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We carry out multifractal analyses of multiple tracers namely the main galaxysample, the LRG sample and the quasar sample from the SDSS to test theassumption of cosmic homogeneity and identify the scale of transition tohomogeneity, if any. We consider the behaviour of the scaled number counts andthe scaling relations of different moments of the galaxy number counts inspheres of varying radius $R$ to calculate the spectrum of theMinkowski-Bouligand general dimension $D_{q} (R)$ for $-4 \leq q \leq 4$. Thepresent analysis provides us the opportunity to study the spectrum of thegeneralized dimension $D_{q}(R)$ for multiple tracers of the cosmic densityfield over a wide range of length scales and allows us to confidently test thevalidity of the assumption of cosmic homogeneity. Our analysis indicates thatthe 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 sampleshow transition to homogeneity on an even larger length scales at $\sim 150\,h^{-1}\, {\rm Mpc}$ and $\sim 230\, h^{-1}\, {\rm Mpc}$ respectively. Thesedifferences in the scale of homogeneity arise due to the effective mass andredshift scales probed by the different tracers in a Universe where structuresform hierarchically. Our results reaffirm the validity of cosmic homogeneity onlarge scales irrespective of the tracers used and strengthens the foundationsof the Standard Model of Cosmology.

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