tokens: panel, the upper right-hand panel and the lower panel of Fig. 3 , respectively. For simplicity, here we consider only four bins. For low z , β is always consistent with a constant value. For SDSS, the 1σ upper bound of β in the last bin deviates from the results given by the full SDSS subsample at 2σ CL, showing that the SDSS subsample favours a decreasing β at high red shift. This conclusion is consistent with the results of Marriner et al. ( 2011 ). For SNLS, the 1σ upper bounds of β in the third and fourth bins deviate from the results of the full SNLS subsample at 1.6σ and 3.3σ CL, respectively. So compared with SDSS, the SNLS subsample favours a time-varying β with a larger decreasing rate. It should be mentioned that this conclusion differs from the results of the full SNLS3 sample (Wang Wang 2013a ). This means that the SNLS3 data set may contain unknown systematic uncertainties (Betoule et al. 2014 ). Figure 3. The 1σ confidence regions of β given by the three subsamples: low z (upper left-hand panel), SDSS (upper right-hand panel) and SNLS (lower panel). The grey region and the grey dashed line are the 1σ region and the best-fitting result given by the full low- z , the full SDSS and the full SNLS subsamples, respectively. The red, green, blue and yellow regions correspond to the 1σ regions of the first, second, third and fourth bins, respectively. In Fig. 4 , we compare the 1σ confidence regions of β given by the JLA without HST data (left-hand panel) with the results given by the full JLA sample (right-hand panel). We can see that the HST subsample only affects the evolution behaviour of β at high red shift. For the case without HST , the 1σ upper bound of β in the last bin deviates from the results given by the full sample at 3.9σ CL. For the full JLA sample, the 1σ upper bound of β in the last bin deviates from the results given by the full JLA sample at 3.6σ CL. This indicates that the HST subsample can slightly slow down the rate of decrease of β at high red shift. Figure 4. The 1σ confidence regions of β given by the JLA without HST data (left-hand panel) and the full JLA sample (right-hand panel) at red-shift region [0,1]. The grey regions and the grey dashed lines denote the 1σ regions and the best-fitting results given by the full samples, respectively. The red, green, blue and yellow regions correspond to the 1σ regions of the first, second, third, and fourth bins, respectively. Next, we discuss the impact of a varying β on parameter estimation. For simplicity, here we consider the standard cosmological model: the ΛCDM model. As shown in Fig. 2 , β prefers a higher value at low red shift and a lower value at high red shift. So we assume that β is related to the red shift by a simple piecewise function: where β 1 and β 2 are two model parameters. In Fig. 5 , using only the full JLA sample, we plot the one-dimensional (1D) marginalized probability distributions of Ω m 0 for constant β and varying β( z ). It can be
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