1. Spherical collapse of warm dark matter
Timothy Chan
March 2, 2015
Abstract
A region in which a small and spherically symmetric density per-
turbation sits on top of a uniform background can be shown to evolve
independently of the background. In this report, the gravitational col-
lapse of a spherical tophat perturbation is numerically simulated. An
attempt is them made to simulate the collapse of warm dark matter
using a sound speed approximation, leading to a discussion of the ve-
locity distribution of matter in the early universe. Finally, the results
of the numerical simulation are compared with the results expected
from Jeans instability analysis.
1 The Spherical Collapse Model
The spherical collapse model, introduced by Gunn and Gott in 1972 [1], as-
sumes that there existed small, spherically symmetric density perturbations
in the early universe. It can be shown that such a perturbation evolves in-
dependently of its surroundings. To characterise the perturbation, we define
the density inside the region of interest as ρ(R) = δ(R)¯ρ, where ¯ρ is the mean
background density of the universe and R is the physical distance from the
perturbation’s center. While ρ can, in principle, be a function of R, we will
assume that our perturbation is initially a sphere of constant density ρi > ¯ρi
surrounded by a shell with density ρi < ¯ρi such that the average density over
the region is equal to ¯ρi.
Since we ignore all non-gravitational effects in section 2, the simulation
presented in that section is only valid for “cold dark matter” (CDM). It turns
out that this is an acceptable model of the universe because dark matter
comprises a large majority of the total matter in the universe. Later, in
section 3, we will give the perturbation a thermal velocity, allowing us to
model the collapse of so-called “warm dark matter” (WDM).
1

2. Before developing any equations of motion, we first introduce the scale
factor a, a measure of the relative size of the universe defined by the relation
d(t) = a(t)d0, where d(t) is the proper distance at time t between two objects
that move with the Hubble flow, and d0 is the proper distance at reference
time t0. For convenience, we choose the reference time t0 to be the present
time. This sets the current value of a as 1, with a decreasing towards 0 as one
goes back in time. We can now relate the mean background matter density
to its present value ¯ρ0 using the equation ¯ρ(t) = ¯ρ0/a(t)3
.
From a, we further define the comoving radius r ≡ R/a, and the con-
formal time τ characterised by the equation dt ≡ adτ. The benefit of using
these definitions is that it allows the expansion of the universe to be easily
absorbed into the equations of motion that we are about to develop.
2 CDM Collapse
2.1 Methodology
We begin with a Lagrangian L(r, ˙r, τ) = a [T − V ] for a test particle with
mass m and velocity ˙r ≡ dr/dτ moving in a spherically symmetric gravita-
tional potential well φ(r, τ) generated by the CDM perturbation. Because
our region is spherically symmetric and carries no angular momentum, the
motion of the test particle is confined to a radial line. This allows us to
express the Lagrangian simply as
L(r, ˙r, τ) = a
1
2
m ˙r2
− mφ(r, τ) . (1)
The canonical momentum conjugate to r is pr ≡ ∂L/∂ ˙r = am ˙r, leading to
the Hamiltonian
H(r, pr, τ) = ˙rpr − L
=
1
2
am ˙r2
+ amφ(r, τ)
=
1
2am
p2
r + amφ(r, τ). (2)
By applying Hamilton’s equations dr/dτ = ∂H/∂pr and dpr/dτ = −∂H/∂r,
we obtain
dr
dτ
=
pr
am
,
dpr
dτ
= −am. (3)
We now choose to rewrite equation 3 in terms of redshift z and the nor-
malised quantity ur ≡ pr/m = a ˙r, thus rendering our equations of motion
2

3. into the form
dr
dz
= −
ur
˙a
,
dur
dz
=
a2
˙a
∂φ
∂r
, (4)
where a = (1 + z)−1
and ˙a ≡ da/dt = H0 a−1 Ωm,0 + a2 ΩΛ,0 are, respec-
tively, the scale factor and the expansion rate.[2]
To obtain the gravitational potential φ(r, τ), we apply the Poisson equa-
tion
2
φ ≡
1
r2
∂
∂r
r2 ∂φ
∂r
= 4πGa2
¯ρ(τ)δ(r, τ), (5)
which can be solved to give
∂φ
∂r
=
4πGa2
r2
r
0
¯ρ(τ)δ(r , τ)r 2
dr =
G
ar2
Mp(r), (6)
where Mp(r) ≡ a3 r
0 ¯ρ(τ)δ(r , τ)r 2
dr can be identified as the enclosed phys-
ical mass of the overdensity ¯ρ(τ)δ(r , τ).
Recall that near the beginning of this report, we postulated a perturba-
tion in the form of an overdense sphere surrounded by an underdense shell.
Since ∂φ/∂r has no dependence on masses lying outside radius r, we can
effectively ignore the underdense shell and focus entirely on the sphere that
it encapsulates.
We proceed with a Lagrangian method by dividing the region into N
shells, each carrying a particular (conserved) mass. Because of spherical
symmetry, the problem is reduced to 1-D and we can simply apply 4 to each
shell separately.
The mass distribution of each shell is assumed to be a gaussian with
standard deviation σ equal to half of the shell’s width. This choice of σ is
large enough to produce the desired tophat density profile on the interior
of the perturbation, but not so large that it obfuscates large-scale density
fluctuations that arise during the collapse.
A brief outline of the simulation is as follows:
1. Input an initial redshift zi, and use a = (1 + z)−1
to find ai.
2. Input a total mass Mtot and an initial (tophat) density fluctuation δi,
using ¯ρi = ¯ρ0/a3
i to find the initial size ri of the region.
3. Break the region into evenly spaced shells, calculating their masses
from their volumes, δi, and ¯ρi.
4. Apply equation 4 to each shell at each timestep using a Runge-Kutta
method, recording radii and times.
3

4. 5. Define the turnaround time tta as the time at which the physical radius
R = ra of the outermost shell begins to decrease.
6. Define the the collapse time tc as the time at which the physical radius
of the outermost shells reaches 0.
7. Plot the physical radius of the region as a function of time.
It is also worth noting several of the computational techniques that were
made in this simulation:
1. Plummer Softening: The equation for potential of the system is mod-
ified by inserting a smoothing factor to prevent divergences in the po-
tential and acceleration during collapse:
φ = −
G
a
√
r2 + 2
Mp(r), (7)
∂φ
∂r
=
Gr
a(r2 + 2)3/2
Mp(r), (8)
where , the “smoothing scale”, is set as half of the width of each shell.
2. Adaptive step size: If any of the properties of the region (time, shell
position/velocity) change too much in a single timestep, the step is
undone and re-simulated using a lower step size. This ensures higher
precision and smoother output curves.
3. Choice of units: It was found that by expressing r in terms of kpc, ve-
locities in terms of m/s, and masses in terms of M , many of constants
and values became easier to numerically manipulate. This reduces the
simulation’s load, freeing up resources that were instead used to im-
prove precision.
2.2 Results
The primary feature of the collapse is that the problem exhibits a scale-
independence: the mass of the region makes no difference to the evolution
of the system. Instead, the collapse is defined entirely by the value of δi and
zi. From figure 1, we see that a lower δi leads to a slower collapse. In all
cases, the collapse is approximately symmetric with tc = 2tta, which agrees
with analytical solutions of CDM collapse.[3, 4]
Unless otherwise stated, we will henceforth assume a zi of 500 and a δi
of 5x10−5
, which corresponds to the red curve in figure 1. The rationale
4

5. 0
10
20
30
40
50
60
0 200 400 600 800 1000 1200 1400
R/Ri
Time (Myr)
Figure 1: Evolution of regions with different overdensities, taking zi = 500.
Green: δi = 10−4
. Red: δi = 5x10−5
. Blue: δi = 10−5
. Complete failure to
collapse occurs at around δi = 3x10−8
.
behind this choice of zi is that it roughly corresponds to the beginning of the
matter-domination era of the universe. δi has then been chosen such that
galactic collapse takes around 500 million years to complete. This is around
the time that the Milky Way took to form, based on the age of its oldest
stars.[5]
3 WDM Collapse
3.1 Methodology
The analysis of dark matter collapse presented in section 2 suggests that for
δi > 3x10−8
, any region, regardless of size, should eventually collapse to form
a bound structure. However, when we map the universe, we do not observe
isolated structures with small mass; instead, we see clusters of galaxies with
mass 1012
M or above. One way to explain this phenomenon is to give
the dark matter in the early universe a thermal velocity distribution (hence
“warm” dark matter). This creates a pressure that opposes the presence of
density gradients. Clearly, such an effect would counteract collapse, slowing
the evolution of the system and possibly preventing collapse altogether.
To include the effect of warm dark matter, it is necessary to modify
equation 4. We proceed from the Navier-Stokes equation for astrophysics,
now considering both gravitational interaction and a new pressure term P:
d ˙r
dτ
+ H ˙r =
P
ρ
+ φ = 0, (9)
5

6. 0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800 900
R/Ri
Time (Myr)
Figure 2: Evolution of regions with different masses but the same initial
sound speed v i = 10−3
. Blue: Mtot = 1015
M , Green: Mtot = 1012
M ,
Red: Mtot = 1011
M . The collapse time for the CDM case is 420Myr.
where H ≡ a−1
(da/dτ) = da/dt = ˙a.
As with equation 4, we convert the equation by introducing z and ur,
leaving us with the WDM equations of motion:
dr
dz
= −
ur
˙a
,
dur
dz
= −
a2
˙a
P
ρ
+ φ = 0 . (10)
We now replace the pressure gradient with the approximation P =
v 2
ρ, where v is the WDM’s average sound speed, which is dependent
on the velocity distribution of the matter in the region. We further assume
that the sound speed is slowed only by momentum loss due to the universe’s
expansion, i.e. v = v i(ai/a), where the subscript i denotes the initial
values of each quantity:
dr
dz
= −
ur
˙a
,
dur
dz
= −
a2
˙a
v 2
i
ai
a
ρ
ρ
+ φ = 0 . (11)
3.2 Results
In the previous section, it was noted that for cold dark matter, the evolution
of the collapse is independent of the mass of the region. However, when the
matter is given a velocity distribution, the nature and speed of the collapse
becomes dependent on the initial size of the region, and therefore, the total
mass of the collapsed object.
For the same sound speed, smaller regions/masses are more greatly im-
pacted by WDM. This occurs because the gravitational term, which depends
6

7. 0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800 900
R/Ri
Time (Myr)
Figure 3: Evolution of a 1011
M region for different initial sound speeds.
Blue: v i = 10−5
. Green: v i = 10−4
. Red: v i = 10−3
. Complete failure
to collapse occurs at around v i = 1.3x10−2
c. The collapse time for the
CDM case is 420Myr.
on the enclosed mass, is reduced, while the pressure term, which depends on
the density gradient, remains unchanged.
Figure 3 shows that up to a particular sound speed, the collapse of the
region is barely affected. We can compare this to the results roughly expected
from Jeans instability analysis. This is a method that involves calculating
a free streaming scale λ, equal to the distance that a particle with velocity
v = v can travel during the duration of the collapse.
Traditionally, this translates to the equation λ = v itc.[6] However, in
our case, we must account for the expansion of the universe, which causes a
decay in the sound speed (see derivation of equation 11), so we instead use
λ =
tc
ti
v dt =
tc
ti
v i
ai
a
dt. (12)
Objects smaller than λ are expected to have their collapse be affected
by the pressure term, whereas the evolutions of objects much larger than λ
should be virtually unchanged. The following table displays some compar-
isons between λ and the initial physical radius Ri, along with the extent to
which the collapse time is affected by v :
The results of the Jeans analysis fit quite well with the data obtained
from the simulation; when λ/Rin < 1, we see a < 10% increase in the collapse
time relative to the CDM case. As v i is increased and λ/Rin grows past 1,
the collapse time quickly diverges towards infinity. If v i is increased even
further, then most particles reach “escape velocity” and the region never
collapses.
7

8. v i λ/Rin tc
0 0 420.9
10−5
c 0.024 421.5
10−4
c 0.247 433.8
10−3
c 3.457 844.1
10−2
c n/a n/a
10−1
c n/a n/a
v i λ/Rin tc
0 0 420.9
10−5
c 0.011 421.1
10−4
c 0.113 421.7
10−3
c 1.327 577.9
10−2
c 36.84 4618
10−1
c n/a n/a
v i λ/Rin tc
0 0 420.9
10−5
c 0.001 420.9
10−4
c 0.011 421.1
10−3
c 0.113 421.7
10−2
c 1.327 577.9
10−1
c 36.84 4618
Table 1: Comparisons of the collapse time (in Myr) of regions with various
sizes and sound speeds. Left: Mtot = 1011
M . Middle: Mtot = 1012
M .
Right: Mtot = 1015
M . We expect tc to be relatively unaffected when
λ/Rin < 1. The label “n/a” indicates that the region fails to collapse at
the selected thermal velocity.
From the table, we see that a 1015
M region evolves the same way as a
1012
M region if v i is raised by a factor of 10. This is because v i acts on
the radius of the region, which is proportional to M
1/3
tot .
As a final observation, note that a 1015
M region collapses even when v i
is set at the extremely high value of 10% of the speed of light c. This suggests
that the formation of large scale structures such as galactic superclusters is
practically guaranteed by the Big Bang model. Meanwhile, by observing the
smallest isolated structures present in the universe, we can attain some rough
lower constraints on the thermal velocity of matter in the early universe.
4 Final discussion
Crucially, the simulation fell victim to what is believed to be numerical in-
stability when the initial sound speed v i was close to the critical value
v i,crit(ti, δi, Mtotal) that leads to non-collapse. A correction of this problem,
perhaps with a more robust iterative algorithm, would give better insight
into the behaviour of regions that barely or almost collapse.
The process of evaluating P at each timestep involved the summation
of N gaussians, where N is the number of Lagrangian shells used. Since
this had to be calculated for each shell, the computational complexity of
the simulation was of at least order N2
. A method of approximating this
process, or simply the application of greater computational resources, would
have allowed more accurate calculations to be performed.
8

9. Acknowledgements
First and foremost, thanks to Dr. Yvonne Wong for her supervision, guid-
ance, and input over the course of this summer research project. Thanks also
to the UNSW Faculty of Science and School of Physics for providing research
opportunities to students.
References
[1] J. E. Gunn and J. R. Gott, III. ApJ, 176:1, August 1972.
[2] A. Ringwald and Y. Y. Y. Wong. Gravitational clustering of relic neu-
trinos and implications for their detection. JCAP 0412, 005, 2004.
[3] G. Knodel. Spherical collapse of dark matter. June 2010.
[4] T. Padmanabhan. Structure Formation in the Universe. Cambridge Uni-
versity Press, 1993.
[5] A. Frebel et al. The Astrophysical Journal 660 (2): L117, 2007.
[6] E. Bertschinger. Cosmological dynamics: Course 1. 1993.
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