How the dark energy can reconcile Planck with local determination of the Hubble constant

We try to reconcile the tension between the local 2.4 % determination of Hubble constant and its global determination by Planck CMB data and BAO data through modeling the dark energy variously. We find that the chi-square is significantly reduced by Δχall2=-6.76\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \chi ^2_\text {all}=-6.76$$\end{document} in the redshift-binned dark energy model where the 68%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$68~\%$$\end{document} limits of the equation of state of dark energy read w(0≤z≤0.1)=-1.958-0.508+0.509\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(0\le z\le 0.1)=-1.958_{-0.508}^{+0.509}$$\end{document}, w(0.1<z≤1.5)=-1.006-0.082+0.092\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(0.1< z\le 1.5)=-1.006_{-0.082}^{+0.092}$$\end{document}, and here w(z>1.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(z>1.5)$$\end{document} is fixed to -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document}.

Since the global determinations of H 0 are highly modeldependent, there is a well-known solution to this tension: adding additional dark radiation to the base CDM model [1,4,[9][10][11]. Recently in [12] the authors found that a variation of equation of state (EOS) of Dark Energy (DE) in a 12parameter extension of the CDM model is more favored than adding dark radiation. Furthermore, they pointed out that the tension between R16 and the combination of Planck 2015 data and BAO data still exists in their 12-parameter extension. In this short paper we try to reconcile the tension between R16 with the combination of Planck CMB data and BAO data through modeling the DE variously.
In the literature there are several well-known DE models: the CDM model where DE is described by a cosmological constant with EOS fixed as −1; the wCDM model where DE has a constant EOS w; the CPL model in which EOS of DE is time-dependent [13,14]. In general the non-parametric reconstructions of the EOS of DE in bins of redshift, like those in [15][16][17], are considered to be model-independent.
We will investigate whether these DE models can be used to reconcile the tension on the determinations of Hubble constant in this short paper.
The rest of the paper is arranged as follows. In Sect. 2, we reveal our methodology and cosmological datasets used in this paper. In Sect. 3, we globally fit all of the cosmological parameters in various extensions to the CDM model by combining R16, Planck and BAO datasets. A brief summary and discussion are included in Sect. 4.

Data and methodology
In this paper, we try to reconcile the tension between R16 and the global determination of H 0 by CMB data released by Planck collaboration in 2015 and BAO measurements.
We focus on a spatially flat universe and the Friedmann equation reads where the energy density of DE is related to its EOS (w(z)) by The global determination of H 0 highly depends on the evolution of the energy density at low redshift, and consequently on the evolution of EOS of DE at low redshift. In the CDM model, w(z) = −1 and there are six base cosmological parameters which are denoted by { b h 2 , c h 2 , 100θ MC , τ, n s , ln(10 10 A s )}. Here b h 2 is the physical density of baryons today, c h 2 is the physical density of cold dark matter today, θ MC is the ratio between the sound horizon and the angular diameter distance at the decoupling epoch, τ is the Thomson scatter optical depth due to reionization, n s is the scalar spectrum index, and A s is the amplitude of the power spectrum of primordial curvature perturbations at the pivot scale k p = 0.05 Mpc −1 .
The wCDM model with an arbitrary constant EOS w is the simplest DE extension of the CDM model. Therefore, there are six base cosmological parameters plus another free parameter w. The CPL model is a widely used DE extension of the CDM model and the EOS of DE is parametrized by where w 0 and w a are two free parameters in addition to the six base free parameters mentioned above. Usually we use w 0 w a CDM to donate this model. Furthermore, we also consider two model-independent redshift-binned DE models. Since the expansion of the universe is insensitive to DE at high redshifts where the energy density of DE becomes negligibly small, we fix the EOS of DE to be −1 for z > 1.5 for convenience. The first redshiftbinned DE model is denoted by w 0.25 w 1.5 CDM in which the EOS is divided into three bins as follows: where w 0.25 and w 1.5 are two free parameters. On the other hand, we notice that the effective redshift for the lowest-redshift BAO is z eff = 0.106. In order to "effectively" exclude the distance scale constraints from BAO datasets, we also consider another redshift-binned DE model (w 0.1 w 1.5 CDM model) in which the EOS of DE is given by where w 0.1 and w 1.5 are two free parameters. In this case, there is no BAO dataset in the first redshift bin (0 ≤ z ≤ 0.1).

Results
Our main results are given in Table 1  Even though a parameter w a for describing the evolution of the EOS of DE is included in the w 0 w a CDM model, the  We see that the cosmological constant is disfavored at more than 95 % CL in w 0.1 w 1.5 CDM model. Notice that there are two BAO data points in the range of 0.1 < z < 0.25, and they are consistent with Planck CMB data in the CDM model. But the combination of the local determination of Hubble constant and the Planck CMB data prefers a phantom-like DE model. Therefore the EOS of DE w 0.25 < −1 in the range of 0 < z < 0.25 enhances the chisquares in w 0.25 w 1.5 CDM model compared to w 0.1 w 1.5 CDM model.

Summary and discussion
To summarize, the tension between the local determination of Hubble constant and the global determination by Planck Fig. 2 The plots of EOS at z = 0 and w 1.5 in w 0.1 w 1.5 CDM model and w 0.25 w 1.5 CDM model. The crossing point of the two gray dashed lines denotes the CDM model. In particular, the cosmological constant is disfavored at more than 95 % CL in w 0.1 w 1.5 CDM model which gives a better fit to the data with χ 2 = −6.76 compared to the CDM model CMB data and BAO data is statistically significant in the CDM model, and the EOS of DE is preferred to be less than −1 at low redshifts when R16 is added to Planck CMB and BAO data. The chi-square for the w 0.1 w 0.25 CDM model is significantly reduced by χ 2 all = −6.76 compared to the CDM model, and this model can reconcile the tension on determination of Hubble constant between R16 and the combination of Planck CMB data and BAO data.
Since the EOS of DE is preferred to be less than −1 at low redshifts, the matter energy density today becomes smaller than that in the CDM model. See Fig. 3.
In particular, m = 0.2589 +0.0169 −0.0199 in the w 0.1 w 0.25 CDM model, and it is much lower than that in the CDM model ( m = 0.3043 ± 0.0062). One may worry that this model cannot fit the Planck CMB data because of the lower matter density today. Actually CMB power spectra are more sensitive to b h 2 and c h 2 , not m . The constraints on b h 2 and c h 2 for different models are showed in Fig. 4, from which we see that the constraints on b h 2 and c h 2 in all of these models are similar to each other.
Finally, with reduction of large-scale systematic effects in HFI polarization maps, Planck collaboration gave some new constraints on the reinoization optical depth in [21,22] recently. For example, for the likelihood method of Lollipop and the conservative cross-spectra estimators of PCL, the constraint on the reionization optical depth was τ = 0.053 +0.011 −0.016 , which is smaller than that given by Planck TT and lowP power spectra. Here the reionization optical depth in the models extended to the CDM models in this paper prefer a lower reionization optical depth which is more con-  Even though a very rare fluctuation might explain the tension between the local determination and the global determination of Hubble constant [23], it still looks quite un-natural. In a word, if all of the datasets including the local determination of Hubble constant, Planck CMB data and BAO data are reliable, it should imply some new physics beyond the six-parameter base CDM model.