A sparse-sampling strategy for the estimation of large-scale clustering from redshift surveys

The form of the power spectrum of galaxy clustering at large wavelengths provides a powerful test of theories for galaxy formation. We show that estimating the two-point correlation function from a complete, magnitude limited redshift survey provides a rather inefficient way to apply this test. By sampling only a randomly chosen fraction of the galaxies, but to a fainter magnitude limit, one can significantly reduce the uncertainty in the two-point function, ξ, for a given investment of telescope time. In order to extend our knowledge of ξ beyond the point at which the currently available estimates fall to a level consistent with vanishing correlations, which we estimate to be around $$r\simeq15\enspace h^{-1}\enspace \text {Mpc}$$, one should sample at a rate of about one in 20 bright galaxies. The signal-to-noise ratio for such a sparse sample is roughly twice that provided by a complete survey of the same cost, and this performance is the same as for a larger complete survey of about seven times the cost. The same increase in performance can be obtained if the redshifts are collected multiply on a wide-field telescope, provided that the survey is sufficiently large that close to full sky coverage can be maintained. For smaller, multiply collected surveys one should ideally sample at the somewhat denser rate of about one in 10 bright galaxies, though the improvement in performance in this case is relatively small. The optimum-sampling fraction for Abell's rich clusters is close to unity and, in this case, no significant improvement in performance is provided by sparsely sampling.