Measuring high-order moments of the galaxy distribution from counts in cells: The edgeworth approximation

To probe the weakly nonlinear regime, past the point where simple linear theory is sufficient to describe the statistics of the density distribution, we measure the skewness (S₃) and kurtosis (S₄) of the count probability distribution function (CPDF) of the IRAS 1.2 Jy sample obtained from counts in cells. These quantities are free parameters in a maximum likelihood fit of an Edgeworth expansion convolved with a Poissonian to the observed CPDF. This method, applicable on scales ≳5 h^-1 Mpc, is appreciably less sensitive to the tail of the distribution than are measurements of S₃ and S₄ from moments of the CPDF. We measure S₃ and S₄ to l ∼ 50 h^-1 Mpc; the data are consistent with scale invariance, yielding averages of 〈S₃〉 = 2.83 ± 0.09 and 〈S₄〉 = 6.89 ± 0.68. These values are higher than those found by Bouchet et al. in 1993 (〈S₃〉 = 1.5 ± 0.5 and 〈S₄〉 = 4.4 ±3.7) using the moments method on the same data set, owing to lack of correction for finite-volume effects in the latter work. Unlike the moments method, our results are quite robust to the fact that IRAS galaxies are underrepresented in cluster cores. We use N-body simulations to show that our method yields unbiased results.