The Hubble constant $H_0$ are in tension between the CMB measurements from Planck^[1-2] and the type Ia supernova measurements from SH0ES (SNe, $H_0$, for the Equation of State of dark energy).^[3-4] The value of $H_0$ can be determined by the local measurements from Hubble Space Telescope (HST) in a model-independent way, and the model-dependent global fitting from CMB data (see Ref. ^[5] for review on determining the Hubble constant). Riess et al. (2016)^[3] reported a local determination of $H_0=73.24 \pm 1.74 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ (2.4% precision) from Cepheids in the hosts of Type Ia supernovae (SNIa). Recently Riess et al. (2018)^[4] improved the precision to 2.3%, yielding $73.48 \pm 1.66 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$. However, the Planck survey reported $H_0=67.27 \pm 0.66 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ (0.98% precision; TT, TE, EE+lowP) in 2015^[1] and $67.27 \pm 0.60 {\rm km} \cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ (0.89% precision; TT,TE,EE+lowE) in 2018^[2] in the $\Lambda$CDM model. There exist a $3.7\sigma$ tension between the latest results of Planck in the $\Lambda$CDM model and SH0ES. Addison et al. (2016)^[6] have discussed the internal tension inferred from the Planck data itself. They have analyzed the Planck TT power spectra in detail and found that the Hubble constant $H_0 = 69.7\pm1.7 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ at the lower multipoles ($\ell < 1000$) and $H_0 = 64.1\pm1.7 {\rm km} \cdot{\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ at the higher multipoles ($\ell \geq 1000$).

At present it is difficult to explain the $H_0$ disagreement in the standard cosmological model. See Refs. [7--26] for more literature on $H_0$. The tensions among datasets could be due to some underestimated systematic error associated with the experiments. On the other hand, we cannot exclude the possibility of new physics beyond the $\Lambda$CDM cosmology,^[27-33] so the extended models are expected. In order to be independent of Planck and SH0ES measurements, we also call for another independent precise CMB power spectrum measurement, namely WMAP. The 9-year WMAP (TT, TE, EE+BB) reported a 3% precision determination of $H_0=70.0 \pm 2.2 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ in a spatially flat $\Lambda$CDM model.^[34] And in Ref. ^[1] the authors reported $H_0=68.0 \pm 0.7 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ in $\Lambda$CDM model for WMAP+BAO datasets, where the BAO included 6dFGS, SDSS DR7 MGS, BOSS DR11 LOWZ and CMASS. Since more BAO datasets were released after Ref. ^[1] and the analysis in Ref. ^[1] depends on the $\Lambda$CDM model, we extend the discussion in Ref. ^[1] in two aspects: one is that we use the latest BAO measurements from the 6dFGS survey,^[35] the SDSS DR7 MGS,^[36] the BOSS DR12 (9-zbin),^[37] and the eBOSS DR14 measurement;^[38] another is that we consider two cosmological models, namely $\Lambda$CDM and $w$CDM models.

Beyond the standard cosmological model and without SH0ES and Planck measurements, it would be necessary to study the Hubble constant constraints using the latest data. In this work we consider the WMAP data of CMB power spectrum, and combine with the latest BAO measurements (6dFGS+MGS+DR12+DR14). We analyze the data sets in detail to place constraints on $H_0$ in $\Lambda$CDM and $w$CDM model. This paper is organized as follows. In Sec. 2, we will introduce the model and data sets used in this work. In Sec. 3, we present our main results. Conclusions of our work are given in Sec. 4.

In this paper we discuss the spatially flat $\Lambda$CDM and $w$CDM model. We firstly use the 9-year WMAP data (TT,TE,EE+BB) only,^[34] and then combine it with the additional BAO data sets (including the 6dFGS survey,^[35] the SDSS DR7 MGS,^[36] the BOSS DR12,^[37] and the eBOSS DR14 measurement.^[38] Their effective redshifts and constraints are listed in Table 1.

The angular diameter distance takes the form of

For the comoving sound horizon at the end of the baryon drag epoch $z_d$, we take $r_d \equiv r_s (z_d)$.$D_V$ is a combination of the angular diameter distance $D_A(z)$ and Hubble parameter $H(z)$,

Our analysis employs the same Markov Chain Monte Carlo (MCMC) formalism used in previous analyses.^[39-44] We use the CosmoMC package^[45] to sample the parameter space. We explore the WMAP-only and WMAP+BAO likelihood with MCMC simulations of the posterior distribution for the six base parameters as given in Planck Collaboration.^[1-2] The six basic parameters are the baryon density today, $\Omega_b h^2$, the cold dark matter density today, $\Omega_c h^2$, 100 $\times$ approximation to $r_*/D_A$, $100 \theta_{\rm MC}$, the reionization optical depth, $\tau$, the log power of the primordial curvature perturbations, $\ln(10^{10} A_s)$, and the scalar spectrum power-law index, $n_s$. This approach naturally generates the likelihoods of parameters, which are marginalized over all other fitting parameters.

Figures 1 and 2 show 2-dimensional marginalized constraints on the basic MCMC sampling parameters in the $\Lambda$CDM model and $w$CDM model. We explore the posterior of parameters and plot against the following derived parameters (the Hubble constant $H_0$, matter density parameter $\Omega_m$, and late-time clustering amplitude $\sigma_8$). Here we plot the results using 9-year WMAP and combine BAO data sets (6dF+MGS+DR12+DR14, labeled by BAO). The blue contours show the constraints using 9-year WMAP data alone, and the red contours include WMAP+BAO datasets. It is easy to see that addition of BAO changes the constraints on $H_0$, $\Omega_m$, and $\sigma_8$, namely, it can effectively improve the WMAP-only constraints.

Figure 3 presents 68% and 95% likelihood contours of the $\Omega_m$-$H_0$ plane for the WMAP+BAO data sets. The red contours correspond to a flat $\Lambda$CDM model and the blue contours correspond to the $w$CDM model. Figure 4 shows the marginalized likelihood distribution of $H_0$ and summarized the $H_0$ measurements from Planck (2018) and SH0ES (2018). Blue line and red line show constraints in $\Lambda$CDM and $w$CDM model corresponding to 9-year WMAP and WMAP+BAO datasets respectively. Clearly, adding the latest BAO as a complementary to WMAP, our measured $H_0$ results are consistent with the recent data on CMB constraints from Planck (2018), which prefer a value lower than 70 ${\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$.

Table 2 gives the 68% confidence level of cosmological parameters in the $\Lambda$CDM model and $w$CDM model using WMAP and WMAP+BAO datasets. The first group is the base six parameters in $\Lambda$CDM model, which are sampled in the MCMC analysis. In $w$CDM model $w$ is an additional sampling parameters. The second group lists the constraints of three representative derived parameters ($H_0$, $\Omega_m$, and $\sigma_8$). The third group shows the $\chi^2$ of WMAP and each BAO datasets. The column labeled "WMAP" is 9-year WMAP only. The first two columns give results of six parameter $\Lambda$CDM from 9-year WMAP data without and within BAO measurements. The last two columns give the results in $w$CDM framework from WMAP data only and when BAO are added. Adding BAO measurements to WMAP, we constrain cosmological parameters $\Omega_m=0.298\pm0.005$, $H_0=68.36^{+0.53}_{-0.52} {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ (0.78% precision), $\sigma_8=0.8170^{+0.0159}_{-0.0175}$ in a spatially flat $\Lambda$CDM model, and $\Omega_m=0.302\pm0.008$, $H_0=67.63\pm1.30 {\rm km}\cdot {\rm s}^{-1}\cdot{\rm Mpc}^{-1}$ (1.93% precision), $\sigma_8=0.7988^{+0.0345}_{-0.0338}$ in a spatially flat wCDM model. The combined constraint on $w$ from WMAP+BAO in a spatially flat $w$CDM model is $w=-0.96\pm0.07$. Compared with the WMAP alone analysis, the WMPA+BAO analysis reduced the error bar of $H_0$ by 75.4% and 95.3% in $\Lambda$CDM model and $w$CDM model, respectively. However, they are still in 3.1 and $3.5\sigma$ tension with local measurements of Riess et al. (2018) in $\Lambda$CDM and $w$CDM framework, respectively.

In this paper, we determine the Hubble constant $H_0$ using the CMB data from WMAP and the latest BAO measurements (6dFGS+MGS+DR12+DR14) in a spatially flat $\Lambda$CDM and $w$CDM cosmology. Adding BAO measurements to WMAP, we constrain cosmological parameters $\Omega_m=0.298\pm0.005$, $H_0=68.36^{+0.53}_{-0.52} {\rm km}\cdot {\rm s}^{-1}\cdot{\rm Mpc}^{-1}$ (0.78% precision), $\sigma_8=0.8170^{+0.0159}_{-0.0175}$ in a spatially flat $\Lambda$CDM model, and $\Omega_m=0.302\pm0.008$, $H_0=67.63\pm1.30 {\rm km}\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$ (1.93% precision), $\sigma_8=0.7988^{+0.0345}_{-0.0338}$ in a spatially flat $w$CDM model. The combined constraint on $w$ in $w$CDM model from WMAP+BAO datasets is $w=-0.96\pm0.07$. By adding the latest BAO as a complementary to WMAP, our measured results of $H_0$ consistent with the constraint from Planck (2018), which prefers a value lower than 70 km$\cdot {\rm s}^{-1}\cdot {\rm Mpc}^{-1}$. However, there are 3.1 and $3.5\sigma$ tension with local measurement of Riess et al. (2018) in $\Lambda$CDM and $w$CDM framework, respectively. Compared with the WMAP-only analysis, the WMPA+BAO analysis reduces the error bar of $H_0$ by 75.4% in $\Lambda$CDM model and 95.3% in $w$CDM model.

Our results indicate that the combination of WMAP and BAO datasets gives a tight constraint on the Hubble constant comparable to that adopting Planck data. In order to soften the model-dependent constraint using CMB and BAO data, we also extend our analysis to more general dark energy model ($w$CDM cosmology), but there is still a significant tension between the global fitting CMB+BAO datasets and local determination.