Design and analysis of redshift surveys

In this paper we consider methods of analysis and optimal design of redshiftsurveys. In the first part, we develop a formalism for analysing galaxyredshift surveys which are essentially two-dimensional, such as thindeclination slices. The formalism is a power spectrum method, using sphericalcoordinates, allowing the distorting effects of galaxy peculiar velocities tobe calculated to linear order on the assumption of statistical isotropy butwithout further approximation. In this paper, we calculate the measuredtwo-dimensional power for a constant declination strip, widely used in redshiftsurveys. We present a likelihood method for estimating the three-dimensionalreal-space power spectrum and the redshift distortion simultaneously, and showthat for thin surveys of reasonable depth, the large-scale 3D power cannot bemeasured with high accuracy. The redshift distortion may be estimatedsuccessfully, and with higher accuracy if the 3D power spectrum can be measuredindependently, for example from a large-scale sky-projected catalogue. In thesecond part, we show how a 3D survey design can be optimized to measure thepower spectrum, considering whether areal coverage is more important thandepth, and whether the survey should be sampled sparsely or not. We show quitegenerally that width is better than depth, and show how the optimalsparse-sampling fraction, f, depends on the power, P, to be measured. For aSchechter luminosity function, a simple optimization fP \simeq 500 h^{-3} Mpc^3is found.Comment: 9 pages (Latex), 6 postscript figures included, MNRAS in pres