Analysis Of Redshift Surveys Research Articles

Boundary corrections in fractal analysis of galaxy surveys

The analysis of redshift surveys with fractal tools requires one to apply some form of statistical correction for galaxies lying near the geometric boundary of the sample. In this paper we compare three different methods of performing such a correction upon estimates of the correlation integral in order to assess the extent to which estimates may be biased by boundary terms. We apply the corrections illustrative examples, including a simple fractal set (L\'{e}vy flight), a random $\beta$-model, and a subset of the CfA2 Southern Cap survey. This study shows that the new ``angular'' correction method we present is more generally applicable than the other methods used to date: the conventional ``capacity'' correction imposes a bias towards homogeneity, and the ``deflation'' method discards large-scale information, and consequently reduces the statistical usefulness of data sets. The ``angular'' correction method is effective at recovering true fractal dimensions, although the extent to which boundary corrections are important depends on the form of fractal distribution assumed as well as the details of the survey geometry. We also show that the CfA2 Southern sample does not show any real evidence of a transition to homogeneity. We then revisit the IRAS PSCz survey and ``mock'' PSCz catalogues made using N-body simulations of two different cosmologies. The results we obtain from the PSCz survey are not significantly affected by the form of boundary correction used, confirming that the transition from fractal to homogeneous behaviour reported by Pan & Coles (2000) is real.

Design and analysis of redshift surveys

In this paper we consider methods of analysis and optimal design of redshift surveys. In the first part, we develop a formalism for analysing galaxy redshift surveys which are essentially two-dimensional, such as thin declination slices. The formalism is a power spectrum method, using spherical coordinates, allowing the distorting effects of galaxy peculiar velocities to be calculated to linear order on the assumption of statistical isotropy but without further approximation. In this paper, we calculate the measured two-dimensional power for a constant declination strip, widely used in redshift surveys. We present a likelihood method for estimating the three-dimensional real-space power spectrum and the redshift distortion simultaneously, and show that for thin surveys of reasonable depth, the large-scale 3D power cannot be measured with high accuracy. The redshift distortion may be estimated successfully, and with higher accuracy if the 3D power spectrum can be measured independently, for example from a large-scale sky-projected catalogue. In the second part, we show how a 3D survey design can be optimized to measure the power spectrum, considering whether areal coverage is more important than depth, and whether the survey should be sampled sparsely or not. We show quite generally that width is better than depth, and show how the optimal sparse-sampling fraction, f, depends on the power, P, to be measured. For a Schechter luminosity function, a simple optimization fP \simeq 500 h^{-3} Mpc^3 is found.