and estimate $H_0$ from CCH and SnIa measurements. The result obtained with this method, $H_0=(68.90\pm 1.96)$ km/s/Mpc, is fully compatible with the GPs ones. Finally, a more conservative GPs+WPR value is also provided, $H_0=(68.45\pm 2.00)$ km/s/Mpc, which is still almost $2σ$ away from $H_0^{\rm HST}$.

Figures (11)

• Figure 1: Relative differences (as defined in the legend, with X=(G,C)) between the reconstructed functions $H(z)$ obtained with the Gaussian (G) and Matérn (M) kernels (dashed blue curve, $\Delta_G$), and between the ones that are obtained with the Cauchy (C) and the Matérn kernels (solid red curve, $\Delta_C$). The corresponding formulas for the kernels are provided in \ref{['eq:Gaussiankernel']}-\ref{['eq:Maternkernel']}. Notice that the two curves are below $0.6$ in absolute value, what clearly indicates that the reconstructions with the three kernels are fully consistent (at $<1\sigma_M$). In this case we have only included the CCH data, but the differences are even lower when the Pantheon+MCT data or the $H_0^{{\rm HST}}$ measurement are also considered.
• Figure 2: Reconstructed $H(z)$ in [km/s/Mpc] with the corresponding $1\sigma$ bands obtained with the GPs method and the Gaussian kernel \ref{['eq:Gaussiankernel']} when only CCH are used (the two figures on the top), and when we add the Pantheon+MCT Hubble rates from Riess2017 (the ones at the bottom). In the two cases we zoom in the redshift range $z\in[0,0.5]$ (see the two plots on the right) in order to better appreciate our determination for $H_0$ and how the uncertainty bands narrow when the SnIa data are included in the analysis. The processed ($E(z_i)\rightarrow H(z_i)$) Pantheon+MCT data obtained in the last iteration of the recursive process are plotted in green, whereas the CCH data (cf. Table 1) are in red. See related comments in the main text.
• Figure 3: Convergence of the CCH+Pantheon+MCT result for the Hubble parameter as a function of the iteration step of the recursive method described in the text (using the Gaussian kernel), together with the value of $H_0^{\rm HST}$ and corresponding $1\sigma$ bands. The values of $H_0$ are given in [km/s/Mpc].
• Figure 4: Histograms of the hyperparameters $\sigma_f$ (in [km/s/Mpc]) and $l_f$ of the Gaussian kernel \ref{['eq:Gaussiankernel']}, obtained by a Monte Carlo sampling of \ref{['eq:logL']} and by considering only the CCH data. Both distributions are far from peaked, and this justifies our analysis of Sect. 3.3.
• Figure 5: Reconstructed curves of $H(z)$ obtained with the GP method (Gaussian kernel) and three different CCH data sets: (i) the one presented in Table 1 (BC03, in blue); (ii) the one that results from exchanging the BC03 data points from Refs. Moresco2012Moresco2016 by the MaStro ones (MaStro, in red); and (iii) the one obtained by exchanging the BC03 data with the weighted average of the BC03 and MaStro values from the same references, as described in the text, with $\rho_i=1\,\forall{i}$ (BC03/MaStro, in black).