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Astron. Astrophys. 320, 365-377 (1997)

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7. Non-Gaussian fluctuations

In all of the above analysis, we have assumed that the fluctuations were Gaussian. This was implicit when we adopted for the function F in Eq. 2 the PS formula, which is known to reproduce the results of numerical simulations in the case of Gaussian fluctuations. Notice, however, that the appearance of the exponential in this formula is not trivial and should be considered as fortuitous since the function F is an extremeley complex quantity to evaluate, and that this has not yet been achieved even in the Gaussian case. It is not our goal to investigate any specific case of non-Gaussian fluctuations, since there is little physical motivation for any specific model. We would rather like to point out some differences that may result in such a case. For non-Gaussian fluctuations it is still possible to write the mass function as :

[EQUATION]

with

[EQUATION]

However, the function [FORMULA] is now arbitrary. It is then rather simple to show that the mass function will mimic a Gaussian fluctuation spectrum [FORMULA] which is related to the non-Gaussian perturbation spectrum by the following relation:

[EQUATION]

Because of the arbitrariness of the function [FORMULA], it is possible to fit the local properties of clusters, whatever the spectrum is, by using an appropriate distribution function. As an illustration, we have computed the distribution function for which an [FORMULA] spectrum would mimic an [FORMULA] spectrum. This is represented on Fig. 6. As is naively expected, the distribution function presents a tail towards high [FORMULA] which favors the formation of massive clusters.

[FIGURE] Fig. 6. Probability distribution functions: the Gaussian case (solid line) and the distribution function for which an [FORMULA] spectrum would mimic the mass function obtained with an [FORMULA] spectrum (dashed line)
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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