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

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Appendix A: expressions of joint cumulants

A.1. General expressions

The statistical quantities described in the main text are expected to be affected by a cosmic variance due to the finite size of the sample. The estimation of such an effect is determined by joint moments between local smoothed convergences taken at two different locations. Indeed these moments provide information on how observed local smoothed convergences are expected to be correlated. The exact derivation of the joint moments of interest would be a difficult task, that requires an explicit calculation of the higher order expression (up to the sixth order!) of the local convergence. As we do not require a very accurate calculation but rather a realistic estimation of these quantities we will make some simplifications in this derivation. First of all we will assume that the high order terms of the local convergence are dominated by the projection along the line-of-sight of the corresponding high order terms of the local density in the dominant lens plane. This is what has been found in Sect. 5 for the skewness, and we expect it to be also true for the cumulants we are interested in. Moreover to do the calculations we will assume that the size of the sample in which the local convergence is available is significantly larger that the smoothing angular scale. This will make the derivation of the joint cumulants more simple and accessible to Perturbation Theory. Similar results were obtained in a slight different context (Bernardeau 1996a) for which the derivation of joint cumulants is given for 3D top-hat smoothed density contrasts. It is shown that when the distance between the smoothing cells is large compared to the smoothing scale, the joint cumulants can be written,

[EQUATION]

at the leading order in Perturbation Theory, and where the [FORMULA] coefficients are finite numbers (they are constant for a power law spectrum). A crucial property for these coefficients is that we have,

[EQUATION]

so that the derivation of these coefficients reduces to the calculations of a single index series of numbers. They have been given in the paper mentioned above for the 3D top-hat smoothing.

These results cannot be used directly to estimate the joint moments we are interested in, but form however the basis of the calculations. Indeed if we assume, as mentioned in the introduction of this appendix, that the high order expressions of the local convergence are dominated by the projection of the local density of the lenses at the same order, then the expressions of the joint moments are given by the integration over the line-of-sight of the relation (A1) where the coefficients [FORMULA] correspond to the top-hat smoothed joint cumulants for the 2D dynamics. This is a situation which is similar to the situation encountered in the derivation of the expression of the high order moments in angular survey.

Using the calculation developed in Bernardeau (1996a) and applying it to the 2D dynamics we get, for a power law spectrum of index n,

[EQUATION]

and

[EQUATION]

Then, as a consequence, the local joint cumulants follow a similar hierarchy

[EQUATION]

where the coefficients [FORMULA] are given by ratios of integrals along the line-of-sight. We give here the expressions of these integrals for an Einstein-de Sitter Universe, and we will limit the explicit numerical calculations of these expressions for this case,

[EQUATION]

with

[EQUATION]

where

[EQUATION]

[FORMULA] is the Bessel function of the first kind and [FORMULA] is the angle between the two directions of the smoothing cells 1 and 2.

The final step is to take the geometrical averages of these joint moments in the sample. It is expected to be dominated by cells being at the distance of the order of the size of the sample, [FORMULA]. Then the averaged moments follow the same hierarchy as in (A5) but for which the function [FORMULA] is replaced by

[EQUATION]

where [FORMULA] is introduced in (45). This result is valid if the sample has a spherical shape, although it is by no means a crucial hypothesis.

In the latter case the integral [FORMULA] is actually the expression of the variance of the convergence in the whole sample. All the considered joint cumulants are also found to be proportional to this variance, and thus the cosmic uncertainties on the results will be all the more important that this integral is large. The results will therefore depend very strongly on the adopted shape of the power spectrum. In the following subsection we give the numerical results for the APM measured power spectrum (5). Compared to a CDM spectrum it contains more power at large scale and is thus less favorable for the estimation of the cosmic errors, but it is probably more realistic.

A.2. Numerical results

In the following we give the results of the previous integrals for the power spectrum given in (5). The index at the 30' smoothing scale is found to be

[EQUATION]

and then the coefficients of interest are found to be

[EQUATION]

for a sample size of 25 [FORMULA], corresponding to a sample size of [FORMULA]. These results are calculated with the distribution of lenses (2) in which [FORMULA].

Appendix B: evaluation of the cosmic errors

In this appendix we present the derivation of the errors, due to the cosmic variance, i.e. the fact that measurements made in a finite sample are affected by systematic and random errors. We will naturally focus our presentation on the second and third moments of the local smoothed convergence.

To present the calculations it is necessary to introduce two different averages. One is the geometrical average, the mean value of a given observed quantity at different locations, and the second is the ensemble average, which corresponds to the expectation value of quantities that depend on cosmic variables (local densities..) or random quantities associated with intrinsic properties of the galaxies such as their intrinsic ellipticities. In principle the two averages of a same quantity coincide in a perfect case, that is if the available data set was infinite, but in practice this is obviously not the case! Actually, the geometrical averages, considered as estimations of the corresponding ensemble averages, are also random variables that are expected to vary from one sample to another. In the following we estimate the properties of these geometrical averages, that is their cosmic expectation values and their variances, assuming that for a sufficiently large sample they obey a Gaussian statistics. We follow here ideas that were developed by Colombi, Bouchet & Schaeffer (1995) and more precisely by Szapudi & Colombi (1996).

B.1. Notations

In the following we denote [FORMULA] the (connected) geometrical average of a quantity X (and as usual, [FORMULA] its ensemble average). Thus the observed moments of the measured convergence are

[EQUATION]

where the summations are made over the [FORMULA] different locations where the smoothed convergences are supposed to have been determined. We actually assume that the local convergences have been measured in a compact sample, of a size significantly larger than the smoothing length. In the geometrical averages it is also not excluded that the smoothing areas overlap, so that [FORMULA] can actually be arbitrarily large. Of course the resulting averages cannot be arbitrarily accurate because the measured [FORMULA] are correlated variables, and will be all the more correlated that they are measured in close directions.

B.2. Modelisation

In the following we not only estimate the errors due to the cosmic variance but also the effects of the intrinsic noise in the measurements due to the use of a limited number of tracers for the shear and of their intrinsic imperfections. So assume that the measured convergence in the direction [FORMULA] is given by,

[EQUATION]

where [FORMULA] is the true cosmological value of the local convergence in the direction [FORMULA] and [FORMULA] is the error made in this same direction due to the intrinsic ellipticities of the galaxies. Assuming the ellipticities of the galaxies are independent from one another, and using results obtained for their intrinsic ellipticities (see for instance Miralda-Escudé 1991b) we get,

[EQUATION]

where [FORMULA] is the number of galaxies in a given smoothing angular cell. Moreover two variables [FORMULA] and [FORMULA] are independent if the corresponding cells do not overlap, and they are all assumed to be independent of the true values of the local convergences. This is probably a rather simple and naive modelisation but we leave more accurate numerical studies of a more realistic modelisation for a further paper.

As the variance of [FORMULA] is independent of i we will denote, [FORMULA] their common RMS fluctuation,

[EQUATION]

and from the expected density of galaxies in a [FORMULA] field (about [FORMULA] for the limit magnitudes [FORMULA] or [FORMULA] currently used in ultra-deep imaging; Tyson 1988, Smail et al. 1995) we can estimate that

[EQUATION]

We then define the random variable [FORMULA] that describes the departure between an observed geometrical average and the corresponding ensemble average,

[EQUATION]

and

[EQUATION]

for [FORMULA]. In the following we will assume that the cosmic random variables [FORMULA] have a small variance compared to the variance at the smoothing scale. Therefore the variables [FORMULA] can be considered to follow a Gaussian statistics and we will do the calculation at their dominant order. The knowledge of the cosmic errors in any of the considered measurements can then be derived from the values of the variances and cross-correlations between these variables.

B.3. Statistical properties of the variables [FORMULA]

To obtain the statistical properties of [FORMULA] one can simply consider the ensemble average of the [FORMULA], of their squares and products. From the assumed properties of the variables [FORMULA] and [FORMULA] we have,

[EQUATION]

where we have identified the geometrical average of [FORMULA] with the variance of the local convergence at the scale of the sample, [FORMULA]. At this scale we assume that the ensemble average of [FORMULA] is negligible compared to the one of [FORMULA]: it decreases like Poisson noise, that is more rapidly than the variance of the convergence.

The ensemble averages for the second moment read,

[EQUATION]

where the calculations have been limited to the quadratic terms in [FORMULA] and [FORMULA]. The calculations make also intervene the geometrical averages of the joint moments [FORMULA] which are evaluated in the previous appendix.

For the third moment we have,

[EQUATION]

We have also to consider the cross-correlations between those moments which give,

[EQUATION]

From these results we have entirely define the statistical properties of the variables [FORMULA], [FORMULA] and [FORMULA]. Simple identifications lead to

[EQUATION]

It is interesting to note that [FORMULA] enters only in the expectation value of [FORMULA].

As a result we can calculate both the expectation value of a measured quantity and the variance of this measure. To do so the quantities we are interested in are expressed in terms of the random variables [FORMULA] and expanded up to the quadratic order. This is a trivial calculation for the variance, and for the measured [FORMULA] we have

[EQUATION]

Of course this expansion is correct only if the sample size is larger than the smoothing scale.

B.4. Results for the variance and for [FORMULA]

Then when we apply the previous rules on the variables [FORMULA] and get

[EQUATION]

One can distinguish two different effects of the cosmic variance for the uncertainties of the results. There is first a systematic shift proportional to the ratio of the variances. And secondly a cosmic scatter is expected to appear, which is directly proportional to the RMS of the fluctuations of the convergence at the sample scale. It is thus crucial to have a catalogue as large as possible!

Using the numerical results of the previous appendix we have

[EQUATION]

The cosmic errors of course decreases with the size of the sample. What the previous results show is that the errors are already quite small when the size of the sample is [FORMULA] and would allow, for a perfectly well known source distribution, a determination of the cosmological parameters at the 5 to 10% level. Note that [FORMULA] appears to be less sensitive to the cosmic variance because it is a ratio of moments.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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