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 } { 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 } { Mpc } } and \sim 230 { h ^ { -1 } { 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 .