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 and the coupling term are given by,
In the following I will estimate this expression for different hypothesis on , , and .
A.1. Four separate directions
Here I assume that
The integral (A3) will be dominated by values of , and k for which , and are about unity. It implies that and are all small quantities thus making the filtering effects negligible, so that,
It implies that
This integral can be eventually integrated from properties of Bessel functions,
with which we find,
is the angle between and and is the angle between and (see Fig. 1).
A.2. When two directions coincide
A.2.1. When =
In this case, is expected to be of the order of , thus larger than k so that
As a result one has,
When one integrates over the angle of the first term vanishes. The second term of the expansion is thus the dominant contribution, which takes the form,
Using the property (A6) we have
Interestingly is now proportional to the angular correlation function of the local magnification and not of the local displacement.
A.2.2. When =
In this case k is expected to be of the order of , thus larger than and . As a result one has,
Here the effect is proportional to the mean displacement in the beam size .
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 and
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 , that here should be written for and . We then have
The integrations over the angles between and and between and give each a factor leading to
which can be expressed in terms of the angular correlation function of the magnification.
A.3.2. When and
This case is a particular case of Appendix A.2.1.
A.4. When the four directions coincide
In this case we have
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,
To complete the calculation it is interesting to have in mind the property (Bernardeau 1995),
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
Then, using this property it is easy to show that
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]),
We have thus
As a result
In the following I assume a power law behavior for both the mass power spectrum and ,
Then using the property,
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 and of interest.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998