## 4. Moments in real space## 4.1. Signature of the normalization and non-Gaussian propertiesWeak 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
, 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 filtersCompensated 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 this latter filter is a compensated filter
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.
## 4.3. Moment estimations and shot noise correctionsDue 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. ResultsWe 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/arcmin
## 4.4.1. Noise correctionThe 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 varianceThe 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 ## 4.4.3. The skewnessThe 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 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.
## 4.5. Comparisons of different observational scenariosThe results presented previously had been obtained from the full
mass reconstruction of 240 maps
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 sizeBy 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 redshiftFig. 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).
## 4.6. BG versus CDM power spectrumIn 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.
The skewness is thus a robust estimator of fairly insensitive to the power spectrum. ## 4.7. Effect of the normalizationThe 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.
## 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
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 |