Power-spectrum analysis of one-dimensional redshift surveys

We consider the statistical properties of power spectra obtained from "pencil-beam" redshift Surveys, with particular attention to the recent study by Broadhurst et al. (BEKS). We discuss tests of the null hypothesis that the long-wavelength one-dimensional power Spectrum found in such surveys is generated by the known small-scale clustering of galaxies. The visual impression of the BEKS redshift data is dominated by a relatively small number of rich clumps, and we first test whether there is any evidence for nonuniformity in the distribution of these objects. We make a "clump catalog, " identifying the richest clumps by eye from the redshift histogram of BEKS. The real clumps appear to have a width in redshift of about 800 km s^-1^; if clumps in synthetic catalogs are given a similar width, this causes the power spectrum to fall quite rapidly with increasing frequency. This can enhance the apparent significance of any spike at k<~0. 1h Mpc^-1^ compared with the exponential power distribution seen at higher k. We generate an ensemble of mock catalogs with a Poissonian distribution of clumps, using the observed clump richness distribution, and find that power spectra with features as striking as the real data arise quite frequently. To a good approximation, the Fourier transform of the one-dimensional galaxy counts is a complex Gaussian random field, and the statistical properties of the low-k power spectrum are calculable in terms of the survey selection functions and the power spectrum of small-scale galaxy clustering. This provides a prediction for the one-dimensional power level which agrees quite well with the Poisson clump model prediction, and allows us to calculate the frequency of peaks in the power spectrum, as well as such quantities as the uncertainty in the power estimated from a finite bandwidth. We generate realizations of power spectra and compare with the surveys used by BEKS. Somewhat surprisingly, we find that the power Spectra for the two deep surveys alone show little evidence for periodicity or extra large-scale power-the visual impression of the counts notwithstanding. With the inclusion of the shallower samples the significance of the spike at k O. 05h Mpc^-1^ increases but is still marginal (such peaks are expected to occur ~6% of the time). The texture of our simulations matches very well with observation and shows that (often quite isolated) spikes are common features. We estimate the level of extra large-scale power which might be detectable with one-dimensional surveys of this kind, and find that it is rather large, although a factor of ~2 improvement in sensitivity can be obtained by a nonlinear clipping of the data. The limited sensitivity of these surveys means that the results are expected to be consistent with a wide range of models for large-scale structure. Most importantly, they are certainly consistent with a galaxy distribution which is very nearly uniform on 100 Mpc scales.