Astron. Astrophys. 342, 15-33 (1999)

4. Moments in real space

4.1. Signature of the normalization and non-Gaussian properties

Weak lensing carries more information than the amplitude and shape of the dark matter power spectrum. Fig. 1 demonstrates that distortion maps of the same amplitude (and with very similar power spectrum as can be checked in Fig. 2) can display very different features. On these maps the variance of the local convergence is the same, but the amount of non-linearities is very different. For low universes, the same amount of distortion can be reached only with a rather large value of thus corresponding to a much more evolved dynamics. As a result the difference between the underdense and the overdense regions is more pronounced. The `voids' tend to occupy a much larger area, whereas the super clusters tend to be sharper. These features appear because of the non-linear couplings contained in the gravitational dynamics. At large scale the use of Perturbation Theory has proved to be extremely good in predicting the emergence of such properties. All these calculations are based on the hypothesis that the initial conditions were Gaussian, which we will assume as well.

It has already been stressed (BvWM) that the departure from a Gaussian statistics is described by the skewness of the probability distribution function (PDF) of the local convergence. We will restrict our analysis on the skewness basically for two reasons: it is beyond the scope of this paper to explore all possible indicators of the non-Gaussian properties, and we know that the approximate dynamics we have adopted reproduces correctly the skewness of the local PDF (see Appendix A). In addition the lens-lens coupling and the Born approximation terms which are known to be small for the third moment are probably more important for higher orders, and this requires a complete dedicated work.

Let us summarize the expected results. For a top-hat window function we expect to have,

where is the rms value of at the scale , n is the index of the power spectrum, is the 3D rms density at 8Mpc scale and is the mean redshift of the sources. The computed skewness is expected to be independent from the normalization of the power spectrum. It is only weakly dependent on the shape of the power spectrum as well as on the cosmological constant (see discussion). The skewness would then be a very robust way ⁶ of determining the density parameter .

Although the moment analysis is generally performed on the basis of a top-hat filter there is a priori no reason to limit our investigations to this filter. In particular SvWJK have proposed the use of an alternative function, the compensated filter that might prove more efficient to constrain , with a lower cosmic variance.

4.2. Top-hat versus compensated filters

Compensated filters were considered by SvWJK as a way to measure the convergence directly from the galaxy shape. They use the filter of size that defines the quantity as

where is the tangential component of the shear field. is the convergence field filtered by ,

where has to be the compensated filter and it is related to the arbitrary filter V through

this latter filter is a compensated filter  ⁷ and, for instance, a convenient filter to use is given by,

The spectral responses (defined as the squared amplitude of the Fourier transforms of the window funstions) of the filter's family used in this work are shown in Fig. 5 (for the top-hat) and by Fig. 6 (for the compensated filter ). Clearly a field smoothed with a top-hat filter of size is sensitive to fluctuations of size larger than that contribute also to the cosmic variance of the moments. On the other hand, a compensated filter integrates the fluctuation modes only around the target frequency and any power at lower or larger scales will affect neither the signal nor its cosmic variance. SvWJK showed that the de-correlation properties of compensated filters are by far better than for top-hat filters because any power at small wavelengths between two disconnected fields is highly suppressed (see Fig. 8 of their paper). Thus the cosmic variance should be smaller for a compensated filter than for a top-hat filter. However this attractive feature comes with a price: a compensated filter needs to be sampled by a larger number of galaxies. In other words, the shot noise has a larger effect on the aperture mass than on the top-hat filtered mass at the same scale. A compromise has to be found, that depends on the functional shape of the compensated filter and on the shape of the power spectrum.

Fig. 5. Spectral response for the family of top-hat filters used in this work. The finite size effects are visible for the largest smoothing scales. The curves are not regularly spaced because of pixelisation effects.

Fig. 6. Spectral response for the family of compensated filters used in this work.

4.3. Moment estimations and shot noise corrections

Due to the intrinsic ellipticities of the galaxies and the cosmic variance (see for instance Szapudi & Colombi 1996, Colombi et al. 1998), estimates of the moments of from the reconstructed map are biased. We show here that the shot noise can be accurately calculated and the estimated moments corrected. It is worth noting that SvWJK has shown that we can find an unbiased estimator of the moments of , which completely cancel the shot noise correction problem. It is based on the measurement of the tangential shear . Unfortunately, the measurable quantity is (the tangential reduced shear) rather than , and unless this is taken into account, the estimator given in SvWJK is no longer unbiased. In other words, the shot noise correction problem is shifted to an estimator correction problem.

It was shown in Sect. 3.1 that the shot noise leads to a pure white noise in the reconstructed convergence maps. The amplitude of this noise can be obtained by measuring the observed ellipticities of galaxies as described for the power spectrum estimation. Therefore, estimates of the variance and the skewness of the convergence in (19) and (20), corrected from the intrinsic ellipticities of the galaxies, are obtained by simply removing the noise term in the second moment. Note that the skewness correction only requires the correction of the variance since the third moment is not affected by the noise. As the analytical calculation of the noise term for compensated filters can be rather cumbersome, it is estimated using Monte-Carlo simulations.

Even after that noise correction, finite sample effects may bias the estimations of the second moment and of and may increase significantly the cosmic variance. This difficulty was partly investigated in BvWM with the use of perturbation theory. They pointed out that the accessible geometrical averages are expected to be smaller than the true ensemble averages, and that a dispersion is expected in the measurements (cosmic variance):

• the bias that affects the expectation values was found to be proportional to the variance at the sample size divided by the one of at the filtering scale;

• the scatter was found to be proportional to the rms of at the sample size.

These estimates were done fully in perturbation theory, with numerous approximations (in particular it was assumed that the sample size was much bigger than the smoothing scale which is probably an erroneous approximation for most of the cases considered here). For accurate investigations of all these effects that take into account both the Poisson noise and the finite volume effects see Szapudi & Colombi (1996).

4.4. Results

We now turn to the measurement of moments in the simulated fields. The same simulations used for the power spectrum analysis are used here. The results are given in Figs. 7 (variance in a degree field with ) and 8 (skewness in a degree field with ). For each of these figures the plots are organized in the same way: the first raw ((a), (b) and (c) plots) is for the flat model, the second raw ((d), (e) and (f) plots) for the open model. It corresponds to the first two raws of Table 1. The first columns (a) and (d) show the estimator measured with a top-hat filter, the second columns (b) and (e) with a compensated filter, and the third columns (c) and (f) show the signal to noise ratio of these estimators. In plots (a), (b), (c) and (d) the solid lines give the estimators measured in the noise-free maps, the dotted lines in the noisy reconstructed maps with gal/arcmin² and the dashed lines with gal/arcmin². The dotted-dashed lines show the estimators measured on the reconstruction with noise with gal/arcmin² corrected from the noise. Since the case gal/arcmin² gives the same results they are not plotted. On the signal to noise plots (c) and (f) the thin solid lines show the results for a top-hat filter and the thick solid lines for a compensated filter. The results obtained for the noisy maps with gal/arcmin² or gal/arcmin² corrected from the noise are respectively given by the dashed and dotted lines (either thin or thick).

Fig. 7a-f. The measured variance of the convergence, and the corresponding signal to noise ratio. The upper panels show the case, while the bottom panels correspond to the case. The left and middle panels are respectively the measured variance with a top-hat and a compensated filter. On these plots, the thick solid line is the true variance measured on the noise-free maps, the dashed line and the dotted line for the noisy reconstructed mass maps (with respectively gal/arcmin2 and gal/arcmin2). The dotted-dashed line is the variance measured from the gal/arcmin2 case and corrected from the noise. The right panels show the signal to noise ratio of the variance detection with the top-hat (thin solid line) and compensated (thick solid line) filters. Dotted and dashed lines have the same meaning as for a , b , d and e but here the variance has been corrected from the noise.

Fig. 8a-f. Same as Fig. 7, but for the skewness of the convergence, .

4.4.1. Noise correction

The noise correction as described in Sect 4.3 gives unbiased results, as it is expected for superimposed white noise. This is true even when the correction is two orders of magnitude higher than the signal (see for example Figs. 8 plots (b) and (e)). This confirms the simple properties of the noise in the reconstructed mass maps already found in Sect. 3.1.

4.4.2. The variance

The variance obtained for a top-hat filter is slowly decreasing with an increasing scatter, as expected. For the compensated filter the curves are almost flat as expected from the shape of the power spectrum. The noisy maps (dotted and dashed lines in Fig. 7a and b display higher values for the variance. Once it is corrected, the results are in perfect agreement with the noise-free simulations. The signal to noise ratio is basically not affected by the shot noise for a top-hat filter, it shows that going deep does not improve the measurement precision. The compensated filter reveals much more sensitive to the shot noise as predicted in Sect. 4.2 since we can see on plot Fig. 7c and f (thick lines) the bell shape of the signal to noise ratio, with a significant reduction of the measurement precision at the smallest available scales. The open and flat cases show basically no differences. It can be seen however that the signal to noise ratio for a compensated filter is sightly lower for the open case. We interpret this effect as due to the presence of more nonlinear couplings in the maps. Remarkably, the precision with which the variance can be measured in some specific k range reaches 5%.

4.4.3. The skewness

The skewness can be accurately measured at the smallest scales (in Fig. 8 only the error bars for the first four points have been drawn). The skewness is decreasing with scale in the two cases. Once again, the noise correction applied to the reconstructed maps allow to recover the skewness with a surprising accuracy. The signal to noise of the skewness is still not affected by the noise for a top-hat filter, while for the compensated filter the situation is worse than for the variance; for instance, at the smallest scale, the signal to noise is almost one order of magnitude smaller on the reconstruction with noise than on the noise-free maps for the case. The two models, open and flat, provide us with very different magnitude for . Their skewness ratio is 3 as expected from perturbation theory ⁸, with a significance of the separation at roughly . This is observed in case of the top-hat as well as the compensated filter and this confirms the fact that the skewness of the convergence can strongly separate low and high density universes.

To be more precise we present the actual histograms of the measured skewness in Fig. 9 which demonstrates clearly that the two cosmologies can be easily separated. One can see that the scatter in is roughly the same in the two cases and that the difference in the relative precision is due to the differences in the expectation values. This plot also shows that the distribution of the measured is quite Gaussian.

Fig. 9. Histograms of the values of , top-hat filter, for (solid lines) and (dashed lines) for a degree survey (thick lines) and a degree survey (thin lines). The angular scale is the pixel size .

4.5. Comparisons of different observational scenarios

The results presented previously had been obtained from the full mass reconstruction of 240 maps  ⁹ as described in Appendix B. Since it was demonstrated in the preceding section that noise acts as a pure de-correlated white noise in the reconstructed maps, we pursue our analyses of large series of simulated fields simply by adding the noise on the initial maps (especially for degree data sets for which convergence reconstruction would take typically one and a half hour on DEC PWS-500 computers). The subsequent analyses are therefore made with this simplified scheme.

To complement the previous cases, we have built and analyzed the cosmic variance on 60 maps for each of the following models: open () and flat () cosmologies, a survey size of degrees for , a survey size of degrees for , with the power spectrum of Eq. (A13) and a survey size of for with a CDM spectrum.

4.5.1. Effect of the survey size

By increasing the total area of a factor 4 we increase the signal to noise on the variance and the skewness with a top-hat filter by almost a factor 1.7. This can be seen in Fig. 9 when comparing the degree case with the case. With a compensated filter, the signal to noise ratios of the variance and of the skewness are increased by exactly a factor 2, thus improving more rapidly with the sample scale than the top-hat window function. This is expected from the de-correlation properties of those filters. It makes the compensated filter actually more attractive for such a large survey.

4.5.2. Effect of the source redshift

Fig. 10 shows the effect of a change in the mean source redshift. For the galaxies that are further away, the variance of is larger since the gravitational distortion is stronger, conversely the skewness is smaller since the accumulated material along the line of sight creates a field that is more and more Gaussian. The surprising result is that the signal to noise of these quantities does not depends strongly on the redshift of the sources. This means that it will be a waste of time to observe at high redshift, while it will basically not improve the precision of the measurement. Things are slightly improved for the compensated filter, but fundamentally, the results with the top-hat filter show that we do not learn more by increasing the redshift. Note that there is no improvement due to the increasing galaxy number density if the survey size is unchanged (see Sect. 4.3). In addition, high redshift surveys may create new problems such as uncertainties due to the Born approximation, the lens-lens coupling or the recently investigated source clustering effect (Bernardeau 1998).

Fig. 10. For a flat cosmological model, comparison of different observational strategies between a survey of size degrees (dashed lines) a degrees survey for sources at mean redshift 1.5 (dotted lines) and a degrees survey for sources at mean redshift 1 (solid lines). Left panels are for the top-hat filter and middle panels for the compensated filter. The thicker lines in the right panels hold for the compensated filter.

4.6. BG versus CDM power spectrum

In order to test the robustness of the skewness as an estimator of independent of the power spectrum we re-ran our simulations for a standard CDM power spectrum. Fig. 11 shows the comparison between the CDM model (dotted line) and BG power spectra, which clearly shows that CDM contains more structures at small scale by looking at the variance plots (a), (b). As predicted in BvWM, the skewness of the convergence if almost unaffected by the change of power spectrum (see Fig. 11 (d), (e)), but there is a small improvement in the signal to noise for the flat model (because of the larger power at small scale for CDM). On the other hand, the variance is strongly affected and in particular there is a significant decrease of power at large scale compare to BG spectrum. Note that the compensated filter yields a more accurate representation of the underlying power spectrum than the top-hat filter (because it is a pass band filter), and leaves the angular dependence of the variance unchanged compared to the spectrum Eq. (A13) excepts at large scales.

Fig. 11. Comparison of different choices of power spectra for a degree survey and a flat cosmological model. The dashed lines correspond to an CDM spectrum with and , dotted lines to a BG spectrum with and , and the solid lines to a BG spectrum with and .

The skewness is thus a robust estimator of fairly insensitive to the power spectrum.

4.7. Effect of the normalization

The skewness is found to be independent of the normalization, as expected from the Perturbation Theory. The signal to noise ratio however is increased by about 40% in the case of high normalization . These results are summarized in Fig. 12 that shows the histograms for various cosmological cases. It demonstrates that the skewness is clearly independent of the shape and normalization of the power spectrum. However there is a strong dependence on the mean redshift of the sources. If the signal to noise ratio for depends on the cosmology it is independent of the mean source redshift. This again is favoring rather shallow surveys.

Fig. 12. Histograms of the values of for degree survey, top-hat filter, for for a BG spectrum with and (thick solid line), and (thin solid line), and (dotted line) and a CDM spectrum with and (dashed line). The angular scale is the pixel size, .

4.8. Beyond the skewness to measure ?

The skewness of the PDF of the local convergence does not entirely characterize the PDF itself. Thus it is natural to measure higher order moments of the convergence to probe the cosmology.

It is clear that the skewness breaks the degeneracy between the power spectrum and the cosmological parameters, and is completely insensitive to the normalization. Bernardeau (1995) already noticed that in the case of cosmic density field or cosmic velocity field, the ratio calculated from the perturbation theory with a top-hat filter is almost a constant, and independent of the underlying cosmological model. This work was recently extended to the lensing case (Bernardeau, 1998) where he found . If all systematics of the gravitational lensing measurement can be controlled, search for such a magic number in our Universe would be a strong indication of validity of the paradigm of the gravitational instability scenario started from Gaussian initial conditions. On the other hand may be a new way to measure the density parameter (but not totally independent of the skewness). In our maps we find that the noise correction still works for the kurtosis, and that a compensated filter is more efficient than the top-hat filter (at least for noise-free data). In addition the kurtosis appears to be a fairly good discriminant for . Unfortunately the signal to noise ratio remains lower than for the skewness, which make it more difficult to measure. Moreover, it is more sensitive to the usual lensing approximations (Born approximation and lens-lens coupling terms) as well as source clustering (Bernardeau 1998). The error bars of the kurtosis found in our simulations are so large with a degree survey that it is impossible to measure it at scales larger than a few arcminutes. It turns out that the ratio is for the flat case and for the open case for a top-hat filter, while it is for the flat case and for the open case for a compensated filter. It is not the scope of this work to compare further the differences between the two filters, but we want to point out that filtering may be an interesting way to change the dependence of an estimator versus the cosmology. This should be studied in order to search for optimal measurements of higher order moments. More generally, the whole shape of the PDF could probably be used with more efficiency than the skewness alone. In a regime of small departure from a Gaussian distribution it can for instance be fruitful to describe the shape of the PDF with an Edgeworth expansion that takes into account the first few moments (Juszkiewicz et al. 1995, Bernardeau & Kofman 1995). For instance, from the Edgeworth expansion it is easy to show that the fraction of values of the convergence that is above the average value, , is

The results obtained from this formulae are in good agreement with those obtained from the direct measurements, however with a slightly larger cosmic variance.

Finally, the non-Gaussian features can also be characterized with topological indicators (which have been shown to be fruitful for the analysis if CMB data already, see for instance Winitzki & Kosowsky 1998, Schmalzing & Gorski 1997). What cosmic variance could be derived from the joint use of topological quantities or/and information on the shape of the PDF is left for further investigations.

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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