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

5. Discussion and conclusions

In this work, I have calculated the expression of the four-point temperature correlation function as induced by weak-lensing effects. For standard CDM model, the amplitude of this correlation function, in units of the square of the second, is found to be of order of . However, this estimation did not take into account the nonlinear evolution of the power spectrum that might significantly amplify this signal at small angular scale. This is for instance what is predicted for the two-point correlation function of the polarization of background galaxies (Jain & Seljak 1996). Unfortunately, in the case of CMB maps, the scale at which this effect might appear cannot be deduced straightforwardly from this work. This effect is indeed the result of a line of sight integration that mix different scales and different redshifts for a given selection function (e.g. Fig. 10) which is itself dependent on the redshift of the sources. Moreover all the intervening quantities have non trivial dependences on the cosmological parameters that should be taken into account. A detailed examination of the nonlinear effects is then left for a forthcoming paper.

I would like to stress, that the amplitude of the lensing effects should be large enough to be detectable, at least marginally, in full sky CMB anisotropy measurements. The possibility of doing such measurements is directly related to the cosmic noise associated with the quantities of interest, fourth moment or four-point correlation function. So far, no precise estimation of the cosmic noise for the four-point correlation function has been made, but following Srednicki (1993), who presented the calculation for the three-point correlation function, one expects the cosmic variance of those quantities to be of the order of where is the typical value of l contributing to the temperature fluctuations. One can observe that the value of is of the order of the signal, however, one should have in mind that a direct and too naive calculation of the cosmic noise may be actually misleading since the long wavelength fluctuations (corresponding to the low l part of the power spectrum) contribute significantly to it, whereas the lensing signal originates mainly from the small angular scales (below 1 degree). It suggests that the weak lensing signal might be more easily detectable in maps where the long wavelength temperature fluctuations have been removed. Moreover the detailed methods used to extract the signal might also be of different robustness against the cosmic noise. In particular, it could be fruitful to take advantage of the a priori knowledge of the geometrical dependence of the four-point correlation function (see for example the factor in the expression [ 30]). In a forthcoming paper, we explore the different possible strategies for the data analysis, and will present detailed estimations of the precision at which such a detection could be made in the future satellite missions.

It also has to be noted that other secondary effects or foregrounds may also contribute to the four-point correlation function, not to mention the case of more exotic cosmological models based on intrinsically non-Gaussian topological defects. In particular the nonlinear Doppler effects could induce a significant four-point correlation function, because it is caused by intrinsically non-Gaussian objects. There are however no reasons for these effects to have the same geometrical dependences, factor and dependence on the shape of the temperature two-point correlation function. Hence, it should be possible to distinguish this effect from other sources.

The most exciting aspect of this analysis is probably that the magnitude of the effect depends on the cosmological parameters, , and in a known way. The detection of the temperature four-point correlation function may thus reveal to be extremely precious to test the global picture of the large-scale structure formation, as it will be unveiled by CMB anisotropy measurements.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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