Abstract
We calculate spatial correlation functions of galaxies, $\xi(r)$, structure
functions, $g(r)=1 +\xi(r)$, and fractal dimension functions, $D(x)=
3+\gamma(r) = 3+ d g(r)/ d r$, using dark matter particles of biased
$Łambda$ cold dark matter (CDM) simulation, observed galaxies of the Sloan
Digital Sky Survey (SDSS), and simulated galaxies of the Millennium and EAGLE
simulations. We analyse how these functions describe fractal and other
geometrical properties of the cosmic web. Correlation functions of biased
$Łambda$CDM model samples describe at small distances (particle/galaxy
separations), galaxy samples only brightest galaxies in clusters are visible,
and the transition from clusters to filaments occurs at distance $r 0.8
- 1.5$~\Mpc. On larger separations correlation functions describe the
distribution of matter and galaxies in the whole cosmic web. The effective
fractal dimension of the cosmic web is a continuous function of the distance
(separation). On small separations, $ r 2$~\Mpc, the fractal dimension
decreases from $D 1.5$ to $D 0$, reflecting the distribution
inside halos/clusters. The minimum of the fractal dimension function $D(r)$
near $r 2$ is deeper for more luminous galaxies. On medium separations,
$2 r 10$~\Mpc, the fractal dimension grows from $0$ to $\approx
2$, and approaches at large separations 3 (random distribution). Real and
simulated galaxies of low luminosity, $M_r -19$, 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.
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