Correlation function: biasing and fractal properties of the cosmic web
  We calculated spatial correlation functions of galaxies, $\xi(r)$, structure
functions, $g(r)=1 +\xi(r)$, gradient functions, $\gamma(r)= d \log g(r)/ d
\log r$, and fractal dimension functions, $D(r)= 3+\gamma(r)$, using dark
matter particles of the biased $\Lambda$ cold dark matter (CDM) simulation,
observed galaxies of the Sloan Digital Sky Survey (SDSS), and simulated
galaxies of the Millennium and EAGLE simulations. We analysed how these
functions describe fractal and biasing properties of the cosmic web. The
correlation functions of the biased $\Lambda$CDM model samples at small
distances (particle and galaxy separations), $r \le 2.25$~\Mpc, describe the
distribution of matter inside dark matter (DM) halos. In real and simulated
galaxy samples, only the brightest galaxies in clusters are visible, and the
transition from clusters to filaments occurs at a distance $r \approx 0.8 -
1.5$~\Mpc. Real and simulated galaxies of low luminosity, $M_r \ge -19$, have
almost identical correlation lengths and amplitudes, indicating that dwarf
galaxies are satellites of brighter galaxies, and do not form a smooth
population in voids. The combination of several physical processes (e.g. the
formation of halos along the caustics of particle trajectories and the phase
synchronisation of density perturbations on various scales) transforms the
initial random density field to the current highly non-random density field.
Galaxy formation is suppressed in voids, which increases the amplitudes of
correlation functions and power spectra of galaxies, and increases the
large-scale bias parameter. The combined evidence leads to the large-scale bias
parameter of $L_\star$ galaxies the value $b_\star =1.85 \pm 0.15$. We find
$r_0(L_\star) = 7.20 \pm 0.19$ for the correlation length of $L_\star$
galaxies.
