Cosmological simulations #7: Limitations and some considerations

Limitations

In the previous posts we encountered some of the limitations of cosmological
simulations. Let’s review these in detail.
First, we can consider a simulation composed of a finite box in a bigger space but to represent a real system, this box shouldn’t be isolated so we use the periodic boundary conditions (here). This means that all the space around the box is filled with images of the box itself: a particle that leaves the box from one side will come in
from the opposite side.

Second, the mass inside the box is not continuous. Instead, it is made by particles of mass of the order of $10^9$ solar masses. These particles represent collisionless fluid elements (made by a huge quantity of real particles) with a certain
volume and can’t be treated as solid spheres. When two simulation particles are
separated by a distance smaller than the radius of the volumes they represents
they must feel less than the force coming from the entire mass (thanks to the
Gauss/Birkhoff’s theorem). To do this we soften (read “we reduce”) the force at
such small scales (here). Third, time is not continuous and its discreteness was also treated (here) with some
criteria to decide the time steps.
Until now, however, we haven’t consider the effects of taking into account
initial density fluctuations over a range of scales that is finite. In
addition to this, the finite size of the box pose a limit on the force
resolution, because fluctuations on scales bigger than the box side will not
included in the simulation due to the way the Fourier transforms act on a
period box. Some tests in literature show that the exclusion of small
scales shouldn’t affect too much large scales when they reach the non linear
regime but this not holds for the exclusion of large scales, those scales bigger
than the box side. Following Bagla, the large scale exclusion should not
disturb the formation of small haloes but could change their distribution.
This effect will appear as an underestimation of the correlation function. Bagla
finds that the best way of quantifying the effects of long wave modes is to
check whether including them in the simulation will change the number of
massive haloes or not and this can be estimated using the Press-Schecther mass
function.
In Tormen&Bertschinger (1996) the missing power on large scales will cause
something like a statistical cosmic bias decreasing the number of high-density
regions, the strength of the clustering and the amplitude of the peculiar
velocities.
Methods have been developed to take the missing “larger than the box” wave modes
into account and we will have a look on these in a future post.

Some considerations

As we have seen (here) N-body cosmological simulations
are useful to understand aspects of non-linear gravitational clustering,
since it’s not possible to carry out laboratory experiments in gravitational
dynamics and the analytic models fail when the system reach the non linear
regime, i.e. when the density contrast overcome the unity. Related with
cosmological simulations there are a pair of aspects that Bagla underlines in its
articles that interesting to consider.
The first issue is whether or not the gravitational clustering
erase memory of initial conditions. Is there a one-to-one correspondence between
some characterization of initial perturbations and the final state?
N-body simulations shows that gravitational clustering does not erase memory of
the initial conditions, the final power spectrum is a function of the initial
power spectrum and this relationship can be written as a one-step mapping and
the functional form of this mapping depends on the initial power spectrum.
However density profiles of massive haloes have a form independent of
initial conditions but there is a considerable scatter in density profiles
obtained from N-body simulations and it is difficult to state whether a given
functional form is always the best fit or not. I must admit that these last concepts are not very clear to me at the moment, and that I trust Bagla but I will deepen them as soon as possible to be able to comfortably master them.
The second question is if it is possible to predict the masses and distribution
of haloes that form as a result of gravitational clustering.
The initial density field is taken to be a Gaussian random field and for
hierarchical models the simple assumption that each peak undergoes collapse
independent of the surrounding density distribution can be used to estimate the
mass function and several related quantities but N-body simulations shows that
this simple set of approximations is incorrect. However, the resulting mass
function estimation is fairly accurate over a wide range of masses. Merger rates
can be thus computed using the extended Press-Schecther formalism. Modifying
some of this assumption can lead to improved predictions.

References:

• J. S. Bagla, Cosmological N-body simulation: Techniques, scope and status
• J.S. Bagla and T. Padmanabham, Cosmological N-body simulations
• Giuseppe Tormen and Edmund Bertschinger, Adding long wavelenght modes to an N-body simulation
• S. Cole, Adding long-wavelength power to N-body simulations