EFFECTS OF SAMPLING ON STATISTICS OF LARGE-SCALE STRUCTURE

The effects of sampling are investigated on measurements of counts-in- cells in three-dimensional magnitude-limited galaxy surveys, with emph asis on moments of the underlying smooth galaxy density field convolve d with a spherical window. A new estimator is proposed for measuring t he kth-order moment [rho(k)]: the weighted factorial moment (F) over t ilde(k)[omega] Since these statistics are corrected for the effects of the varying selection function, they can extract the moments in one p ass without the need to construct a series of volume-limited samples. The cosmic error on the measurement of (F) over tilde(k)[omega] is com puted via the formalism of Szapudi & Colombi, which is generalized to include the effects of the selection function. The integral equation f or finding the minimum variance weight is solved numerically, and an a ccurate and intuitive analytical approximation is derived, omega(optim al)(r) proportional to 1/Delta(r), where Delta(r) is the cosmic error as a function of the distance from the observer. The resulting estimat or is more accurate than the traditional method of counts-in-cells in volume-limited samples, which discards useful information. As a practi cal example, it is demonstrated that, unless unforeseen systematics pr event it, the proposed method will extract moments of the galaxy distr ibution in the future Sloan Digital Sky Survey (hereafter SDSS) with a ccuracy of order a few per cent for k = 2, 3 and better than 10 per ce nt for k = 4 in the scale range of 1 less than or equal to l less than or equal to 50 h(-1) Mpc. In the particular case of the SDSS, a homog eneous (spatial) weight omega = 1 is reasonably close to optimal. Opti mal sampling strategies for designing magnitude-limited redshift surve ys are investigated as well. The arguments of Kaiser are extended to h igher order moments, and it is found that the optimal strategy depends greatly on the statistics and scales considered. A sampling rate f si milar to 1/3 - 1/10 is appropriate to measure low-order moments with k less than or equal to 4 in the scale range 1 less than or similar to l less than or similar to 50 h(-1) Mpc. However, the optimal sampling rate increases with the order considered, k, and with 1/l. Therefore c ounts-in-cells statistics in general, such as the shape of the distrib ution function, high-order moments, cluster selection, etc., require f ull sampling, especially at small, highly non-linear scales l similar to 1 h(-1) Mpc. Another design issue is the optimal geometry of a cata logue, when it covers only a small fraction of the sky. Similarly to K aiser, we find that a survey composed of several compact subsamples of angular size Omega(F) spread over the sky on a glass-like structure w ould do better, with regards to the cosmic error, than the compact or the traditional slice-like configurations, at least at small scales. T he required dynamic range of the measurements determines the character istic size of the subsamples, It is, however, difficult to estimate, s ince an accurate and cumbersome calculation of edge effects would be r equired at scales comparable to the size of a subsample.