Topological analysis of the CfA redshift survey

We study the topology of large-scale structure in the Center for Astrophysics Redshift Survey, which now includes approximately 12,000 galaxies with limiting magnitude m(sub B) is less than or equal to 15.5. The dense sampling and large volume of this survey allow us to compute the topology on smoothing scales from 6 to 20/h Mpc; we thus examine the topology of structure in both 'nonlinear' and 'linear' regimes. On smoothing scales less than or equal to 10/h Mpc this sample has 3 times the number of resolution elements of samples examined in previous studies. Isodensity surface of the smoothed galaxy density field demonstrate that coherent high-density structures and large voids dominate the galaxy distribution. We compute the genus-threshold density relation for isodensity surfaces of the CfA survey. To quantify phase correlation in these data, we compare the CfA genus with the genus of realizations of Gaussian random fields with the power spectrum measured for the CfA survey. On scales less than or equal to 10/h Mpc the observed genus amplitude is smaller than random phase (96% confidence level). This decrement reflects the degree of phase coherence in the observed galaxy distribution. In other words the genus amplitude on these scales is not good measure of the power spectrum slope. On scales greater than 10/h Mpc, where the galaxy distribution is rougly in the 'linear' regime, the genus ampitude is consistent with the random phase amplitude. The shape of the genus curve reflects the strong coherence in the observed structure; the observed genus curve appears broader than random phase (94% confidence level for smoothing scales less than or equal to 10/h Mpc) because the topolgoy is spongelike over a very large range of density threshold. This departre from random phase consistent with a distribution like a filamentary net of 'walls with holes.' On smoothing scales approaching approximately 20/h Mpc the shape of the CfA genus curve is consistent with random phase. There is very weak evidence for a shift of the genus toward a 'bubble-like' topology. To test cosmological models, we compute the genus for mock CfA surveys drawn from large (L greater than or approximately 400/h Mpc) N-body simulations of three variants of the cold dark matter (CDM) cosmogony. The genus amplitude of the 'standard' CDM model (omega h = 0.5, b = 1.5) differs from the observations (96% confidence level) on smoothing scales is less than or approximately 10/h Mpc. An open CDM model (omega h = 0.2) and a CDM model with nonzero cosmological constant (omega h = 0.24, lambda (sub 0) = 0.6) are consistent with the observed genus amplitude over the full range of smoothing scales. All of these models fail (97% confidence level) to match the broadness of the observed genus curve on smoothing scales is less than or equal to 10/h Mpc.