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\def\inv{$^{-1}$} \def\hmpc{\rmh \inv~Mpc} \def\lteq{\leq}
To quantify clustering in the large-scale distribution of galaxies and to test theories for the formation of structure in the universe, we apply statistical measures to the CfA Redshift Survey. This survey is complete to $m_{B(0)}=15.5$ over two contiguous regions which cover one-quarter of the sky and include $\sim 11,000$ galaxies. The salient features of these data are voids with diameter $30-50$\hmpc~ and coherent dense structures with a scale $\sim 100$\hmpc.
Comparison with N-body simulations rules out the ``standard" CDM model ($\Omega=1$, $b=1.5$, $\sigma_8=1$) at the 99\% confidence level because this model has insufficient power on scales $\lambda>30$\hmpc. An unbiased open universe CDM model ($\Omega h =0.2$) and a biased CDM model with non-zero cosmological constant ($\Omega h =0.24$, $\lambda_0=0.6$) match the observed power spectrum. The amplitude of the power spectrum depends on the luminosity of galaxies in the sample; bright ($L>L^*$) galaxies are more strongly clustered than faint galaxies. The paucity of bright galaxies in low-density regions may explain this dependence.
To measure the topology of large-scale structure, we compute the genus of isodensity surfaces of the smoothed density field. On scales in the ``non-linear" regime, $\lteq 10$\hmpc, the high- and low-density regions are multiply-connected over a broad range of density threshold, as in a filamentary net. On smoothing scales $>10$\hmpc, the topology is consistent with statistics of a Gaussian random field. Simulations of CDM models fail to produce the observed coherence of structure on non-linear scales ($>95\%$ confidence level).
The underdensity probability (the frequency of regions with density contrast $\delta\rho/\overline{\rho}=-0.8$) depends strongly on the luminosity of galaxies; underdense regions are significantly more common ($>2\sigma$) in bright ($L>L^*$) galaxy samples than in samples which include fainter galaxies.
