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 Lambda CDM 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 Delta v 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 Delta v and cluster multiplicity A(C) reflect (at similar to 3 sigma) the presence of the Sloan Great Wall and an X-shaped structure that runs for several hundred megaparsecs across the survey volume.