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

3. The effects of weak lensing on temperature correlation functions

3.1. Statistical properties

From the first equation it is easy to see that the temperature, , observed in the direction is in fact the unaffected temperature coming from a slightly changed direction,

In the following I assume that the displacement is small compared to the angular scale at which the observations are made. This is a fair assumption since the displacement is at most of (for cores of clusters) and that the angular resolution of the future satellite missions does not go below . As a result it is always possible to expand the relation (6) with respect to the displacement,

where the Einstein index summation prescription is used.

It is important to have in mind that both quantities and are independent Gaussian fields. The primordial temperature field is Gaussian for inflationary scenario, and independent of the lens potential field because the last scattering surface is far from the intervening lenses (that will be found to be at redshift ). The lens potential field will be assumed to be in the linear regime, although this is not a crucial hypothesis ³.

In the following the correlation functions or moments will be calculated in the small angle approximation, for which the plane approximation for the last scattering surface can be made. Thus one can write,

where the coefficients obey Gaussian statistical rules. In particular,

where the are the "famous" describing the angular power spectrum.

On the other hand the random variables obey the statistics,

where is normalized to the present day. In order to produce a consistent set of power spectra it is important to have a consistent normalization for and . This can be obtained from the small l behavior of corresponding to the small k behavior of . In the following I will assume an Harrison-Zel'dovich initial spectrum, so that,

The coefficient A can be related to the small l behavior of l (see Hu 1995 for instance),

3.2. The two-point correlation function

The dominant corrective term for the angular two-point correlation function can be calculated from the expansion (7),

The corrective terms have been written up to the quadratic term in the large-scale structure density field. The previous expression can be written in Fourier space,

where is the angular distance between and , . This formula has been obtained using Eqs. (9, 10) defining the two power spectra. To complete the calculations one can use the small-angle approximation, well verified below 1 degree scale, which implies that

Then the integral over leads to a Dirac function in . We eventually have,

with

This result does not coincide apparently with the one of Seljak (1996, Eq. A.6), but simply because the exponential was not expanded in his expression. That this expansion can be done is amply justified by the fact that the displacements are small compared to the angular resolution scale. Then one recovers exactly the same expression. Note that the autocorrelation function of the displacement field is automatically introduced by the third term in (13).

3.3. Higher-order correlation function

It is quite easy to see that the weak-lensing effects do not introduce a three-point correlation function. It is indeed impossible to build a term of non-zero ensemble average involving three factors.

The first non trivial high order correlation function is thus the four-point correlation function. At this stage it is important to have in mind that the observable quantity is the connected part, , of the ensemble average, , that is the part which is obtained when the products of two point correlation functions that can be built have been subtracted out,

The connected part is obviously zero for the primordial field: it is a direct consequence of its Gaussian nature. The dominant term, in terms of weak lensing effects, is thus given by,

Roughly speaking it means that the four-point correlation function, in units of the square of the second, is proportional to the weak lensing angular correlation function. Although at this stage it is difficult to give definitive quantitative predictions, the magnitude of the fourth order correlation function should be about (the order of the corrective term in [ 13]), which should be easily detectable in full sky coverage CMB maps. The expression of the four-point correlation function can be given in terms of the power spectra,

This expression can be calculated with an integration over the angles between and and and respectively. It yields the Bessel functions and . The results can thus be expressed in terms of the angular derivative of the two-point correlation function,

and with quantities associated with the angular correlation of the displacement field,

leading to (Appendix A),

where is the angle between and and is the angle between and (see Fig. 1). Two terms are thus involved. The a priori dominant term is the one in , and it is weighted by the cosine of the angle (see Fig. 1), that is the angle between the directions and on the sky. It gives a clear geometrical dependence for the four point-correlation function. However, one should have in mind that 11 other terms have to be taken into account in this calculation. This signal may therefore be masked by other geometric dependences.

Fig. 1. Description of the angles intervening in the expression (24) of the four-point correlation function. The thick solid line materializes the factor, whereas the hatched lines represents the correlation functions of the displacement field.

Quantitative calculations can be done for specific cosmological models (see next section). However, one obvious problem for a practical determination of this correlation function is that it depends on 5 different variables. It is thus crucial to reduce the number of variables by considering simplified geometries.

3.4. Peculiar geometries

3.4.1. When two directions coincide

The first geometry one may think of is when two points are merged together, that is the expression of . This notation is actually a bit oversimplified since the local temperature fluctuations are actually filtered by the used apparatus. One should thus have in mind that the two directions denoted are actually close random directions in a beam centered on .

Of course, once again, many terms are contributing to this ensemble average but I will first concentrate on the case where the connection between the two is made by the lens coupling term (Appendix 2.2). In such a case one can see that is given by the angle between and and is not affected by the smoothing. This is not the case for the term in which is expected to vanish because it is averaged to zero (more precise derivations are given in the Appendix). This contribution is thus proportional to the cosine of the angle, and to the autocorrelation function of the displacement field.

Fig. 2. Geometrical representation of the terms intervening in the expression (25)

What about the other terms? Their geometrical representations are given in Fig. 2. The (2a) diagram is the term that has just been considered and is expected to dominate the final expression. Note that all these diagrams have a symmetry factor of 2 compared to what is given in the Appendix. At first view the (2b) diagram vanishes because takes a random value averaging both and to zero. A more detailed calculation is proposed in the Appendix. It shows that it gives a contribution proportional to the angular correlation function of the local weak lensing convergence (see Blandford et al. 1991, Villumsen 1996, Bernardeau et al. 1996),

The two other diagrams are simpler since they can be simply obtained from the general expression (24). Taking all these terms into account we have,

where is shown on Fig. (2a) ( corresponds to the diagram obtained when the roles of and are inverted) and is the smoothing angle of the experiment. In this expression, the terms have been neglected, has been changed in for diagram (2a) and C in in (2b). This is due to the filtering effects. One should indeed take into account the average displacement within the beam size of the experiment in the first case, the rms temperature fluctuations in the second. More precisely we have,

and

with, for a top-hat window function for instance,

It is then interesting to define the function,

which is a dimensionless quantity. It does not depend in particular on the magnitude of the CMB temperature fluctuations and it is directly proportional to the large-scale structure power spectrum, with a known dependence on the shape of the anisotropy power spectrum. This quantity would thus be a measure the weak lensing effects in CMB maps. Taking into account the fact that the diagram (2b) generally leads to a negligible contribution we have,

A 2D contour plot of the function is proposed in Fig. 6 for a peculiar cosmological case.

The main contribution to this expression is the term coming from the diagram (2a), but the contributions from (2c) and (2b) cannot be neglected because the correlation function of the displacement field is only slowly decreasing with the angle (see next section). It is interesting to have in mind the physical effect described by this result. It corresponds indeed to a shear effect that is the deformation of two nearby temperature patches by a single lens. It can thus be seen as the excess of close temperature peaks that are elongated in the same direction. It is therefore logical that this effect is proportional to the correlation function of the displacement field. However, it does not always dominated the four point correlation function. This is the case in particular at the degree scale since the derivative of the temperature correlation function vanishes (see Fig. 4 in the next section). The next subsection is devoted to a simpler geometry where this case is more specifically investigated.

3.4.2. When two pairs coincide

This case is obtained from the previous case when the directions and coincide. The geometrical representations of the involved terms are presented in Fig. 3. The a priori dominant term corresponds to the diagram (3b). However, as noted previously, both the diagrams (3b) and (3c)) vanish for specific values of the distance between and . In such a case we are left with the diagram (3a). A crude evaluation of this diagram gives zero because the averages over the angles and are expected to vanish. However this is true only when the smoothing angle is negligible compared to the angular distance between and . More precise calculations (Appendix A.3) show that this term is proportional to the logarithmic derivative of the temperature variance in the beam. The resulting value of the correlation function is (taking into account a symmetry factor of 4),

Fig. 3. Geometrical representation of the terms intervening in the expression (31).

Note that the first term is no more proportional to the angular correlation of the displacement field, but to the angular correlation of the convergence (that is, within a factor 2, of the magnification in the weak lensing regime). Here the physical mechanism has changed. The correlation function is not due to a local shear, but to single lens amplification of patches on the primordial sky that create an excess of close bright peaks. It is thus clearly a magnification effect.

3.4.3. When the four directions coincide

The last case I consider is when the four directions coincide. In this case the magnification effect always dominates and the results, obtained in Appendix A.4, are similar to the one discussed in the previous subsection, but with small changes introduced by the filtering effects,

with

This result has been properly demonstrated for a top-hat window function, and should be roughly correct for other window functions. Note that this expression is also the fourth cumulant of the local filtered temperature probability distribution function. We can as well define the dimensionless quantity,

which tells us that the dimensionless kurtosis of the local filtered CMB temperature probability distribution function is proportional to the variance of the local filtered convergence.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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