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Astron. Astrophys. 364, 1-16 (2000)

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6. Statistics of the aperture mass [FORMULA]

It now suffices to plug the numerical expressions we have obtained for [FORMULA] in (34) to get the shape of the [FORMULA] PDF. More precisely we have,


where [FORMULA] is the variance of the aperture mass.

In (124) the integral over y has to be made in the complex plane. The integration path in the y plane is built in such a way that the argument of the exponential is always a real negative number thus avoiding oscillations (see Fig. 5). Moreover, one must make sure that the integration path does not cross the branch cuts of [FORMULA]. The singularities of [FORMULA] induce non-analytic parts for [FORMULA] as well. They are located at positions,




where the maximum value of [FORMULA] is taken along the line-of-sight (and is indeed finite). As noticed in Valageas (2000a) the exponent [FORMULA] of the singularity of [FORMULA] (as defined in (95)) leads to the exponent [FORMULA] for the projected generating function [FORMULA]. However, in both the quasi-linear and highly non-linear regimes we have [FORMULA], see (84). In this case, as shown in App. A, for [FORMULA] the singularity is of the form [FORMULA].

[FIGURE] Fig. 5. Integration paths in the y complex plane. The paths (dashed lines) are dynamically built so that the argument of the exponential in Eq. (124) is always a negative real number. The thick half straight lines represent the locations of the non-analytic parts of [FORMULA] in this plane. Of course, the paths must not cross these branch cuts.

The existence of these branch cuts is directly responsible for two exponential cut-offs in the shape of the PDF of [FORMULA],


It can be noted that the [FORMULA] dependence of [FORMULA] will induce a strong [FORMULA] dependence in the position of the exponential cut-offs. The variation of [FORMULA] with [FORMULA] is thus to be compared with the theoretical uncertainties on [FORMULA].

6.1. The PDF shape

In Figs. 6 we present the resulting shape of the one point PDF obtained for different cases. They have been obtained from the parameterization of [FORMULA] described in the previous sections. In particular we assume a power law spectrum with index [FORMULA]. The variance adopted for the plots is [FORMULA]. In these investigations we did not try to put a realistic source distribution, but we assume all the sources to be at redshift unity. However, all our results can be extended in a straightforward fashion for any redshift distribution of the sources (the latter is simply absorbed by a redefinition of the selection function [FORMULA]). Note also that we present the PDF for the correctly normalized aperture mass [FORMULA], that is without dividing the local convergence by [FORMULA]. We can check in Fig. 6 that the tails of the PDF are stronger in the non-linear regime than in the quasi-linear regime independently of the variance [FORMULA] (which is the same in both plots). This is related to the smaller values of the singularities [FORMULA] in the non-linear case, as shown by the expression (127). Of course, this is due to the smaller value of the singularity [FORMULA] of the 3D density field in the non-linear regime (in a similar fashion the coefficients [FORMULA] are larger).

[FIGURE] Fig. 6. Shape of the one-point PDF of the aperture mass from the quasi-linear regime model (top panel) and in the non-linear regime (bottom panel). The sources are at redshift unity, the variance of [FORMULA] is 0.01, for the filter given by Eq. 15. the solid lines correspond to Einstein-de Sitter case, the dashed lines to flat universe with [FORMULA].

In Fig. 7 we show that the positions of the cut-off depend crucially on the window shape. In particular it is clear that when the disc radius ratio is larger the PDF is more asymmetric and bears more resemblance with the [FORMULA]-PDF for a top-hat window function. Indeed, when the inner disk is much smaller than the outer radius the fluctuations of [FORMULA] are dominated by those of the convergence [FORMULA] which corresponds to this small inner window while [FORMULA] which is governed by larger scales shows lower amplitude fluctuations.

[FIGURE] Fig. 7. Same as previously with the nonlinear model but with cell radius ratio of 10 instead of 2.

6.2. The [FORMULA] dependence of the PDF

It has been stressed in the literature (e.g., Bernardeau et al. 1997) that the non-Gaussian properties of the convergence maps are expected to exhibit a strong [FORMULA] dependence. This is due in particular to the normalization factor. Such dependence is apparent in [FORMULA] that depends crucially on the value of [FORMULA]. This property naturally extends to the shape of the one-point PDF. In particular it is important to have in mind that the [FORMULA] dependence is negligible in [FORMULA] (for a fixed shape of the power spectrum). The [FORMULA] dependence is therefore entirely contained in the projection effect through the shape and amplitude of the efficiency function.

In Fig. 6 we show how low values of [FORMULA] amplify the non-Gaussian features contained in the PDF. Whether such a parameter can be constrained more efficiently with the PDF than with simply the local skewness is not yet clear. Such a study is however beyond the objective of this paper and is left for further works.

6.3. Comparison with numerical simulations

Finally, we compare our predictions for the PDF [FORMULA] of the aperture mass [FORMULA] with the results of N-body simulations (Jain et al. 2000) in Fig. 8. Note that for all these comparisons we exclusively use the [FORMULA] filter . We consider the cosmological models defined in Table 2: a standard CDM (SCDM) and a [FORMULA]CDM scenario in a critical density universe, a low-density open universe (OCDM) and a low-density flat universe with a non-zero cosmological constant ([FORMULA]CDM). Here [FORMULA] is the usual shape parameter of the power-spectrum. We use the fit given by Bardeen et al. (1986) for [FORMULA]. We only consider the weak lensing distortions which affect a source at redshift [FORMULA], with angular window characteristic scale [FORMULA].

[FIGURE] Fig. 8. The aperture mass PDF for a source at redshift [FORMULA] for 4 cosmologies and with the angular window [FORMULA]. The dashed line shows the Gaussian which has the same variance. The points show the results of N-body simulations from Jain et al. (2000). Results for sCDM and [FORMULA]CDM models have been directly taken from Reblinsky et al. (1999).


Table 2. Cosmological models and results obtained with [FORMULA] filter. The sixth and seventh lines show the variance and the skewness of [FORMULA] from numerical simulations (Jain et al. 2000). Results for sCDM and [FORMULA]CDM models have been directly taken from Reblinsky et al. (1999) and have been obtained with one realization only. The eight and ninth lines show the reconstructed PDF properties.

In the numerical calculations, we discretize the integral (34) over redshift and we solve for the system (117 - 118). That is we take into account the redshift dependence of the generating function [FORMULA]. Moreover, we use the relation (122) to get the value of the parameter [FORMULA], where for n we take the local slope of the linear-power spectrum at the wavenumber [FORMULA]. This corresponds to the Fourier modes which are probed by the filter of angular radius [FORMULA], see Fig. 1. For [FORMULA] we obtain [FORMULA] and [FORMULA].

First, we can check in Fig. 8 that we recover the right trend for [FORMULA], with two asymmetric tails for large [FORMULA]. In particular, the exponential cutoff is stronger for negative values of the aperture mass than for positive values. We can also note that the PDF is significantly different from a Gaussian as it shows a clear exponential cutoff, much smoother than the Gaussian falloff, especially for large positive [FORMULA]. On the other hand, we note that Reblinsky et al. (1999) obtained a good match to the tail of the PDF [FORMULA] using a description of the density field as a collection of virialized halos (Kruse & Schneider 1999). However, such a method is restricted to the far tail of the PDF (large positive [FORMULA]) while our approach provides in principle a model for the full PDF [FORMULA]. There seems to be a small discrepancy with the simulations for the [FORMULA]CDM scenario. It is not clear whether this is due to a limitation of HEPT or of our formulation. To clarify this problem one should test the statistics of the 3D density field and of the projected density in the same simulation. However, this is beyond the scope of our paper. Nevertheless, the overall agreement appears to be quite reasonable. Note that the shape of the PDF is governed by only one parameter [FORMULA], which is uniquely related to the local slope of the power-spectrum, independently of scale and of the cosmology. On the other hand, the inaccuracy of the numerical simulations in the tail of the PDF might be somewhat underestimated.

In order to see more clearly the difference of the aperture mass PDF [FORMULA] with respect to the Gaussian we display in Fig. 9 the relative difference [FORMULA]. Here [FORMULA] is the Gaussian with the same variance as the numerical simulations. We can check that we recover a reasonable agreement with the numerical results, since Fig. 9 is directly related to Fig. 8. To get an estimate of the sensivity of our predictions with respect to the parameterization (119) and (122) we also plot in Fig. 9 our results for the same cosmologies when we use [FORMULA] (this would correspond to an initial power spectrum index [FORMULA]) in (122). We can see in the figure that our predictions are not too sensitive to [FORMULA] (for reasonable values of [FORMULA]) within the range [FORMULA] (the difference would look even smaller in Fig. 8). In particular, the variation with [FORMULA] of our results is much smaller than the difference between both cosmologies. This suggests that one could use the deviation of the PDF with respect to a Gaussian to estimate the cosmological parameters. A well-known tool to measure this signature is the skewness but one could devise other statistics which would take advantage of the expected shape of the PDF to maximize their dependence on cosmology. However, such a study is beyond the scope of this article. We can see in Table 2 that we underestimate somewhat the skewness of [FORMULA]. However, it is not clear whether this is due to the parameterization (119) or to the use of HEPT in (122). We note that the tail of [FORMULA] for large negative values of [FORMULA] appears to be slightly more sensitive to [FORMULA] than the tail at positive [FORMULA]. This could be related to the fact that the behaviour of [FORMULA] for [FORMULA] is more sensitive to the detailed properties of the [FORMULA]point correlation functions (see Valageas 2000c for a study of this point).

[FIGURE] Fig. 9. The relative difference [FORMULA] of the aperture mass PDF [FORMULA] with respect to the Gaussian [FORMULA]. The variance of the Gaussian is taken from the simulations. As in Fig. 8 we consider a source at redshift [FORMULA] with the angular window [FORMULA]. We display our results for the [FORMULA]CDM (solid line) and OCDM (dashed line) cosmologies. The dotted lines correspond to the same cosmologies with [FORMULA] (see main text). The points show the results of N-body simulations from Jain et al. (2000).

Finally, we note that although [FORMULA] corresponds to non-linear scales it is not very far from the quasi-linear regime. However, the aperture mass [FORMULA] probes the non-linear density field for filters with larger angular scale than the convergence [FORMULA]. Indeed, since the aperture mass involves compensated filters the contribution from low-k modes is strongly suppressed (see Fig. 1) so that [FORMULA] is governed by the properties of the density field at the comoving wavenumber [FORMULA]. In contrast, the convergence [FORMULA] shows a more important contribution from larger wavelengths which implies that in order to probe non-linear scales only one must set the filter size [FORMULA] farther away into the small-scale non-linear regime. This also means that in principle the aperture mass could be a more convenient tool than the convergence since it should be easier to separate the non-linear and quasi-linear regimes, while for an important range of angular scales the convergence should be sensitive to the transitory regime between both domains. However, a possible caveat is that the statistics of the aperture mass depend on the detailed behaviour of the [FORMULA]point correlation functions (and not on their average over spherical cells only), and therefore requires a better understanding of them.

As discussed above, this property of the aperture mass to probe a narrow range of wavenumbers makes it easier to avoid the intermediate regime as one can select observation windows which are either in the quasi-linear or in the strongly non-linear regime. However, it would clearly be interesting to obtain a model which would also cover this transitory range. Unfortunately, this is rather difficult as one cannot use the simplifications which appear in the two extreme regimes. An alternative to a rigorous calculation would be to use an ad-hoc parameterization which would smoothly join the quasi-linear regime to the highly non-linear regime, for instance in the spirit of HEPT as described in van Waerbeke et al. (2000b) for the skewness of the convergence (note that our model for non-linear scales is based on a simple ansatz which is not rigorously derived). However, although the quasi-linear and strongly non-linear regimes share the same gross features, like the relation (34) which describes the projection effects, their detailed properties are different. In particular, although in both cases we have the scalings (32) the correlation functions obtained in the quasi-linear regime are not given by a tree-model as in (100). This leads to the difference between the two-variable system (77)-(80) and the integral relation (117)-(118). Nevertheless, a simple prescription would be to recast the relations (117)-(118) into the form (77)-(80) by approximating the integrals over [FORMULA] by the difference between two mean values, corresponding to the inner and outer regions of the filter [FORMULA] and characterized by two averages [FORMULA] and [FORMULA]. Then, the shift from the quasi-linear to the highly non-linear regime would simply be described by a smooth interpolation of the generating function [FORMULA], i.e. of the sole parameter [FORMULA]. However, such a study is left for a future work as in this article we preferred to lay out the formalism needed to study the statistics of the aperture mass and to focus on the two regimes which have already been tested in details against numerical simulations, so as not to introduce a new specific parameterization.

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Online publication: December 15, 2000