Investigation of Homogeneity and Matter Distribution on Large Scales Using Large Quasar Groups

We use 20 large quasar group (LQG) samples in Park et al. (2015) to investigate the homogeneity of the 0.3 z 1.6 Universe (z denotes the redshift). For comparison, we also employ the 12 LQGs samples at 0.5 z 2 in Komberg et al. (1996) to do the analysis. We calculate the bias factor b and the two-point correlation function ξ_LQG for such groups for three different density profiles of the LQG dark matter halos, i.e. the isothermal profile, the Navarro-Frenk-White (NFW) profile, and the (gravitational) lensing profile. We consider the CDM (cold dark matter plus a cosmological constant ) underlying matter power spectrum with Ω_m = 0.28, Ω_Λ= 0.72, the Hubble constant H₀ = 100 h·km·s^-1·Mpc^-1 with h = 0.72. Dividing the samples into three redshift bins, we find that the LQGs with higher redshift are more biased and correlated than those with lower redshift. The homogeneity scale RH of the LQG distribution is also deduced from theory. It is defined as the comoving radius of the sphere inside which the number of LQGs N(< r) is proportional to r³ within 1%, or equivalently above which the correlation dimension of the sample D₂ is within 1% of D₂ = 3. For Park et al.'s samples and the NFW dark matter halo profile, the homogeneity scales of the LQG distribution are R_H ～ 247 h^-1·Mpc for 0.2 < z ≤ 0.6, R_H ～ 360 h^-1·Mpc for 0.6 < z ≤ 1.2, and R_H ～ 480 h^-1·Mpc for 1.2 < z 1.6. The maximum extent of the LQG samples are beyond RH in each bin, showing that the LQG samples are not homogeneously distributed on such a scale, i.e. a length range of ～ 500 h^-1·Mpc and a mass scale of ～ 10¹⁴M_⊙. The possibilities of a top-down structure formation process as was predicted by the hot/warm dark matter (WDM) scenarios and the redshift evolution of bias factor b and correlation amplitude ξ_LQG of the LQGs as a consequence of the cosmic expansion are both discussed. Different results were obtained based on the LQG sample in Komberg et al. (1996) and the possible reasons for such differences were discussed.