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

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6. Error evaluations due to the cosmic variance

In this section we investigate the effects of the cosmic variance on the precision of the measurements of the second moment and on [FORMULA]. The origin of this error is the finite size of the sample in which the measurement could be done. Typically we will assume that the sample size is about [FORMULA], which could be accessible in the coming years for the now available technologies. In this section we describe the measured local convergence, [FORMULA] as the sum of the exact local convergence and a noise [FORMULA],

[EQUATION]

The random noise [FORMULA] components are assumed to be all independent of each other, independent of the true values of the local convergence and to have the same variance [FORMULA],

[EQUATION]

It is reasonable to assume that [FORMULA] is of the order of [FORMULA] (see Appendix B). We then introduce the notation [FORMULA] which is the (connected) geometrical average of the quantity X in the sample. These quantities are also cosmic random variables and are thus expected to vary from one sample to another. Our goal is then to estimate the ensemble averages and variances of those geometrical averages for the quantities of interest, i.e. to know how good they are as estimates of the true cosmic quantities.

To do such calculations we have to introduce the expression of joint moments of the local convergence. Luckily, this is not too complicated! Indeed in the quasi-linear regime one can show that, when the separation between two cells is large compared to the smoothing angle, we have

[EQUATION]

where [FORMULA] are finite quantities. This is the application to 2D statistics of what was developed in 3D by Bernardeau (1996a). However contrary to the 3D case, and due to projection effects, these coefficients depend on the details of the adopted model. They can anyway be calculated numerically (see Appendix A). Using these results we can calculate both the expectation value of a measured quantity and the variance of this measure. Thus we have

[EQUATION]

The numerical calculations give the values of the required coefficients [FORMULA] (Appendix A), from which we have

[EQUATION]

The cosmic errors of course decrease with the size of the sample. What the previous results show is that the errors are already quite small when the size of the sample is [FORMULA] and would allow, for a perfectly well known source distribution, a determination of the cosmological parameters at the 10 to 15% level.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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