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Astron. Astrophys. 322, 1-18 (1997)

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4. The variance of the smoothed convergence

We are interested in the statistical properties of the local convergence at a given scale. We are more particularly interested in the variance at a scale corresponding to the linear or the quasi-linear regime where the interpretation in terms of the cosmological models is easier. For the usual cases, sources at a redshift of about unity, it corresponds to angular scales of about [FORMULA], or more.

4.1. The convolution with the window function

Throughout this paper we assume that the geometrical filtering is done with a top-hat window function of angular scale [FORMULA]. In the expression (42) the smoothing applies to the [FORMULA] factor and it gives in the small angle approximation (see Bernardeau 1995),


where [FORMULA] is the component of [FORMULA] along the line-of-sight, [FORMULA] its component in the transverse directions and [FORMULA] is the Fourier transform of the 2D top-hat window function,


([FORMULA] is the Bessel function of the first kind). Note that the expression (44) involves both the distance [FORMULA] and the variable [FORMULA]. These two quantities are identical only for cosmological models with zero curvature.

4.2. The variance in the linear regime

Using the definition (3) of the power spectrum and the relations (44, 42) we get


where [FORMULA] is the growing mode of the density contrast (cf. Eq. [3]).

We use again the assumption that the deflecting angle and the smoothing angle are small. It implies that


Then the integral over [FORMULA] yields a Dirac delta function in [FORMULA] leading to the expression,


In the following section we explore the properties of this function with different hypotheses for the power spectrum, the redshift distribution of the sources, and the cosmological parameters.

4.3. Realistic results for an Einstein-de Sitter universe

For an Einstein-de Sitter Universe, the distance takes a simple expression as a function of the redshift,


and we have,


We also know that the linear growth rate of the fluctuation is proportional to the expansion factor so that


Using the power spectrum given in (5) and the redshift distribution (2) for [FORMULA] and [FORMULA] we have calculated the angular dependence of the variance of the local smoothed convergence. The results are presented in Fig. 2. We can see that the typical amplitude of the convergence is of the order of a few percent. It starts to bend at an angular scale of about 5 to 10 degrees corresponding to the physical scale where the power spectrum bends down. This scale will be of crucial importance for the evaluation of the cosmic errors on the measured moments.

[FIGURE] Fig. 2. The expected variance of the convergence as a function of the smoothing angle (in degrees) for [FORMULA] (solid line) and [FORMULA] (dashed line) in (2) and assuming an Einstein-de Sitter Universe.

We can also notice that the results have a significant dependence on the adopted redshift distribution of the sources. In the next subsection we discuss in more detail this point.

4.4. Dependence with the redshift of the sources

We explore here the dependence of the second moment with the redshift of the sources assuming that they are all at a given redshift [FORMULA]. We also assume that the power spectrum is given by a power law spectrum. For an Einstein-de Sitter Universe the result reads,


where the angular distance [FORMULA] is given in the Eq. (27). For sources of redshift of the order of 1 or 2, and for a power law index of [FORMULA], we have approximately,


This result confirms the trend observed in the previous subsection. It already indicates that the redshift dependence of the second moment is rather large. Therefore any interpretation of a measurement of such a moment in terms of magnitude of the power spectrum would require a precise knowledge of the sources that have been used. Moreover the magnitude of the second moment depends as well on the values of the cosmological parameters [FORMULA] and [FORMULA]. The dependence of the results on those parameters is explored in the next subsection.

4.5. Dependence on the cosmological parameters

In this subsection we assume that the power spectrum is a power law and that the sources are all at redshift [FORMULA] or [FORMULA]. In Fig. 3 we then plot the dependence of the second moment on [FORMULA] for [FORMULA]. We have approximately


for [FORMULA] and


for [FORMULA]. The resulting [FORMULA] dependence is thus weaker than what one would naively expect from Eq. (48) (e.g. [FORMULA]). The fact that the [FORMULA] dependence is actually weaker is mainly due to the growth factor of the linear mode. Indeed at high redshift, for a given normalization at [FORMULA], the lower [FORMULA] the larger the density fluctuations are.

[FIGURE] Fig. 3. The expected dependence of the magnitude of the variance on [FORMULA] for [FORMULA] (solid line) and [FORMULA] (dashed line). The overall normalization is arbitrary.

In Fig. 4 we present a contour plot of the [FORMULA] and [FORMULA] dependence of the second moment. We can see that the expected magnitude of the second moment depends essentially on [FORMULA]. It is however degenerate with [FORMULA] when [FORMULA] is large. As expected, the [FORMULA] dependence is more important when the redshift of the sources is larger.

[FIGURE] Fig. 4a and b. Contour plots of the expected dependence of the magnitude of the variance on [FORMULA] (horizontal axis) and [FORMULA] (vertical axis) for [FORMULA] (left panel) and [FORMULA] (right panel). The lines are iso-values of the variance, normalized to the Einstein-de Sitter case and regularly spaced in linear scale. The variance is stronger at the top right corner of the panels.

4.6. The variance with two populations of sources

Interestingly when one considers jointly two different populations of sources, here with [FORMULA] and [FORMULA], the ratio of the two variances depends on [FORMULA] and [FORMULA] (see contour plot of Fig. 5) but independent of the normalization of the power spectrum. (The dependence on the power law index is weak).

[FIGURE] Fig. 5. Contour plot of the expected dependence of the magnitude of the ratio of the two variances on [FORMULA] (horizontal axis) and [FORMULA] (vertical axis).

This is thus, a priori, a way to disentangle the different parameters.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998