Consistency of the nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model with the new result from BOSS

Using 137,562 quasars in the redshift range $2.1\le z\le 3.5$ from the data release 11 (DR11) of the baryon oscillation spectroscopic survey (BOSS) of Sloan Digital Sky Survey (SDSS)-III, the BOSS-SDSS collaboration estimated the expansion rate $H(z=2.34)=222\pm 7\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ of the Universe, and reported that this value is in tension with the predictions of flat $\mathrm{\Lambda }\mathrm{CDM}$ model at around a $2.5\sigma$ level. In this paper, we briefly describe some attempts made in the literature to relieve the tension, and show that the tension can naturally be alleviated in a nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model with positive curvature. We also perform the observational consistency check by considering the constraints on the nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model from Planck, WP and BAO data. We find that the nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model constrained with $\text{Planck}+\mathrm{WP}$ data fits better to the line of sight measurement $H(z=2.34)=222\pm 7\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$, but only at the expense of still having a poor fit to the BAO transverse measurements.

Figure 1

Plots of $H(z)/(1+z)$ curves for various models are shown in the redshift range $0\le z\le 3$. Solid black curve: flat $\mathrm{\Lambda }\mathrm{CDM}$ model (1). Dotted purple curve: interacting dark sector model (2). Dashed blue curve: braneworld model (4). Dot-dashed green curve: nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model (5). The error bars in red color represent $H(z=0.57)=92.4\pm 4.5\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (left) [6] and $H(z=2.34)=222\pm 7\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (right) [5], both scaled by the factor $1/(1+z)$.

Figure 2

The one-dimensional marginalized distribution, and two-dimensional contours with 68% C.L. and 95% C.L. for the parameters of the nonflat $\mathrm{\Lambda }\mathrm{CDM}$ model. The vertical dotted green line stands for ${\mathrm{\Omega }}_k=0$.

Figure 3

Plots of $H(z)/(1+z)$ curve with $1\sigma$ and $2\sigma$ error bands for nonflat (left panel) and flat (right panel) $\mathrm{\Lambda }\mathrm{CDM}$ models constrained by $\text{Planck}+\mathrm{WP}$ data.

Figure 4

Plots of $c\ln (1+z)/D_M(z)$ curve with $1\sigma$ and $2\sigma$ error bands for nonflat (left panel) and flat (right panel) $\mathrm{\Lambda }\mathrm{CDM}$ models constrained by $\text{Planck}+\mathrm{WP}$ data.

Figure 5

Plots of $H(z)/(1+z)$ curve with $1\sigma$ and $2\sigma$ error bands for nonflat (left panel) and flat (right panel) $\mathrm{\Lambda }\mathrm{CDM}$ models constrained by $\text{Planck}+\mathrm{WP}+\mathrm{BAO}$ data.

Figure 6

Plots of $c\ln (1+z)/D_M(z)$ curve with $1\sigma$ and $2\sigma$ error bands for nonflat (left panel) and flat (right panel) $\mathrm{\Lambda }\mathrm{CDM}$ models constrained by $\text{Planck}+\mathrm{WP}+\mathrm{BAO}$ data.