Fractal analysis of the galaxy distribution in the redshift range 0.45 < z < 5.0
Evidence is presented that the galaxy distribution can be described as a fractal system in the redshift range of the FDF galaxy survey. The fractal dimension $D$ was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with $Ω_{m_0}=0.3$, $Ω_{Λ_0}=0.7$ and $H_0=70 \; \mbox{km} \; {\mbox{s}}^{-1} \; {\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number densities $γ$ and $γ^\ast$ obtained from the red and blue FDF galaxies provides a direct method to estimate $D$, implying that $γ$ and $γ^\ast$ vary as power-laws with the cosmological distances. The luminosity distance $d_{\scriptscriptstyle L}$, galaxy area distance $d_{\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against their respective number densities to calculate $D$ by linear fitting. It was found that the FDF galaxy distribution is characterized by two single fractal dimensions at successive distance ranges. Two straight lines were fitted to the data, whose slopes change at $z \approx 1.3$ or $z \approx 1.9$ depending on the chosen cosmological distance. The average fractal dimension calculated using $γ^\ast$ changes from $\langle D \rangle=1.4^{\scriptscriptstyle +0.7}_{\scriptscriptstyle -0.6}$ to $\langle D \rangle=0.5^{\scriptscriptstyle +1.2}_{\scriptscriptstyle -0.4}$ for all galaxies, and $D$ decreases as $z$ increases. Small values of $D$ at high $z$ mean that in the past galaxies were distributed much more sparsely and the large-scale galaxy structure was then possibly dominated by voids. Results of Iribarrem et al. (2014, arXiv:1401.6572) indicating similar fractal features with $\langle D \rangle =0.6 \pm 0.1$ in the far-infrared sources of the Herschel/PACS evolutionary probe (PEP) at $1.5 \lesssim z \lesssim 3.2$ are also mentioned.