Test $Λ$CDM model with High Redshift data from Baryon Acoustic Oscillations
Abstract
The Baryon Acoustic Oscillations (BAO) provide a standard ruler for studying cosmic expansion. The recent observations of BAO in SDSS DR9 and DR11 take measurements of at several different redshifts. It is argued that the behavior of dark energy could be constrained more effectively by adding highredshift Hubble parameter data, such as the SDSS DR11 measurement of km/sec/Mpc at z = 2.34. In this paper, we investigate the significance of these BAO data in the flat CDM model, by combining them with the recent observational data of the Hubble constant from local distance ladder and the Cosmic Microwave Background (CMB) measurements from Planck+WP. We perform a detailed data analysis on these datasets and find that the recent observations of BAO in SDSS DR9 and DR11 have considerable tension with the Planck + WP measurements in the framework of the standard CDM model. The fitting results show that the main contribution to the tension comes from the Hubble parameter measurement at redshift of . But there is no visible tension once the joint data analysis by combining the datasets of SDSS and Planck+WP is performed. Thus in order to see whether dark energy does evolve, we need more independent measurements of the Hubble parameter at high redshifts.
Subject headings:
expansion history — cosmology: observations — methods: statistical1. Introduction
The cosmic acceleration was discovered in 1998 (Riess et al., 1998; Perlmutter et al., 1999), and it has been confirmed by several independent observations. The standard CDM model provides a succinct description of the cosmic acceleration matching current Cosmic Microwave Background (CMB) observations (Planck XVI, 2013). Besides the CMB measurements, the Baryon Acoustic Oscillation (BAO) measurements are recognized as robust and independent probes of cosmology, since they provide a standard ruler for studying the cosmic expansion. Recently, Delubac et al. (2014) reported a detection of the BAO feature in the fluxcorrelation function of the Ly of highredshift quasars from the Data Release 11 (DR11) of the Baryon Oscillation Spectroscopic Survey (BOSS) (Dawson et al., 2013) of SDSSIII (Eisenstein et al., 2011). By adopting the value of sound horizon at the drag epoch Mpc from the Planck+WP fitting of the concordance cosmology, Delubac et al. (2014) derive a highredshift Hubble parameter km/sec/Mpc. As the measurements of parameter at different redshifts are statistically independent, they are suitable for studying the evolution of the cosmic geometry. With these highredshift data, Sahni et al. (2014) test the cosmological constant hypothesis by employing the evolution of according to the recent observations of BAO’s in SDSS DR9 (Samushia et al., 2013) and DR11 (Delubac et al., 2014). By adopting an improved version of the diagnostic, Sahni et al. (2014) find that there is considerable tension with the Planck+WP measurements in the framework of the standard CDM model. By the diagnostic, the value is independent of the redshift if dark energy is a constant. However, they find that , while the Planck result is (Planck XVI, 2013). Therefore, the CDM model is in conflict with the Hubble parameter measurement at redshift , at the level of two standard deviation.
In this paper, we investigate this tension through joint data analysis instead of the diagnostic, in order to compare the significance of these highredshift BAO data with that of the Planck data.
2. Methodology & Results
For comparison, we use the same dataset of (Sahni et al., 2014), including the Hubble constant from local distance ladder ( km/sec/Mpc (Efstathiou, 2014)), the Hubble parameter from anisotropic clustering of SDSS DR9 ( km/sec/Mpc (Reid et al., 2012)) and the new data from the forest of SDSS DR11 quasars ( km/sec/Mpc at (Delubac et al., 2014)).
As we noted above, the highredshift Hubble parameter is derived by scaling at a from the Planck+WP measurements, so the value of is correlated with the Planck+WP dataset. To avoid this correlation, we also adopt the most precisely determined combination from (Delubac et al., 2014), namely
(1) 
where and are defined as
(2) 
where the fiducial values and at are 8.708 and 11.59 respectively. For the SDSS DR9 dataset, the result that obtained by Anderson et al. (2012) will be used. For a further comparison, we also employ the improved measurements of SDSS DR11 dataset, which gives and (Anderson et al., 2013) respectively.
In the following context, we will use “H0”, “Hz”, “DR9”, “DR11”,“DR11Ly”, and “Planck+WP” to represent the Hubble constant, the three data, of SDSS DR9, and of SDSS DR11, of SDSS DR11 at reshift and the Planck+WP dataset respectively. For convenience, we will use “SDSS” to represent the combination of “Hz”, “DR9”, “DR11” and “DR11Lya” datasets.
With these datasets, we perform analysis and explore the parameter space using Markov Chain Monte Carlo (MCMC) algorithm by modifying the CosmoMC package (Lewis & Bridle, 2002).
Table 1 and Table 2 summarize the fitting results of the parameter constraints (the 68% CL limits) and corresponding ’s. In Figure 1, in order to make a comparison, we plot the likelihood distributions of with two different selections : the Hz and Planck+WP datasets (corresponding the upper panel) ; the DR9, DR11 and Planck+WP datasets (corresponding the below panel).
Dataset  Hz  +WP  H0+H(z=0.57)++WP  Hz++WP 

We list the 68% CL limits.
Dataset  H0+DR9+DR11Ly  H0+DR9++WP  H0+DR9+DR11Ly++WP  H0+DR11+DR11Ly++WP 

We list the 68% CL limits.
Sahni et al. (2014) show that the estimation of the new diagnostic from SDSS DR9 and DR11 data gives , having tension with the value determined for CDM from Planck+WP at over . From Table 1 and the upper panel of Figure 1, we can see that the Hz and the Planck+WP give constraints and respectively, which means the Hz dataset does have tension with the Planck+WP dataset at over . However, when using the Hz+Planck+WP and H0+H(z=0.57)+Planck+WP datasets, the constraints are and respectively. Apparently, they are both consistent with the constraint of Planck+WP dataset.
Table 2 and the lower panel of Figure 1 show the constraints with the corresponding DR9, DR11, DR11Ly and Planck+WP datasets. When we use the measurement of SDSS DR9 in place of the Hubble parameter , the tension is alleviated, this is because the measurement of of (Reid et al., 2012) includes the anisotropic information of galaxy clustering while DR9 measurement of (Anderson et al., 2012) does not. Thus the constraint on of is tighter than DR9 measurement. For a further comparison, we replace the DR9 data with the improved measurements and of DR11 data, the constraint becomes a little tighter, but the tension is still alleviated when compared with the constraint by data.
An obvious feature can be seen from both Table 1 and Table 2, that is the contribution to ’s is mainly dominated by the Planck+WP data. These ’s just change relatively small values by adding these SDSS data. The results reveal that the highredshift BAO data is of less importance compared with the Planck+WP data on constraining the standard CDM model.
For the CDM model,the bestfit (black line) and 2 constraints (filled region) of the reconstructed evolution of constrained by the Hz+Planck+WP data are plotted in Figure 2 respectively. The three data points with their error bars are marked as well. We find that the value given by high redshit BAO data deviates from the reconstructed 2 region of constrained by the Hz+Planck+WP dataset.
We notice that there are similar features described in (Sahni et al., 2014). This is consistent with the results shown in Figure 1: the difference between the constraints by Planck+WP data and that by combining with other SDSS datasets is not much. So the constraints of are also mainly dominated by the Planck+WP data. We conclude that the significance of these highredshift BAO data is less than that of the Planck data.
3. Summary & Conclusion
Using only the SDSS data, the contribution to the Hubble parameter constraint is mainly dominated by the data. However, when we combine the SDSS and the Planck results, the fitting is found to be mostly influenced by the low redshift data and the Planck data, so the CDM model looks fine. These results indicate that the single high redshift measurement of is at variance with other data, in order to see whether dark energy does evolve, we need more independent measurements of the Hubble parameter at hight redshifts.
Footnotes
 affiliation: State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190
 affiliation: Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing, 100190
 affiliation: School of Astronomy and Space Science, Sun YatSen University, Guangzhou 510275, People’s Republic of China
 affiliation: State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190
 affiliation: State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190
 affiliation: Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing, 100190
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