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Astron. Astrophys. 322, 1-18 (1997)

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5. Higher-order moments

In this section we explore the possibility of considering higher order moments of the distribution of the local convergence. The idea is that the higher order moments are sensitive to nonlinear aspects of the dynamics and thus could provide independent constraints on the cosmological parameters.

The nonlinear terms in the expression of the fields can be calculated using Perturbation Theory. Similar calculations have been done for the density field and for the local velocity divergence. Comparisons with numerical results have shown that Perturbation Theory calculations give excellent quantitative predictions of the behavior of the high order moments (see Bernardeau 1996b and references therein).

5.1. Second order perturbation theory for the local convergence

The basic assumption of perturbation theory is that the local convergence can be expanded in terms of the initial density field,

[EQUATION]

where [FORMULA] is linear in the initial density field, so in the variables [FORMULA] (this is the term calculated previously), and [FORMULA] is quadratic in those variables, etc... For calculating the expression of [FORMULA] we need to use the differential equation (23) up to its second order introducing the expression of

[EQUATION]

with

[EQUATION]

To find the expression of the local convergence at the second order it is thus necessary to have the expression of the local density at the same order.

The calculation of the second order term for the local density has been investigated in many papers. It has now been calculated for any cosmological model, and a very useful description is given by,

[EQUATION]

where the [FORMULA] factors are the Fourier transform of the initial density field (cf. [3]).

The coefficients appearing in this expression given for an Einstein-de Sitter universe, are in fact slightly [FORMULA] and [FORMULA] dependent (Bouchet et al. 1992, Bernardeau 1994). These dependences are actually so weak (less than 1%) that they can be safely neglected in the subsequent calculations.

The differential equation for [FORMULA] can be obtained from the Eq. (23), when it is developed up to its second order. We then get,

[EQUATION]

with

[EQUATION]

The expression of the local convergence can be deduced from the previous two equations,

[EQUATION]

As it can be observed two terms are contributing to this expression. One term is given by the second order density field, the other one is a combination of the linear order of the local density and the linear order in the local amplification. The latter corresponds to the nonlinear coupling introduced by two subsequent deflecting lenses. In the following we will see that the contribution to the third moment is dominated by the first contribution.

5.2. The expression of the local skewness

The principle of such a calculation has been developed in previous papers for the density or the velocity field. It relies on the assumption that the initial density field is Gaussian. As a consequence we have,

[EQUATION]

and

[EQUATION]

where the sum over the permutations is made in such a way that all possible pair associations are taken into account.

Then, applying those properties to the initial density field one can calculate the leading term for the expression of the third moment of the local convergence. Using the expansion (56) we have

[EQUATION]

The first term of this expression is identically zero in case of Gaussian initial conditions. The dominant contribution to the skewness is thus given by the next-to-leading term, [FORMULA], that combines both the expression of the linear amplification, and its second order expression.

To compute the previous expression it is worthwhile to know that (Bernardeau 1995)

[EQUATION]

Using this property we get,

[EQUATION]

This integral can thus be integrated numerically without more technical difficulties than for the second moment 3.

At this stage it is crucial to notice that the ratio,

[EQUATION]

is expected to be independent of the normalization of the power spectrum. In the following we will explore in more detail the dependence of this ratio on the different cosmological parameters and physical hypothesis.

To start with let us consider a very simple case in which we assume that we have a power law spectrum, that all sources are at the same redshift [FORMULA] and that we live in an Einstein-de Sitter Universe. In such a case we find that

[EQUATION]

which gives,

[EQUATION]

This is a quantity a priori accessible to a measurement in a reasonably large catalogue. In the last section we will explore in more detail the expected magnitude of the errors for doing such a measurement. Let us start with a study of the dependence of this quantity on the cosmological parameters.

5.3. Numerical results for a realistic case

We present in Fig. 6 the results obtained for the function [FORMULA] for the power spectrum (5), the source distribution (2) with [FORMULA] (solid line) and [FORMULA] =2 (dashed line) in case of an Einstein-de Sitter Universe. We can see that the function [FORMULA] is rather flat, with little variation with the smoothing angle.

[FIGURE] Fig. 6. The expected parameter [FORMULA] as a function of the smoothing angle (in degrees) for [FORMULA] (solid line) and [FORMULA] (dashed line) in (2), the power spectrum (5) and assuming an Einstein-de Sitter Universe.

5.4. Dependence on the redshift

From Eq. (69) we easily can visualize the [FORMULA] dependence of [FORMULA]. This dependence is shown in Fig. 7. We can see that the result is almost independent of the index n, and we have approximately,

[EQUATION]

[FIGURE] Fig. 7. The expected dependence of [FORMULA] on the redshift of the sources for an Einstein-de Sitter Universe. The curves have been plotted for different power spectrum indices, [FORMULA] with the thick grey line for the intermediate value

As for the variance, we found that the dependence of the skewness on the redshift of the sources is also quite large.

5.5. Dependence on the cosmological parameters

From the general formulation we can also explore the dependence of the results on the cosmological parameters. In Fig. 8 we present the [FORMULA] dependence of the result for [FORMULA]. We can see that the dependence is rather large, approximatively as

[EQUATION]

for [FORMULA]. The [FORMULA] dependence is actually slightly weaker when the sources are at higher redshift (when the sources are at very small z we recover the natural expectation that [FORMULA] varies inversely with [FORMULA].)

[FIGURE] Fig. 8. The expected [FORMULA] dependence of [FORMULA] for [FORMULA] (solid line) and [FORMULA] (dashed line).

In Fig. 9 we then present the contour plot showing the dependence of the parameter on [FORMULA] and [FORMULA]. We can see that [FORMULA] is significantly less [FORMULA] dependent.

[FIGURE] Fig. 9a and b. Contour plot showing the dependence of [FORMULA] on [FORMULA] (horizontal axis) and [FORMULA] (vertical axis) for [FORMULA] (left panel) and [FORMULA] (right panel). The contour lines are the logarithmic values of [FORMULA] regularly spaced in a logarithmic scale.

5.6. Discussion, simplified model

A this stage it is worth to compare the qualitative results obtained here with those obtained for an a priori similar case: the statistics in an angular survey. In both cases indeed the observed quantities are proportional to the cosmic density integrated along the line-of-sight. The expected properties of the moments of the one-point PDF are however qualitatively different. The reason is that for a 2D angular survey, the density fluctuations are calculated with a selection function implicitly normalized to unity, whereas the efficiency function in the case of the weak lensing is not normalized (see Fig. 1). As a consequence the local convergence is the summation of the effects of each lens plane, whereas the local measured galaxy over-density identifies with the average of all contributions present on the line-of-sight. The behavior of these quantities is thus dramatically different when the depth of the respective catalogues is changed. The variance of the local convergence increases with the depth (as for a random walk, the mean distance from the center increases as the square root of the number of steps), but it becomes more and more Gaussian, so that [FORMULA] decreases. On the other hand the variance of the local galaxy density contrast decreases with the depth, since the density fluctuations tend to be smoothed out. The parameter [FORMULA] is on the other hand almost independent of this depth. This is exactly what one would obtain by considering respectively the summation or the average of a given number of quasi-Gaussian variables.

From these general remarks one can build a very simple model for the moments of the local convergence that reproduces qualitatively the results discussed in the previous sections. Let us define [FORMULA] as the average of the efficiency function when the sources are at redshift [FORMULA],

[EQUATION]

Then, from Eqs. (48) and (67) we simply expect that,

[EQUATION]

which show the dependence of these statistical quantities on the efficiency function and the depth of the catalogue. These relations are certainly crude approximations, but explain most of the features encountered previously, such as the dependence of the results on [FORMULA] and on the redshift of the sources. It also suggests the existence of a combination of [FORMULA] and [FORMULA] that would be almost independent of the redshift of the sources. This combination (which is almost the product of these two quantities, since [FORMULA] is proportional to the distance) would contain information of purely cosmological nature.

5.7. Systematics due to other quadratic couplings

As it can be seen from Eq. (65) the skewness is induced by any quadratic coupling in the observed convergence. In particular it implies that the convergence should not be measured from the local shear using the linear approximation. The method proposed by Kaiser (1995), or a similar method, that takes into account the nonlinear relation between the local shear and the local convergence has to be used.

Spurious couplings can appear also through:

  • nonlinear coupling between two deflecting lenses;
  • coupling between two deflecting lenses due to the induced displacement of the light path;
  • coupling between the population of selected sources and the local convergence;
  • density fluctuations of sources.

In the following we subsequently examine the importance of these different effects.

5.7.1. Coupling between two deflecting lenses

This coupling is due to the fact that when the light path intercepts two lenses the resulting effect is given by the product of the amplification matrices. The resulting convergence [FORMULA] is given by the trace of this product, but coincides with the sum of the convergences induced by each lens separately only in the linear regime. In general it is indeed given by

[EQUATION]

which contain quadratic terms in the local lens densities (the indices A and B correspond to quantities associated with the lenses A and B respectively).

Actually this nonlinear coupling between two lenses appears in the neglected term in (62). This term induces a corrective expression for the second order convergence term,

[EQUATION]

The other source of coupling is due to the failure of the Born approximation (i.e. the densities are calculated along the unperturbed light path) when two lenses are involved. Indeed the convergence induced by a given lens is not due to the local projected density at the observed angular position if a foreground lens shifts the apparent position of the background galaxies. It implies that in (40) the local over-density can be taken at the unperturbed position only at the linear order but in general should be taken at the position [FORMULA] where [FORMULA] is the displacement induced by the foreground lenses. To get the quadratic coupling one can simply write,

[EQUATION]

This term induces a term similar to (77),

[EQUATION]

These two terms obviously induce a corrective term for the skewness. It can be easily calculated, and we estimate it here for an Einstein-de Sitter universe for sources at a fixed redshift [FORMULA]. In such a case the second order corrective term for the convergence reads,

[EQUATION]

The resulting corrective term for [FORMULA] is

[EQUATION]

so that 4

[EQUATION]

independently of the redshift of the sources. It is completely negligible compared to the main contribution (69), and note that this effect is expected, at first glance, to be mainly independent of [FORMULA].

It might be also worth to note that the resulting deformation field induced by two lenses is no more potential (the amplification matrix is not symmetric). As a result, the method proposed by Kaiser (1995) to build the local convergence from the local shear is expected to fail at this level. Once again, this is not expected to be crucial since the effects of this coupling are seen to be small.

5.7.2. Coupling between the population of sources and the convergence

The origin of this effect is the fact that the population of sources on which the shear is measured may change with the local magnification. In this case

[EQUATION]

with

[EQUATION]

where [FORMULA] is the partial derivative of the number density of sources at a given redshift with the local convergence (therefore the local magnification) for a fixed normalization. It can be evaluated a priori from the population of objects used to do the measurements, and on the selection procedure. For instance the mean depth of the sources is expected to change with the local magnification. We can build a simple model in which we assume that the sources are always at a given redshift, but that its corresponding angular distance is a function of the local magnification,

[EQUATION]

It would imply, for an Einstein-de Sitter universe,

[EQUATION]

In such a case the corrective term for [FORMULA] can be calculated straightforwardly and is given by,

[EQUATION]

It is not necessarily completely negligible, depending on the value of [FORMULA], that is the sensitivity of the selected population of sources with the local magnification (this will be explored in more details in the second paper). So, one way to reduce the effect of this coupling is to select the galaxies used to measure the shear from a surface brightness criterion (Bartelmann & Narayan 1995). In practice however, it is probably impossible to suppress all influence in the selection procedure of the total luminosity of the objects.

5.7.3. Coupling due to density fluctuations of sources

Throughout this study we have always implicitly assumed that the sources formed a uniform background population. This is actually far from being true, and one naturally expects the sources to have cosmic density fluctuations. The effects of these fluctuations of the measurements of the local convergences strongly depend on the adopted method. For a method based on the estimation of the local magnification by the depletion of the galaxy number density (magnification bias), this can be a very strong effect on both moments. For methods in which the local convergence is built from the measured shear, this effect is weak. Simple considerations show that it introduces a significant coupling only at the level of the fourth moment. This is due to the fact that the populations of sources and lenses can be considered as two independent populations so there are no direct couplings between their density fluctuations.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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