and Cosmology

7.5 Non-Linear Structure Evolution
and the behavior on these small scales is affected by
numerical artefacts.
Computation of the Force Field. The computation of
the force acting on individual particles by summation,
as in (7.36), is not feasible in practice, as can be seen
as follows. Assume the simulation to trace 10 8 particles,
then in total 10 16 terms need to be calculated using
(7.36) – for each time step. Even on the most powerful
computers this is not feasible today. To handle this
problem, one evaluates the force in an approximate way.
One first notes that the force experienced by the i-th particle,
exerted by the j-th particle, is not very sensitive to
small variations in the separation vector r i − r j , as long
as these variations are much smaller than the separation
itself. Except for the nearest particles, the force on
the i-th particle can then be computed by introducing
a grid into the cube and shifting the particles in the simulation
to the closest grid point. 7 With this, a discrete
mass distribution on a regular grid is obtained. The force
field of this mass distribution can then be computed by
means of a Fast Fourier Transform (FFT), a fast and
very efficient algorithm. However, the introduction of
the grid establishes a lower limit to the spatial force
resolution; this is often chosen such that it agrees with
the softening length. Because the size of the grid cells
also defines the spatial resolution of the force field, it
is chosen to be roughly the mean separation between
two particles, so that the number of grid points is typically
of the same order as the number of particles. This is
called the PM (particle–mesh) method. To achieve better
spatial resolution, the interaction of closely neighboring
particles is considered separately. Of course, this
force component first needs to be removed from the
force field as computed by FFT. This kind of calculation
of the force is called the P 3 M (particle–particle
particle–mesh) method.
Initial Conditions and Evolution. The initial conditions
for the simulation are set at very high redshift.
The particles are then distributed such that the power
spectrum of the resulting mass distribution resembles
a Gaussian random field with the theoretical (linear)
power spectrum P(k, z) of the cosmological model. The
7 In practice, the mass of a particle is distributed to all 8 neighboring
grid points, with the relative proportion of the mass depending on the
distance of the particle to each of these grid points.
equations of motion for the particles with the force field
described above are then integrated in time. The choice
of the time step is a critical issue in this integration,
as can be seen from the fact that the force on particles
with relatively close neighbors will change more
quickly than that on rather isolated particles. Hence, the
time step is either chosen such that it is short enough for
the former particles – which requires substantial computation
time – or the time step is varied for different
particles individually, which is clearly the more efficient
strategy. For different times in the evolution, the particle
positions and velocities are stored; these results are
then available for subsequent analysis.
Examples of Simulations. The size of the simulations,
measured by the number of particles considered, has
increased enormously in recent years with the corresponding
increase in computing capacities and the
development of efficient algorithms. In modern simulations,
512 3 or even more particles are traced. One
example of such a simulation is presented in Fig. 7.10,
where the structure evolution was computed for four different
cosmological models. The parameters for these
simulations and the initial conditions (i.e., the initial realization
of the random field) were chosen such that the
resulting density distributions for the current epoch (at
z = 0) are as similar as possible; by this, the dependence
of the redshift evolution of the density field on
the cosmological parameters can be recognized clearly.
Comparing simulations like these with observations has
contributed substantially to our realizing that the matter
density in our Universe is considerably smaller than
the critical density.
Massive clusters of galaxies have a very low number
density, which can be seen from the fact that the massive
cluster closest to us (Coma) is about 90 Mpc away.
This is directly related to the exponential decrease of
the abundance of dark matter halos with mass, as described
by the Press–Schechter model (see Sect. 7.5.2).
In simulations such as that shown in Fig. 7.10, the simulated
volume is still too small to derive statistically
meaningful results on such sparse mass concentrations.
This difficulty has been one of the reasons for simulating
considerably larger volumes. The Hubble Volume
Simulations (see Fig. 7.11) use a cube with a side length
of 3000h −1 Mpc, not much less than the currently visible
Universe. This simulation is particularly well-suited
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