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

## Abstract

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 \(\Delta \chi ^2_\text {all}=-6.76\) in the redshift-binned dark energy model where the \(68~\%\) limits of the equation of state of dark energy read \(w(0\le z\le 0.1)=-1.958_{-0.508}^{+0.509}\), \(w(0.1< z\le 1.5)=-1.006_{-0.082}^{+0.092}\), and here \(w(z>1.5)\) is fixed to \(-1\).

## Keywords

Dark Energy Dark Energy Model Hubble Constant Local Determination Large Magellic Cloud## 1 Introduction

*HST*-based 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 \(H_0\): \(H_0=72.5 \, \pm \, 2.5~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) using three anchors (MW, LMC and NGC 4258) and \(H_0=70.6 \, \pm \, 3.3~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) 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, \(H_0=73.00 \, \pm \, 1.75~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) 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 the \(\Lambda \)CDM model: \(H_0=69.7 \,\pm \, 2.1~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) from WMAP9 [4];^{1} \(H_0=68.0\, \pm \,0.7~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) from WMAP9 + BAO [4]; \(H_0=69.3 \, \pm \, 0.7~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) from WMAP9 + ACT + SPT + BAO [7]; \(H_0=67.3 \, \pm \, 1.0~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) from *Planck* TT + lowP [4]; \(H_0=67.6 \, \pm \, 0.6~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) 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 \(H_0=68.17_{-1.56}^{+1.55}~\text {km}~\text {s}^{-1}~\text {Mpc}^{-1}\) which is consistent with the results from CMB data. Comparing R16 with these cosmological 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 \(\Lambda \)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 12-parameter extension of the \(\Lambda \)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 \(\Lambda \)CDM model where DE is described by a cosmological constant with EOS fixed as \(-1\); the *w*CDM 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 \(\Lambda \)CDM model by combining R16, *Planck* and BAO datasets. A brief summary and discussion are included in Sect. 4.

## 2 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. Here we add the R16 prior to the data combination of *Planck* 2015 data (*Planck* TT, TE, EE + lowP + lensing) [4] and the BAO data including 6dFGS (\(z_{\text {eff}}=0.106\)) [18], MGS (\(z_{\text {eff}}=0.15\)) [19], BOSS DR12 LOWZ (\(z_{\text {eff}}=0.32\)) [20] and CMASS (\(z_{\text {eff}}=0.57\)) [20].

*w*(

*z*)) by

In the \(\Lambda \)CDM model, \(w(z)=-1\) and there are six base cosmological parameters which are denoted by {\(\Omega _bh^2\), \(\Omega _ch^2\), \(100\theta _{\text {MC}}\), \(\tau ,n_s\), \(\text {ln}(10^{10}A_s)\)}. Here \(\Omega _bh^2\) is the physical density of baryons today, \(\Omega _ch^2\) is the physical density of cold dark matter today, \(\theta _{\text {MC}}\) is the ratio between the sound horizon and the angular diameter distance at the decoupling epoch, \(\tau \) 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\(^{-1}\).

*w*CDM model with an arbitrary constant EOS

*w*is the simplest DE extension of the \(\Lambda \)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 \(\Lambda \)CDM model and the EOS of DE is parametrized by

## 3 Results

*Planck*TT, TE, EE + lowP + lensing + BAO + R16.

The \(68~\%\) limits for the cosmological parameters in different DE extensions to \(\Lambda \)CDM model from the data combination of *Planck* TT, TE, EE + lowP + lensing + BAO + R16. The \(\chi ^2\) for different models against individual data are also listed explicitly

– | \(\Lambda \)CDM | | \(w_0w_a\)CDM | \(w_{0.1}w_{1.5}\)CDM | \(w_{0.25}w_{1.5}\)CDM |
---|---|---|---|---|---|

\(\Omega _bh^2\) | \(0.02236_{-0.00015}^{+0.00014}\) | \(0.02223_{-0.00016}^{+0.00015}\) | \(0.02222\pm 0.00015\) | \(0.02227\pm 0.00015\) | \(0.02225_{-0.00016}^{+0.00015}\) |

\(\Omega _ch^2\) | \(0.1180_{-0.0010}^{+0.0011}\) | \(0.1197_{-0.0013}^{+0.0012}\) | \(0.1196\pm 0.0013\) | \(0.1191_{-0.0014}^{+0.0013}\) | \(0.1193\pm 0.0013\) |

\(100\theta _{MC }\) | \(1.04101\pm 0.00029\) | \(1.04082_{-0.00030}^{+0.00031}\) | \(1.04082_{-0.00032}^{+0.00031}\) | \(1.04089\pm 0.00031\) | \(1.04086_{-0.00031}^{+0.00030}\) |

\(\tau \) | \(0.071\pm 0.012\) | \(0.057_{-0.014}^{+0.013}\) | \(0.058\pm 0.015\) | \(0.064\pm 0.014\) | \(0.061\pm 0.014\) |

\({\text {ln}}(10^{10}A_s)\) | \(3.072\pm 0.023\) | \(3.047\pm 0.025\) | \(3.050\pm 0.028\) | \(3.060_{-0.026}^{+0.027}\) | \(3.055_{-0.025}^{+0.026}\) |

\(n_s\) | \(0.9686\pm 0.0041\) | \(0.9643\pm 0.0044\) | \(0.9643_{-0.0044}^{+0.0045}\) | \(0.9659\pm 0.0046\) | \(0.9652\pm 0.0045\) |

EOS\((z=0)\) | – | \(w=-1.113_{-0.055}^{+0.056}\) | \(w_0=-1.185_{-0.211}^{+0.185}\) | \(w_{0.1}=-1.958_{-0.508}^{+0.509}\) | \(w_{0.25}=-1.296_{-0.202}^{+0.203}\) |

– | – | – | \(w_a=0.196_{-0.485}^{+0.664}\) | \(w_{1.5}=-1.006_{-0.082}^{+0.092}\) | \(w_{1.5}=-1.037_{-0.113}^{+0.112}\) |

\({H_0}({\text {km}\cdot \text {s}^{-1}\cdot \text {Mpc}^{-1}})\) | \(68.08_{-0.48}^{+0.47}\) | \(70.71_{-1.49}^{+1.35}\) | \(71.29_{-1.88}^{+1.90}\) | \(74.18_{-2.51}^{+2.54}\) | \(72.07_{-2.01}^{+1.98}\) |

\(\Omega _{\Lambda }\) | \(0.6957\pm 0.0062\) | \(0.7146\pm 0.0107\) | \(0.7190_{-0.0145}^{+0.0163}\) | \(0.7411_{-0.0169}^{+0.0199}\) | \(0.7256_{-0.0146}^{+0.0171}\) |

\(\Omega _m\) | \(0.3043\pm 0.0062\) | \(0.2854\pm 0.0107\) | \(0.2810_{-0.0163}^{+0.0145}\) | \(0.2589_{-0.0199}^{+0.0169}\) | \(0.2744_{-0.0171}^{+0.0146}\) |

\(\chi ^2_{\text {all}}\) | 12,966.04 | 12,962.42 | 12,962.26 | 12,959.28 | 12,962.28 |

\(\chi ^2_{\text {CMB}}\) | 12,957.07 | 12,955.64 | 12,955.50 | 12,954.74 | 12,956.44 |

\(\chi ^2_{\text {BAO}}\) | 2.40 | 4.38 | 5.10 | 4.48 | 5.24 |

\(\chi ^2_{H_0}\) | 6.57 | 2.40 | 1.66 | 0.06 | 0.60 |

Combining R16 with the *Planck* 2015 and BAO data, the constraints on the six parameters in the \(\Lambda \)CDM model almost do not change with respect to the case without adding a local \(H_0\) prior, but \(\chi ^2_{H_0}=6.57\) indicates the existence of tension between the local and global determinations of \(H_0\). For the \(\Lambda \)CDM model, \(\chi ^2_\mathrm{BAO}=2.40\) implies that the \({ Planck}\) 2015 and BAO data are consistent with each other.

Compared to the \(\Lambda \)CDM model, the chi-square is reduced by \(\Delta \chi ^2_{\text {all}}=-3.62\) in *w*CDM model, and a phantom-like DE with EOS of \(w=-1.113_{-0.055}^{+0.056}\) is preferred at more than \(95~\%\) CL. However, in the *w*CDM model, the Hubble constant reads \(70.71_{-1.49}^{+1.35}\) \(\text {km}\cdot \text {s}^{-1}\cdot \text {Mpc}^{-1}\), which is still small compared to the local determination.

*w*CDM 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 \(95~\%\) CL, and \(w_0=-1.185_{-0.211}^{+0.185}\) is consistent with the constraint on *w* in *w*CDM model. Actually, the \(w_{0}w_{a}\)CDM model provides a little bit better fit to the Hubble constant (\(\chi ^2_{H_0}=1.66\), and \(\Delta \chi ^2_{H_0}=-0.74\) compared to *w*CDM model), but the fitting to the BAO datasets becomes worse by \(\Delta \chi ^2_\text {BAO}=0.72\). Thus, from the statistic point of view, the data do not prefer this model.

*w*CDM model. Furthermore, \(H_0=74.18_{-2.51}^{+2.54}\) \(\text {km}\cdot \text {s}^{-1}\cdot \text {Mpc}^{-1}\) and \(\chi ^2_{H_0}=0.06\) indicate that the global fitting is consistent with the local determination of Hubble constant in \(w_{0.1}w_{1.5}\)CDM model. However, the fitting becomes worse in \(w_{0.25}w_{1.5}\)CDM model (\(\Delta \chi ^2_\text {all}=3.00\), \(\Delta \chi ^2_\text {CMB}=1.70\), \(\Delta \chi ^2_\text {BAO}=0.76\), and \(\Delta \chi ^2_{H_0}=0.54\) compared to \(w_{0.1}w_{1.5}\)CDM model). That is to say, all of the individual datasets prefer \(w_{0.1}w_{1.5}\)CDM model compared to \(w_{0.25}w_{1.5}\)CDM model. 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 are illustrated in Fig. 2.

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 \(\Lambda \)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 chi-squares in \(w_{0.25}w_{1.5}\)CDM model compared to \(w_{0.1}w_{1.5}\)CDM model.

## 4 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 \(\Lambda \)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 \(\Delta \chi ^2_\text {all}=-6.76\) compared to the \(\Lambda \)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.

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 \(\tau =0.053_{-0.016}^{+0.011}\), which is smaller than that given by *Planck* TT and lowP power spectra. Here the reionization optical depth in the models extended to the \(\Lambda \)CDM models in this paper prefer a lower reionization optical depth which is more consistent with the recent result released by the *Planck* collaboration.

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 \(\Lambda \)CDM model.

## Footnotes

## Notes

### Acknowledgments

We would like to thank Sai Wang for the helpful conversations. This work is supported by Top-Notch Young Talents Program of China, Grants from NSFC (Grant Nos. 11322545, 11335012 and 11575271) and a Grant from Chinese Academy of Sciences (No. QYZDJ-SSW-SYS006).

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