The many scales to cosmic homogeneity: Use of multiple tracers from the SDSS

We carry out multifractal analyses of multiple tracers namely the main galaxy sample, the LRG sample and the quasar sample from the SDSS to test the assumption of cosmic homogeneity and identify the scale of transition to homogeneity, if any. We consider the behaviour of the scaled number counts and the scaling relations of different moments of the galaxy number counts in spheres of varying radius $R$ to calculate the spectrum of the Minkowski-Bouligand general dimension $D_{q} (R)$ for $-4 \leq q \leq 4$. The present analysis provides us the opportunity to study the spectrum of the generalized dimension $D_{q}(R)$ for multiple tracers of the cosmic density field over a wide range of length scales and allows us to confidently test the validity of the assumption of cosmic homogeneity. Our analysis indicates that the SDSS main galaxy sample is homogeneous on a length scales of $80\, h^{-1}\, {\rm Mpc} $ and beyond whereas the SDSS quasar sample and the SDSS LRG sample show transition to homogeneity on an even larger length scales at $\sim 150\, h^{-1}\, {\rm Mpc}$ and $\sim 230\, h^{-1}\, {\rm Mpc}$ respectively. These differences in the scale of homogeneity arise due to the effective mass and redshift scales probed by the different tracers in a Universe where structures form hierarchically. Our results reaffirm the validity of cosmic homogeneity on large scales irrespective of the tracers used and strengthens the foundations of the Standard Model of Cosmology.