## 3. Special clustering models in closed formIn 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 (
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 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 () we have analyzed a generic model (Model II) with the following initial potential (here, the coordinates are normalized by ): The random coefficients have been determined by the requirement that the power spectrum had the slope 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). © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |