How the Dark Energy Can Reconcile \textit{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 $\textit{Planck}$ CMB data and BAO data through modeling the dark energy variously. We find that the chi-square is significantly reduced by $\Delta\chi^2_\text{all}=-6.76$ in the redshift-binned dark energy model where the $68\%$ limits of the equation of state of dark energy read $w(0\leq z\leq 0.1)=-1.958_{-0.508}^{+0.509}$, $w(0.11.5)$ is fixed to $-1$.


I. INTRODUCTION
Recently Riess et al. in [1] confirmed and improved their former determination of the Hubble constant (H 0 = 73.8 ± 2.4 km s −1 Mpc −1 [2]): H 0 = 73.00 ± 1.75 km s −1 Mpc −1 (1) at 68% confidence level (C.L.) by still using three distance anchors (Milky Way (MW), Large Magellic Cloud (LMC) and NGC 4258). The improvement in uncertainty is due to not only an enlarged number of SN hosts, Cepheids (in the LMC) and MW Cepheids (with HSTbased trigonometric parallaxes) but also a revised distance to NGC 4258 and the LMC. In 2013 Efstathiou revisited the dataset of [2] and yielded another two different value of H 0 : H 0 = 72.5 ± 2.5 km s −1 Mpc −1 using three anchors (MW, LMC and NGC 4258) and H 0 = 70.6 ± 3.3 km s −1 Mpc −1 with NGC 4258 as only anchor in [3]. As emphasized in [1], although these three local determinations of H 0 with the same three anchors are consistent with each other, H 0 = 73.00±1.75 km s −1 Mpc −1 is considered to be the best one (R16, hereafter) due to the aforementioned improvement. On the other hand, there are some widely accepted global determinations of H 0 mainly derived from CMB data under the assumption of ΛCDM model: [4]; H 0 = 67.6 ± 0.6 km s −1 Mpc −1 from Planck TT+lowP+BAO [4]. In addition, in [8] combining the low-redshift and high-redshift isotropic BAO data, the Hubble constant was given by H 0 = 68.17 +1.55 −1.56 km s −1 Mpc −1 which is consistent with the results from CMB data. Comparing R16 with these cos- * huangqg@itp.ac.cn † wangke@itp.ac.cn 1 The constraint from WMAP9 is different the original ones in [5,6] which does not include 0.06 eV neutrino mass.
mological estimates, we find that there is a longstanding tension which becomes more significant now.
Since the global determinations of H 0 are highly model-dependent, there is a well known solution to this tension: adding additional dark radiation to the base ΛCDM 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 12-parameter extension of ΛCDM model is more favored than adding dark radiation. Furthermore, they pointed out that the tension between R16 and the combination of Plank 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 literatures there are several well-known DE models: the ΛCDM 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 Sec. II, we reveal our methodology and cosmological data sets used in this paper. In Sec. III, we globally fit all of the cosmological parameters in various extensions to the ΛCDM model by combining R16, Planck and BAO datasets. A brief summary and discussion are included in Sec. IV.
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 ΛCDM model, w(z) = −1 and there are six base cosmological parameters which are denoted by Here Ω b h 2 is the physical density of baryons today, Ω 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 ΛCDM model. Therefore, there are six base cosmological parameters plus another free parameter w. The CPL model is a widely used DE extension of ΛCDM 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 become negligibly small, we fix the EOS of DE to be −1 for z > 1.5 for convenience. The first redshift-binned 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).

III. RESULTS
Our main results are given in Tab Compared to the ΛCDM model, the chi-square is reduced by ∆χ 2 all = −3.62 in wCDM model, and a phantom-like DE with EOS of w = −1.113 +0.056 −0.055 is preferred at more than 95% C.L.. However, in wCDM model, the Hubble constant reads 70.71 +1.35 −1.49 km · s −1 · Mpc −1 which is still small compared to the local determination.
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 chi-square is almost the same as that in wCDM model. The contour plot of w 0 and w a shows up in Fig. 1. From Fig. 1, the cosmological constant is disfavored at around 95% C.L., and w 0 = −1.185 +0.185 −0.211 is consistent with the constraint on w in wCDM model. Actually, the w 0 w a CDM model provides a little bit better fit to the Hubble constant (χ 2 H0 = 1.66, and ∆χ 2 H0 = −0.74 compared to wCDM model), but the fitting to the BAO datasets becomes worse by ∆χ 2 BAO = 0.72. Thus, from the statistic point of view, the data do not prefer this model.
It is interesting to see that the chi-square for w 0.1 w 1.5 CDM model is significantly reduced by ∆χ 2 all = −6.76 compared to ΛCDM model, and ∆χ 2 all = −3.14 compared to wCDM model.

IV. SUMMARY AND DISCUSSION
To summarize, the tension between the local determination of Hubble constant and the global determination by Planck CMB data and BAO data is statistically significant in the ΛCDM 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 ΛCDM 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 ΛCDM model. See Fig. 3. In particular, Ω 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 ΛCDM model (Ω 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 Ω b h 2 and Ω c h 2 , not Ω m . The constraints on Ω b h 2 and Ω c h 2 for different models are showed in Fig. 4 from which we see that the constraints on Ω b h 2 and Ω 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 reinoization optical depth was τ = 0.053 +0.011 −0.016 which is smaller than that given by Planck TT and lowP power spectra. However, the reionization optical depth in the models extended to the ΛCDM models in this paper prefer a lower reionization optical depth which is more consistent with the recent result released by the Planck collaboration.
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 ΛCDM model.