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

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4. Quantitative predictions

4.1. The cosmological models

The cosmological models used to illustrate the previous results by quantitative predictions are standard CDM models with [FORMULA], [FORMULA] Mpc with an initial Harrison-Zel'dovich spectrum. Two cases have been chosen, [FORMULA], [FORMULA] (model 1) and [FORMULA], [FORMULA] (model 2). The transfer function and temperature power spectrum were both computed with the code of Seljak & Zaldarriaga (1996).

For convenience the mass fluctuation power spectra were approximated by simple analytic fits (similar to the ones proposed by Bond & Efstathiou 1984),






The normalization factor is given by the relation (12) with observational constraints,


The measured values of [FORMULA] and [FORMULA] by the COBE satellite (Mather et al. 1994, Gorski et al. 1994)


provide the normalization constraint for A,


Note that in the previous equations, distance units have been chosen so that [FORMULA].

4.2. Numerical results

I present the derivatives of the temperature angular correlation functions in Fig. 4. They exhibit a remarkable property, since they both drop to zero at a scale slightly larger than 1 degree. This property can be of great help to disentangle various contributions. For instance it implies that specific geometries produce a vanishing four-point correlation function. This is the case for instance when three of the four directions form an equilateral triangle with [FORMULA] of side length. Such a property could be of crucial interest to ascertain the origin of an observed four-point correlation function.

The two other quantities of interest are the functions [FORMULA], [FORMULA] and [FORMULA] (Eqs.  22, 26) that describe the magnitude of the lens effects. They are presented in Fig. 5. One can see that [FORMULA] and [FORMULA] dominate at small scale. The resulting shape of the dimensionless four-point correlation function [FORMULA] (Eq.  29) is presented in Fig. 6. It exhibits specific features induced by the [FORMULA] factor and by the derivative of the temperature angular correlation function. The fact that the latter vanishes is clearly present with a significant circular feature at 1 degree scale. It implies that the kurtosis is maximum for angular distances below 1 degree. Note that in this figure, all contributions have been included, but the contributions from (2b) are found to be negligible and not to affect the global features of this plot.

[FIGURE] Fig. 4. The function [FORMULA] as a function of the angle. The thick line is for model 1 and the thin line for model 2
[FIGURE] Fig. 5. The functions [FORMULA] (solid lines), [FORMULA] (dashed lines) and [FORMULA] (long dashed lines) for model 1 (thick lines) and model 2 (thin lines).
[FIGURE] Fig. 6. Contour plot of the function [FORMULA] (Eq. 29) as a function of the relative position (in degrees) of [FORMULA] when [FORMULA] is the central point of the graph and [FORMULA] is at the coordinates [FORMULA], [FORMULA]. The value of [FORMULA] has been multiplied by 1000.

I also present the quantities intervening in the expression of the dimensionless kurtosis of the CMB temperature PDF. The index of the local temperature fluctuations is given in Fig. 7 for the two cosmological models. The variance of the convergence is presented in Fig. 8, and the resulting value of [FORMULA] in Fig. 9.

[FIGURE] Fig. 7. The functions [FORMULA] for model 1 (thick line) and model 2 (thin line)
[FIGURE] Fig. 8. The functions [FORMULA] (solid lines) and [FORMULA] (dashed lines) for model 1 (thick lines) and model 2 (thin lines)
[FIGURE] Fig. 9. The function [FORMULA] (Eq.  34) for model 1 (thick line) and model 2 (thin line)

The resulting coefficient [FORMULA] can be as high as [FORMULA]. It depends however a lot on the cosmological models and more particularly on the amount of power at about [FORMULA] Mpc scale. The reason why [FORMULA], and consequently [FORMULA], is smaller for the [FORMULA] case is thus actually due to the change of shape of [FORMULA] (i.e., a lower value of [FORMULA]), and not directly to the low value of [FORMULA]. The dependence on the cosmological parameters is investigated in more details in the next subsection.

4.3. Dependence on the cosmological parameters

The dependence on the cosmological parameters enters in various aspects. It is important in particular for the relation between the [FORMULA] and the mass fluctuation power spectrum [FORMULA]. This is not the aim of this paper to explore in great details all these dependences. There is however a dependence which is specific of the lens effects, that is the shape of the efficiency function [FORMULA] (Eq.  5). In Fig. 10, I present the function [FORMULA] for different cosmological models. We can see that both the shape and the amplitude of this function strongly depend on the cosmological parameters.

[FIGURE] Fig. 10. The function [FORMULA] (Eq.  5) for the Einstein-de Sitter case (thick solid line), a model with [FORMULA], [FORMULA] (thin solid line) and a model with [FORMULA], [FORMULA] (dashed line).

The main dependence is actually due to the factor [FORMULA] appearing in the expression of [FORMULA]. This is however quite misleading since this factor cancels out with the one intervening in the expression of the normalization factor A (Eq.  39). The dependence on the cosmological parameters is thus mainly contained in the shape of the power spectrum, that is in the ratio between the amount of power at relatively small scale to the power at very large scale, although they enter significantly in the efficiency function [FORMULA], even when the overall [FORMULA] factor has been dropped. The way the two cosmological parameters [FORMULA] and [FORMULA] enter in this expression is unfortunately rather cumbersome. It is however clear that the CMB four-point correlation function contains informations on the cosmological parameters in a quite different combination compared to the one intervening in the temperature power spectrum. In particular the detection of such an affect may allow to better disentangle what is specific of [FORMULA] from what contribute directly to [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998