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Astron. Astrophys. 318, 1-10 (1997)

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3. Special clustering models in closed form

In the appendix we demonstrate how to construct special models by using "local forms" for the displacement vectors. Although analytically interesting, this procedure is cumbersome if applied to more complex initial data. In this section we describe how we can automate the process of finding closed form expressions for the perturbation potentials.

In general, we are interested in a class of initial data which can be represented by a finite Fourier sum of plane waves having random amplitudes and random phases. The random variables can, e.g., be specified in terms of a power spectrum of a Gaussian random density field. Usually, such initial conditions are generated by FFT (Fast Fourier Transform), a method which was also used to realize the generic model in (Buchert et al. 1994, Melott et al. 1995, Weiß et al. 1996). However, there are two limitations of this method which both restrict the power of spatial resolution, an advantage which is in principle offered by analytical solutions. One of these limitations is due to the limited CPU storage for employing the FFT routine, the other is due to a lack of precision which may arise by constructing the initially small displacements from a given density field, or by interpolating the particle displacements into a smooth density field (using, e.g., CIC binning), respectively. As an alternative, we suggest to solve the Poisson equations in eqs. (2) algebraically by comparing the coefficients of Fourier sums in the source terms and the perturbation potentials. This way the solutions can be calculated to high accuracy without hitting on CPU storage limitations. Since the model is a one-timestep mapping, the CPU time needed for the realization is still reasonably small ([FORMULA] particles require CPU times of a few hours for the generic model discussed below). However, we admit that the algebraic procedure to solve for all seven perturbation potentials in (2) is still limited by the CPU storage for the algebraic program, and the compilation time of, e.g., plot routines can be large for a large number of Fourier modes. Using the manipulation system Mathematica, we are able to construct all perturbation potentials for [FORMULA] Fourier modes on a workstation with 256M storage. The results obtained with this method have also been checked to solve the original equations using two algebraic manipulation systems (Reduce and Mathematica).

For the special models constructed algebraically in this way we have also run the previous code (using FFT), which constructs displacements from given density fields (A.G. Weiß , priv. comm.), and found as expected that the result is a slightly smoothed variant of a direct calculation pursued in the present work. At the same time, this was an independent check of the third-order program used in previous work (compare Weiß & Buchert 1993).

Besides the special model given in the appendix (Model I), i.e., for the initial potential ([FORMULA])

[EQUATION]

we have analyzed a generic model (Model II) with the following initial potential (here, the coordinates are normalized by [FORMULA]):

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The random coefficients have been determined by the requirement that the power spectrum had the slope [FORMULA] down to the smallest wavelength, and the r.m.s. density contrast had the same value as Model I. Model I is the model studied by Moutarde et al. (1991); it has also been used by Mo & Buchert (1990) (at first order) as a statistical toy-model, and by Buchert & Ehlers (1993) (at second order) to demonstrate secondary shell-crossings; Matarrese et al. (1992) and Kate Croudace (priv. comm.) have compared general relativistic with Newtonian dynamics with the help of this model.

All seven perturbation potentials and the corresponding displacement vectors are listed in the appendix for Model I. For Model II the potentials and the displacement vectors can be obtained on request (buchert@stat.physik.uni-muenchen.de).

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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