Investigation of homogeneity and matter distribution on large scales
  using large quasar groups
  We use 12 large quasar group (LQG) samples to investigate the homogeneity of
$0.5\lesssim z \lesssim 2$ Universe ($z$ denotes the redshift). We calculate
the bias factor $b$ and the two-point correlation function $\xi_{{\rm LQG}}$
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 $\Lambda$CDM concordance
model of our Universe with $\Omega_m=0.28$, $\Omega_\Lambda=0.72$, the Hubble
constant $H_0=100h~$km s$^{-1}$ Mpc$^{-1}$ with $h=0.72$ in our calculations.
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 redshift-increasing LQG correlation amplitudes we find is incompatible with
that predicted by the standard theory of structure growth. The homogeneity
scale $R_H$ of the LQG distribution is also deduced. It is defined as the
comoving radius of the sphere inside which the number of LQGs $N(<r)$ is
proportional to $r^3$ within $1\%$, or equivalently above which the correlation
dimension of the sample $D_2$ is within $1\%$ of $D_2=3$. For the NFW dark
matter halo profile, the homogeneity scales of the LQG distribution are
$R_H\simeq 224$ $h^{-1}$Mpc for $0.5< z\leq 1$, $R_H\simeq 316$ $h^{-1}$Mpc for
$1< z\leq 1.5$, and $R_H\simeq 390$ $h^{-1}$Mpc for $1.5< z\lesssim 2$. These
values are above the characteristic sizes of the LQG samples in each bin,
implying the validity of the cosmological principle on the LQG scale, i.e. a
length range of $200\sim 400~h^{-1}$Mpc and a mass scale of $\sim
10^{14}$M$_\odot$. ...
