Topology analysis of the sloan digital sky survey. I. Scale and luminosity dependence

We measure the topology of volume-limited galaxy samples selected from a parent sample of 314,050 galaxies in the Sloan Digital Sky Survey (SDSS), which is now complete enough to describe the fully three-dimensional topology and its dependence on galaxy properties. We compare the observed genus statistic G(v_f) to predictions for a Gaussian random field and to the genus measured for mock surveys constructed from new large-volume simulations of the ΔCDM cosmology. In this analysis we carefully examine the dependence of the observed genus statistic on the Gaussian smoothing scale R_G from 3.5 to 11 h^-1 Mpc and on the luminosity of galaxies over the range -22.50 < M_r < - 18.5. The void multiplicity A_V is less than unity at all smoothing scales. Because A_V cannot become less than 1 through gravitational evolution, this result provides strong evidence for biased galaxy formation in low-density environments. We also find clear evidence of luminosity bias of topology within the volume-limited subsamples. The shift parameter Δv indicates that the genus of brighter galaxies shows a negative shift toward a "meatball" (i.e., cluster dominated) topology, while faint galaxies show a positive shift toward a "bubble" (i.e., void dominated) topology. The transition from negative to positive shift occurs approximately at the characteristic absolute magnitude M_r* = -20.4. Even in this analysis of the largest galaxy sample to date, we detect the influence of individual large-scale structures, as the shift parameter Δv and cluster multiplicity A_C reflect (at ∼ 3 σ) the presence of the Sloan Great Wall and an X-shaped structure that runs for several hundred megaparsecs across the survey volume.