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Astron. Astrophys. 340, 335-342 (1998)

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4. Skewness and kurtosis spectra of Ly[FORMULA] forests

4.1. Data preparation

The observational QSO Ly[FORMULA] 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 [FORMULA]. 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 [FORMULA], one can extend it to [FORMULA] by adding zeros to the data in ranges [FORMULA] and [FORMULA]. The wavelet coefficients in the interval [FORMULA] will not be affected by the addition of zero in [FORMULA] and [FORMULA]. Any statistics can then be computed by the ensemble of wavelet coefficients, [FORMULA], by simply dropping all wavelet coefficients with l in the added zero ranges.

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 [FORMULA] is localized in the space interval [FORMULA] for [FORMULA], the influence cone centered at [FORMULA] is given by [FORMULA] (Farge 1992). Wavelet coefficients corresponding to positions outside [FORMULA] 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 [FORMULA] h-1 Mpc to [FORMULA] h-1 Mpc, if [FORMULA] and h [FORMULA] km s-1 Mpc-1. To eliminate the proximity effect, all lines with [FORMULA] are deleted from our samples. These two samples are extended to [FORMULA] as described above, and binned into in 512 bins with comoving size [FORMULA] 2.5 h-1 Mpc, which is about the scale where the effect of line blending occurs. If a line is not located at the center of a bin, it is separated into the two neighboring bins, weighted according to the distance to each of the centers.

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 [FORMULA] 2.5 h-1 Mpc. Thus it is possible to compare all 3 sets for scales down to [FORMULA].

4.2. Skewness and kurtosis spectra of real data

As opposed to [FORMULA] discussed in Sect. 3, real data provide only histograms of Ly[FORMULA] line distributions, i.e. the data are not a continuous function of x, but a point process in x space. In this case, the WFCs [FORMULA] will not be calculated by an integral like Eq. (3), but by the wavelet transformation matrix (Press et al. 1992.)

It is well known that point processes generally are non-Gaussian. For instance, if the Ly[FORMULA] clouds do not distribute structurally, then [FORMULA] 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 [FORMULA] (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.

[FIGURE] Fig. 2. Frequency distributions of [FORMULA] from 5000 random samples. It can be clearly seen that this distribution is non-Gaussian.

[FIGURE] Fig. 3. Skewness spectrum for LWT ([FORMULA] Å), JB ([FORMULA] Å) and HKCSR+KT samples. The red-shift evolution model (RSEM) given by Eq. (1), and the 95% confidence limits are also shown. The physical scale is related to j by [FORMULA] h-1 Mpc.

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[FORMULA] Å), JB (W[FORMULA] Å) and HKCST+KT (b [FORMULA] km s-1) data are shown in Figs. 3 and 4, respectively. The skewness and kurtosis spectra of random data and the 95% confidence levels are also shown in these figures. Figs. 3 and 4 show that the skewness and the kurtosis spectra of the LWT (W[FORMULA] Å) and JB (W[FORMULA] Å) are almost the same. The kurtosis spectra are [FORMULA], significantly different from the random samples on all scales, and, on scales j= 5, the skewness spectra are also [FORMULA] and significantly different from the random samples. The Keck data, i.e. HKCSR+KT, have qualitatively the same non-Gaussian behavior, especially at [FORMULA] and [FORMULA], where the HKCSR+KT data are the same as that of LWT and JB. The HKCSR+KT gives lower values for [FORMULA] and [FORMULA] than the LWT and JB, while it gives higher [FORMULA] and [FORMULA] than LWT and JB. With the current data it is not possible to determine whether these differences are due to the higher resolution of the HKCSR+KT data. It is important to realize that these observational data sets were compiled by different groups, on different instruments, and as much as 6 years apart. These data sets are very independent and makes more convincing the case for the existence of the non-Gaussianity at least on scales of 5-10 h-1 Mpc, and confirms that the features shown in the non-Gaussian spectrum are intrinsic features of the density field traced by Ly[FORMULA] forests.

[FIGURE] Fig. 4. Kurtosis spectrum for LWT ([FORMULA] Å), JB ([FORMULA] Å) and HKCSR+KT samples. The red-shift evolution model (RSEM) given by Eq. (1), and the 95% confidence limits are also shown.

We can directly describe the non-Gaussianity by the one-point distributions of the WFCs. Fig. 5 plots the one-point distributions of [FORMULA] for the JB sample with [FORMULA] Å. For each scale j, the corresponding Gaussian distribution is plotted such that it has the same normalization and variance as the one-point distribution. This figure clearly shows that all the distributions on scales [FORMULA] (or less than about 80 h-1 Mpc) are significantly non-Gaussian. These distributions are also asymmetric, with fewer positive wavelet coefficients.

[FIGURE] Fig. 5. Histogram of one-point distribution of wavelet coefficients for JB (W[FORMULA] Å). At each scale j, the Gaussian distribution (dashed line) has the same variance and normalization as the wavelet coefficient distribution.

This asymmetry is at least partially due to the red-shift-dependence of the Ly[FORMULA] clouds. The wavelet coefficient [FORMULA] is mainly determined by the difference of (positive) densities at [FORMULA] and [FORMULA] (PF). For a clump in red-shift space, the density change on the lower red-shift or lower l side contributes negative wavelet coefficients, while the higher red-shift side gives positive wavelet coefficients. If as shown in PF, the number of Ly[FORMULA] clumps decreases with increasing red-shift, the change in clustering amplitudes (wavelet coefficients) on the higher red-shift side (positive wavelet coefficients) should be less than the lower side (negative wavelet coefficients). That is, the number of positive wavelet coefficients will be less than negative wavelet coefficients. This is consistent with a small, but positive skewness.

4.3. Removing degeneracy by the non-Gaussian spectrum

We have shown that cluster identification by a wavelet decomposition is a useful tool for discriminating among models of Ly[FORMULA] clouds (PF). The spectrum of non-Gaussianity can play the same role. Namely, [FORMULA] and [FORMULA] are effective measures for removing the degeneracy that exists at second order among models.

As an example, we examined the BGF simulated Ly[FORMULA] 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[FORMULA] forests: a.) the number density of Ly[FORMULA] 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[FORMULA] lines are chosen with width [FORMULA] Å. The error bars in Fig. 6 are given by the distribution of 20 realizations for each model.

[FIGURE] Fig. 6. Kurtosis spectrum of BGF samples in the SCDM, LCDM, and CHDM models with [FORMULA] Å.

Fig. 6 shows that even though the BGF simulation is based on the linear power spectrum of the density perturbations, the Ly[FORMULA] 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 [FORMULA] amplitudes of all three models are different on all scales. The [FORMULA] of the SCDM model is significantly larger than zero for all scales j. Yet, the LCDM model gives much lower [FORMULA] for all scales j. The non-Gaussian spectrum provides a extremely effective method for removing the degeneracy among models.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998