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Astron. Astrophys. 324, 15-26 (1997)

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Appendix A: calculation of the filtering effects

In this appendix I explicitly take into account the filtering effects to compute the expressions of the four-point correlation function in different geometries. The filtering can be the one due to the angular resolution of the apparatus or due to a subsequent filtering of the temperature field. When it needs to be specified the adopted window function will always be the angular top-hat window function.

The expression I am interested in is the expression (19) where [FORMULA] and the coupling term [FORMULA] are given by,

[EQUATION]

and,

[EQUATION]

where [FORMULA] is the smoothing angle. The quantity of interest is thus

[EQUATION]

In the following I will estimate this expression for different hypothesis on [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

A.1. Four separate directions

Here I assume that

[EQUATION]

The integral (A3) will be dominated by values of [FORMULA], [FORMULA] and k for which [FORMULA], [FORMULA] and [FORMULA] are about unity. It implies that [FORMULA] and [FORMULA] are all small quantities thus making the filtering effects negligible, so that,

[EQUATION]

An interesting property to be used is that

[EQUATION]

It implies that

[EQUATION]

with,

[EQUATION]

This integral can be eventually integrated from properties of Bessel functions,

[EQUATION]

with which we find,

[EQUATION]

where,

[EQUATION]

[FORMULA] is the angle between [FORMULA] and [FORMULA] and [FORMULA] is the angle between [FORMULA] and [FORMULA] (see Fig. 1).

A.2. When two directions coincide

A.2.1. When [FORMULA] = [FORMULA]

In this case, [FORMULA] is expected to be of the order of [FORMULA], thus larger than k so that

[EQUATION]

As a result one has,

[EQUATION]

When one integrates over the angle of [FORMULA] the first term vanishes. The second term of the expansion is thus the dominant contribution, which takes the form,

[EQUATION]

Using the property (A6) we have

[EQUATION]

Interestingly [FORMULA] is now proportional to the angular correlation function of the local magnification and not of the local displacement.

A.2.2. When [FORMULA] = [FORMULA]

In this case k is expected to be of the order of [FORMULA], thus larger than [FORMULA] and [FORMULA]. As a result one has,

[EQUATION]

leading to

[EQUATION]

where

[EQUATION]

Here the effect is proportional to the mean displacement in the beam size [FORMULA].

A.2.3. Other cases

The other cases do not give specific formulae and can be derived from results of Appendix A.1.

A.3. When two pairs coincide

A.3.1. When [FORMULA] and [FORMULA]

This case is similar to the Appendix A.2.2 case where the result is dominated by the second order term of an expansion in [FORMULA], that here should be written for [FORMULA] and [FORMULA]. We then have

[EQUATION]

The integrations over the angles between [FORMULA] and [FORMULA] and between [FORMULA] and [FORMULA] give each a factor [FORMULA] leading to

[EQUATION]

which can be expressed in terms of the angular correlation function of the magnification.

A.3.2. When [FORMULA] and [FORMULA]

This case is a particular case of Appendix A.2.1.

A.4. When the four directions coincide

In this case we have

[EQUATION]

This expression cannot be simplified furthermore if the window function is not specified. In this paragraph I assume that W is the top-hat window function,

[EQUATION]

To complete the calculation it is interesting to have in mind the property (Bernardeau 1995),

[EQUATION]

This property is rigorously exact. In principle it is not possible to separate the two terms. Both relations are however good approximation as it will be shown in the following. It is thus reasonable to assume that

[EQUATION]

Then, using this property it is easy to show that

[EQUATION]

A.5. Validity of the top-hat window function property (A26)

To examine the property (A26), an interesting property of the Bessel function to use is that (Gradshteyn & Ryzhik, 1980, Eq. [8.532.1]),

[EQUATION]

with

[EQUATION]

We have thus

[EQUATION]

and

[EQUATION]

As a result

[EQUATION]

In the following I assume a power law behavior for both the mass power spectrum and [FORMULA],

[EQUATION]

Then using the property,

[EQUATION]

one can easily compute the few terms of the previous series. It is found to converge very rapidly. The resulting ratio,

[EQUATION]

with

[EQUATION]

is plotting in Fig. 11. It shows that the error made by using the approximation is at most of a few percent for the values of [FORMULA] and [FORMULA] of interest.

[FIGURE] Fig. 11. The ratio (A36) as a function of [FORMULA] and [FORMULA].
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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