equilibrium and therefore can be approximated by hydrodynamic equations. These evolve projected distribution functions that are only a function of spacex, but not velocity (like f). However, we will not further worry about the influence of baryons in this thesis, but instead focus on the gravity-only or dark matter only case ρb = 0. We just want to note here that all qualitative statements in this thesis about the phase space distribution of dark matter still hold in the case that baryons were included. That is so, because dark matter still follows the collisionless Boltzmann equation and the Baryons have only a very indirect impact by modifying the source term in the Poisson equation. However, baryons can modify quantitative results.
2.3 The N-body method 35 Discretize ⇒
Continuous initial conditions Particle Ensemble
z ∼100
x
v
Sheet
x
v
N-body Particles
⇓ Evolve
z ∼0
x
v
Sheet
x
v
N-body Particles
True continuous solution ∼Monte-Carlo realization
Figure 2.7: Illustration of the N-body method in phase space. From linear theory we have given a very accurate continuous representation of the initial conditions around z ∼ 100 (top left). These initial conditions are discretized to a set of particles. This can be done for example by using the Eulerian xi,vi coordinates of a Lagrangian qi grid (top right).
The particle ensemble is then evolved as a Hamiltonian system using (softened) point-mass interactions to reach a final set of Eulerian coordinates (bottom right). These coordinates are typically interpreted as a Monte-Carlo realization of the true continuous solution of the Vlasov-Poisson system (bottom left).
approximation on a Lagrangian grid
qijk= ∆q
i j k
(2.56)
xijk=xza(qijk) (2.57)
vijk=vza(qijk) (2.58)
mijk=ρ0∆q3 (2.59)
These particles then follow the Newtonian equations of motion like given in (2.45) and
(2.46). The potential is inferred from a density field which is approximated as ρ(x) =X
miδ(x−xi) (2.60)
where δ is normally not chosen as an actual Dirac δ-function, but instead some softened version of this, e.g. in Springel (2005a) as
δ(r) = 1 h3W
|r|
h
(2.61)
W(u) = 8 π
1−6u2+ 6u3, u≤ 12 2(1−u)3, 12 < u≤1
0, u >1
(2.62)
where h = 2.8 is the radius of the softening kernel and is the Plummer equivalent softening parameter. With these definitions it is also possible to write the system as a Hamiltonian
X
i
mivi2 2a2(t) +1
2 X
i,j
mimjψ(xi−xj)
a(t) (2.63)
where ψ(x) is the pair potential which incorporates the softening and possibly periodic boundary conditions.
The density estimate as in (2.60) (which we will refer to as N-body density estimate in the future) is a very crude approximation to the actual density field. Typically the softening kernel is chosen much smaller than the mean particle separation h ∆q. As a consequence of this the N-body density is zero in the majority of the volume. As we shall see later, the actual density cannot be zero anywhere if dark matter is a continuous fluid (see chapter 3). However, the density estimate does not need to be accurate to get a reasonable evolution of positions and velocities. For those the force field is more relevant.
The force field is the density field convolved with a long range kernel which drops as 1/r2 and is not very sensitive to the immediate (noisy) surrounding of each particle.
2.3.2 Visualizations
The N-body method has been extremely successful at predicting the large scale structure of the universe as well as smaller scale properties of the dark matter density field. As such it was found that the density profiles of dark matter haloes can be described by the NFW-profile (Navarro et al., 1996)
ρ(r) = ρc r rs
1 + rr
s
2 (2.64)
This two-parameter profile can be seen as an elementary building block of the non-linear dark matter density field. There have been a large variety of predictions and successes which
2.3 The N-body method 37
Figure2.8:Equirectangularprojectionofthenon-lineardarkmatterdensityfield.Thisisasingleframefroma360◦ “virtualreality”moviethatcanbefoundunderhttps://youtu.be/uMBvgCYiUiI.Thenon-lineardensityfieldcontainsa largevarietyofdifferentstructuresonalargevarietyofdifferentlengthscales.Mostprominentlyseencanbeherehaloes andfilaments–aligninginacosmicweb.
were achieved on the back of N-body simulations (Frenk & White, 2012, for an overview).
Further they have provided us with intuitive and beautiful images and visualizations of the dark matter density field. Most famous are here probably the images and movies of the Millennium simulation (Springel et al., 2005a).
Since I have some of my own visualization material, I will use this here to provide the reader with an intuitive impression of how the dark matter density field looks in the nonlinear regime. In Figure 2.8 we show an equirectangular projection of the non-linear dark matter density field from a L= 155 Mpc/hcold dark matter N-body simulation with 5123 particles. This image is a frame from a 360◦ virtual reality (vr) - video which has been created in collaboration with Raul Angulo and can be found underhttps://youtu.
be/uMBvgCYiUiI. If opened on youtube, it is possible to change the direction of the view (with the mouse on a computer or by turning around with a phone). With a google card board device1 one can even watch this movie as a full vr movie. I can highly recommend trying this. I also provide a non-vr version under https://youtu.be/5r8iY4_7FRI. The videos use some of the interpolation techniques that will be described in section 2.4 to get a high quality density estimate.
The non-linear density field shows a variety of different structures - such as halos, filaments and pancakes - which assemble on a spider-web like pattern - also known as the cosmic web.
2.3.3 Artificial fragments
While cold dark matter simulations are mostly believed to give reliable results, simulations of warm dark matter create artificial numerical fragments during the early nonlinear stages of evolution. In Figure 2.9 we show examples of these. Wang & White (2007) have shown that these small haloes which align like “beads in a string” inside the filaments are numerical artefacts of the discreteness of N-body simulations. They are not expected in the true continuum solution of the Vlasov Poisson system. As a simple numerical experiment Wang
& White (2007) set up the collapse of the N-body realization of a perfectly homogeneous filament which, however, also produced such numerical fragments.
It is important to point out here again that N-body simulations of warm dark matter and cold dark matter only differ in their choice of initial conditions. CDM simulations use a power-spectrum that creates physical halos on all mass scales whereas WDM simulations have power spectra that do not create physical halos below some finite mass scale. It is therefore not entirely clear whether CDM are completely unaffected by this type of problem, or whether the problem is simply not so visible because it is overshadowed by actually physically forming halos.
The reason for the artificial fragmentation is not entirely clear. A possible argumen-tation (largely from Oliver Hahn, personal conversation) goes like this: When the N-body method is applied to a hot three-dimensional system like a halo, it profits from ergodicity.
In a time averaged sense the distribution function of the N-body system gives an accurate
1which can be obtained for less than 20 euros if one has a compatible phone
2.3 The N-body method 39
Figure 2.9: Artificial fragmentation in N-body simulations of warm dark matter. Left:
Figure 2 from Wang & White (2007) showing regularly spaced fragments in a filament.
Right: (figure from my master thesis St¨ucker (2015)) the filaments around a zoom-in halo are fragmenting into small lumps. In these warm dark matter simulations there should be no such small structures.
representation of the underlying distribution function, even if at any particular moment in time the representation is not quite perfect. However, this is only the case for systems which are hot in every dimension, i.e. which have a non-zero velocity dispersion in every direction. During the early stages of anisotropic collapse, however, systems are formed which only have a velocity dispersion support in one dimension (pancakes) or along two dimensions (filaments). The remaining dimensions remain cold in these cases. In these dimensions the realization-dependent noise does not cancel out in a time-averaged sense.
Instead any imperfect perturbation can grow over time and then form for example collapsed artificial halos.
The discreteness effects in warm dark matter simulations, put a question mark on the validity of the N-body scheme in general. In fragmented warm dark matter simulations the Monte-Carlo view is certainly not correct. That is the final N-body distribution can not be seen as a random realization of the true distribution function. Instead it contains additional features that remain after coarse graining. However, it is not clear how far the final distribution deviates from a Poisson sampling of the true distribution. Possibly it is good enough to identify problematic regions (e.g. artificial haloes) and ignore them in the analysis (Lovell et al., 2014) while the Monte-Carlo view might still apply in other regions - like e.g. the centers of massive haloes. It is important to understand this problem quantitatively to test the validity of the N-body scheme and its predictions. Therefore we will discuss in the next sections alternative simulation approaches that do not suffer from discreteness effects. They can be used to benchmark the N-body method.