## 4. Skewness and kurtosis spectra of Ly forests## 4.1. Data preparationThe observational QSO Ly forests data have a
complex geometry. By this we mean that different forests cover
different spatial ranges, and no one of the forests distributes on the
entire range . At the very least, a complicated
weighting scheme is needed in order to form an ensemble. The DWT
provides a way around this problem. Because wavelets are localized,
the wavelet coefficients are only determined by the local
distribution. If a forest sample lies in range ,
one can extend it to by adding zeros to the
data in ranges and . The
wavelet coefficients in the interval will not
be affected by the addition of zero in and
. Any statistics can then be computed by the
ensemble of wavelet coefficients, , by simply
dropping all wavelet coefficients with More precisely, the effect of zero padding on the wavelet coefficients is described by the so-called "influence cone" which consists of the spatial support of all dilated wavelet functions. For instance, if is localized in the space interval for , the influence cone centered at is given by (Farge 1992). Wavelet coefficients corresponding to positions outside will not corrupt information within the influence cone. The LWT and JB samples cover a red-shift range of 1.7 to 4.1, i.e.
the spatial range in comoving distance is from about
h The HKCSR and KT samples (HKCSR+KT), can be treated the same way.
Since HKCSR and KT cover a smaller red-shift range from about 2.4 to
3.1, the forests of HKCSR+KT are extended to 128 bins of comoving size
2.5 h ## 4.2. Skewness and kurtosis spectra of real dataAs opposed to discussed in Sect. 3, real
data provide only histograms of Ly line
distributions, i.e. the data are not a continuous function of
It is well known that point processes generally are non-Gaussian. For instance, if the Ly clouds do not distribute structurally, then should be a 1-D Poisson process. However, Poisson processes are non-Gaussian. Obviously, this non-Gaussianity is not what we are seeking to measure because it is not the result of non-linear evolution or physical processes related to structure formation. Therefore, we should carefully distinguish the clustering related non-Gaussianity with that from sampling and binning. For this purpose, we generate 5000 random realizations in which the mean number density follows Eq. (1). Each realization is treated in the same way as the real data sets. From these random samples we can produce frequency distributions for each statistic, say (Fig. 3). These frequency distributions are fair estimators of the underlying probability distribution. It can clearly be seen from Fig. 2 that the random data are non-Gaussian.
Using these frequency distributions, the confidence levels for the mean values of the random data can be estimated. These are then to be compared with the real data. The skewness and kurtosis spectra of the LWT
(W Å), JB
(W Å) and HKCST+KT (b
km s
We can directly describe the non-Gaussianity by the one-point
distributions of the WFCs. Fig. 5 plots the one-point distributions of
for the JB sample with
Å. For each scale
This asymmetry is at least partially due to the
red-shift-dependence of the Ly clouds. The
wavelet coefficient is mainly determined by
the difference of (positive) densities at and
(PF). For a clump in red-shift space, the
density change on the lower red-shift or lower ## 4.3. Removing degeneracy by the non-Gaussian spectrumWe have shown that cluster identification by a wavelet decomposition is a useful tool for discriminating among models of Ly clouds (PF). The spectrum of non-Gaussianity can play the same role. Namely, and are effective measures for removing the degeneracy that exists at second order among models. As an example, we examined the BGF simulated Ly forest samples. This simulation shows that two models, SCDM and LCDM, are degenerate if only the first (number density) and second (variance, or power spectrum) order statistics are considered. That is, both SCDM and LCDM give about the same predictions for the following features of the Ly forests: a.) the number density of Ly lines and its dependencies on red-shift and equivalent width; b.) the distribution of equivalent widths and its red-shift dependence; c.) the two-point correlation function. This degeneracy can be removed by the non-Gaussian spectrum. Fig. 6 plots the kurtosis spectra for the LCDM, SCDM, and CHDM models, in which the Ly lines are chosen with width Å. The error bars in Fig. 6 are given by the distribution of 20 realizations for each model.
Fig. 6 shows that even though the BGF simulation is based on the
linear power spectrum of the density perturbations, the
Ly distribution is non-Gaussian. This is because
the selection of peaks from the density field is a non-Gaussian
processes. More important, Fig. 7 clearly shows the significant
difference among the three models. With 95% confidence, the
amplitudes of all three models are different on
all scales. The of the SCDM model is
significantly larger than zero for all scales © European Southern Observatory (ESO) 1998 Online publication: November 9, 1998 |