5. Discussion and conclusions
The DWT skewness and kurtosis spectra of real and simulated samples of Ly forests has been calculated. Deviations from a Gaussian state in these samples are detected for all data on scales of 5 to 10 h-1 Mpc with confidence level larger than 95%. For the two sets which are available on large scales, the non-Gaussianity is significant even on scales as large as about 80 h-1 Mpc. The scale-dependence (or spectrum) for a The higher order DWT cumulant is especially effective in distinguishing the real samples from random data, and in discriminating among models.
It is possible for non-Gaussianity to result from systematic errors hiding in the original data reduction. For instance, discrete sampling or binning of a continuous distribution into a histogram with bin size leads to non-Gaussianity at least on scale . Many selection effects actually are some kind of sampling, and therefore, they will also give rise to non-Gaussianity. All these non-Gaussianities are "spurious".
Fortunately, in most cases these spurious non-Gaussian signals are significant only at one special scale. For a Poisson process, it is the scale of the mean distance of nearest neighbor clouds. For binning, it is . On scales larger than , the spurious non-Gaussianities will decay out. That is, the scale-dependent behavior of sampling and binning is useful in recognizing spurious non-Gaussianity. In the case of the Ly forests, the mean nearest neighbor distance along with the binning scale are much less than the scales of the detected non-Gaussianities, and therefore, the detected non-Gaussianities are inherent to the distributions. There is also evidence for intrinsic scales in the Ly spectra of magnitude 25 h-1 Mpc (Rauch 1998). However this contamination can be reduced by studying and on scales different from 25 h-1 Mpc.
Nevertheless, high-resolution samples which can cover the spatial range as l arge as 80 h-1 Mpc are needed before a final determination can be made. Recently a scale-scale correlation analysis also shows a non-Gaussianity in the Ly line distribution on such large scales (Pando et al. 1998). It seems to support our conclusion that cosmic matter underwent a non-Gaussian process on quite large scales at quite early times.
Last but not least, the numerical work of calculating DWT higher order moment spectrum is not any more difficult than calculating the second order moments. Generally, the calculation of third and higher order correlation functions of large scale structure samples is very strenuous work. But the numerical work involved in calculating the DWT is about the same as the FFT, and can even be faster. The FFT requires N Log N calculations, while the DWT, using a "pyramid" scheme, requires only order N calculations (Press et al. 1992).
This method can easily be generalized to 2- and 3-dimensions. The kurtosis and skewness spectrum opens a new window for looking at the statistical features of large scale structures. It is an important and necessary addition to the existing methods of describing the clustering and correlation of the cosmic density field, and for discriminating among models of structure formation.
© European Southern Observatory (ESO) 1998
Online publication: November 9, 1998