Investigating Cosmic Homogeneity Using Multi-fractal Analysis of the SDSS-IV eBOSS DR16 Quasar Catalog

We analyze the volume-limited subsamples extracted from the sixteenth data release of the SDSS-IV eBOSS quasar survey spanning a redshift interval of $0.8<z<2.2$, to estimate the scale of transition to homogeneity in the Universe. The multi-fractal analysis used for this purpose considers the scaling behavior of different moments of quasar distribution in different density environments. This analysis gives the spectrum of generalized dimension $D_q$, where positive values of $q$ characterize the scaling behavior in over-dense regions and the negative ones in under-dense regions. We expect fractal correlation dimension $D_q(r)=3$, for a homogeneous, random point distribution in 3-Dimensions. The fractal correlation dimension $D_q(r)$, corresponding to $q=2$ obtained in our study stabilizes in the range (2.8-2.9) for scales $r>80$ $h^{-1}$ Mpc. The observed quasar distribution shows consistency with the simulated mock data and the random distribution of quasars within one sigma. Further, the generalized dimension spectrum $D_q(r)$ also reveals transition to homogeneity beyond $>110$ $h^{-1}$ Mpc, and the dominance of clustering at small scales $r<80$ $h^{-1}$ Mpc. Consequently, our study provides strong evidence for the homogeneity in SDSS quasar distribution, offering insights into large-scale structure properties and, thus can play a pivotal role in scrutinizing the clustering properties of quasars and its evolution in various upcoming surveys such as Dark Energy Spectroscopic Instrument (DESI) and Extremely Large Telescope (ELT).