Structure Formation with Scalar Field Dark Matter: The Fluid Approach
Abril Su´ arez∗1, ∗and Tonatiuh Matos∗1, †
1Departamento de F´ ısica, Centro de Investigaci´ on y de Estudios Avanzados del IPN, A.P. 14-740, 07000 M´ exico D.F., M´ exico.
(Dated: January 24, 2011)
The properties of nearby galaxies that can be observed in great detail suggest that a better theory
rather than Cold Dark Matter would describe in a better way a mechanism by which matter is more
rapidly gathered into Large Scale Structure such as galaxies and groups of galaxies. In this work
we develope and simulate a hydrodynamical approach for the early formation of structure in the
Universe, this approach is also based on the fact that Dark Matter is on the form of some kind of
Scalar Field with a potencial that goes as 1/2m2Φ2+ 1/4λΦ4, the fluctuations on the SF will then
give us some information about the matter distribution we observe these days.
One of the most fundamental problems in modern Cos-
mology is to know the nature of dark matter. It is known
that in the standard cosmology the relativistic Big Bang
theory is a good description for our expanding Universe.
In this model, 4% of the mass in the Universe is in the
baryons, 22% is non-baryonic dark matter and the rest
in some form of cosmological constant.
Another idea that has predominated over the last
years, is that of a homogeneous and isotropic Universe,
although it has always been clear that this homogene-
ity and isotropy are only found until certain level. Now
we know that these anisotropies are very important and
can grow as big as the large scale structure we see today.
Nowadays, the most accepted model of dark matter (DM)
in the Universe is cold dark matter (CDM) because most
of the cosmological observations, like the anisotropies of
Cosmic Microwave Background (CMB) support it. How-
ever, this paradigm has some problems at galactic scales.
In the Big Bang model, gravity plays an essential role,
it collects the dark matter in concentrated regions de-
nominated “Dark matter halos”. In the large dark mat-
ter halos, the baryons are believed to be dense enough so
to radiate enough energy for them to collapse into galax-
ies and stars. The most massive halos, that are natural
hosts for the brightest galaxies, are formed in regions
were the local mass density is the highest. Less massive
halos, which are hosts for the less bright galaxies, appear
in regions with lower local densities, i.e, in regions were
the local density is not well defined . These situa-
tions appear to behave as in our immediate extragalactic
neighborhood. In the Local Sheet, SGZ=0 in extragalac-
tic coordinates, all seems to work alright, but there are
Observations seem to point to a better understanding
of the theory which begins in the less occupied space
called the “Local Void”, which contains just a few galax-
ies, which are bigger than the expected for this region.
This problem would be solved, if the structure grew faster
than it does in the standard theory, therefore filling the
local void and giving rise to more matter in the surround-
Another problem arises for the so called “Pure disk
galaxies”, which do not appear in numerical simulations
of structure formation in the standard theory, because it
is believed that their formation which is relatively slow
began in the thick stellar bulges. Again this problem
would be solved for the early formation of structure, with
the early rain of extragalactic debris ending into galaxies.
Recently, several alternative hypothesis have been pro-
posed in order to solve the potential problems of the
standard ΛCDM model. One of these models is called
Scalar Field Dark Matter (SFDM) . This model sup-
poses that dark matter, which only interacts gravitation-
ally with the rest of the matter, is a real scalar field
Φ minimally coupled to gravity that is endowed with a
scalar potential V (Φ),  used a potential of the form
V (Φ) = V0[cosh(ξΦ) − 1], where V0and ξ are constants,
to perform a first cosmological analysis in the context of
SFDM. They showed that the expansion rate and evo-
lution of the Universe and the linear perturbations in
this model are identical as those derived in the stan-
dard model. Recently, in  we developed a formal-
ism to show that a scalar field with a quadratic potential
V (Φ) = m2Φ2/2 can reproduce the cosmological evolu-
tion of the Universe.
In the SFDM model, the dark matter particle is a spin-
0 boson. Therefore they can form Bose-Einstein Con-
densates (BEC). SFDM forms a BEC if the mass of the
associated particle, m, is < 10−17eV , . Several
authors have studied numerically the collapse and virial-
ization of SFDM/BEC.  found that the critical mass
for collapse in the SFDM/BEC model is of the order of a
Milky Way-sized halo mass , , . This suggests
that SFDM/BEC can be a plausible candidate to dark
matter in galactic halos. Our thermodynamical analysis
of BEC indicates that gravitational structures of SFDM
can be formed at earlier times than CDM structures .
In a recent paper,  studied the conditions for the for-
mation of a SFDM/BEC in the Universe, also concluding
that SFDM/BEC particles must be ultra light bosons.
arXiv:1101.4039v1 [gr-qc] 20 Jan 2011
Other studies show that SFDM/BEC predicts intriguing
phenomena at galactic scales. For instance, it has been
pointed out that SFDM/BEC can explain the recent ob-
servations, X-ray maps and weak gravitational lensing,
of the spatial separation of the dark matter from visible
matter in collisions of galaxy clusters, such as, the Bullet
Summarizing, with the mass mΦ∼ 10−22eV and only
one free parameter, the SFDM model fits the following
1. The cosmological evolution of the density parame-
ters of all the components of the Universe .
2. The rotation curves of galaxies  and the central
density profile of LSB galaxies ,
3. With this mass, the critical mass of collapse for a
real scalar field is just 1012M?, i.e., the one ob-
served in galactic haloes .
4. The central density profile of the dark matter is flat
5. The scalar field has a natural cut off, thus the sub-
structures in clusters of galaxies is avoided natu-
rally. With a scalar field mass of mΦ ∼ 10−22eV
the amount of substructures is compatible with the
observed one .
6. SFDM predicts galaxy formation earlier than the
cold dark matter model, because they form BEC at
a critical temperature Tc>>TeV. So, if SFDM is
right, we have to see big galaxies at high redshifts.
Here we assume that the dark matter is described by a
scalar field Φ endowed with the self interacting, massive
scalar field potential V (Φ) = m2/2Φ2+λ/4Φ4. The pa-
per is organized as follows. In section 1 we analyze the
analytical evolution of our SF, then in section 2 we treat
our SF as a hydrodynamical fluid in order to study its
evolution for the density contrast, in section 3 we com-
pare the results of a previous section with those obtained
by CDM for the density contrast in the radiation domi-
nated era just before recombination and finally our con-
1. THE BACKGROUND
In this sections we perform a transformation in order
to solve the Friedman equations analytically with the ap-
proximation m << H. The scalar field (SF) we deal with
depends only on time, Φ = Φ0(t), and of course the back-
ground is only time dependent as well.
We use the Friedmann-Lemaˆ ıtre-Robertson-Walker
(FLRW) metric with scale factor a(t). Our background
Universe is composed only by SFDM (Φ0) endowed with
a scalar potential. We begin by recalling the basic back-
ground equations. From the energy-momentum tensor T
for a scalar field, the scalar energy density T0
scalar pressure Ti
jare given by
0= −ρΦ0= −
where the dots stand for the derivative with respect to the
cosmological time and δi
jis the Kronecker delta. Thus,
the Equation of State (EoS) for the scalar field is pΦ0=
Notice that background scalar quantities at zero order
have the subscript 0. Now the following dimensionless
variables are defined
being κ2≡ 8πG, H ≡ ˙ a/a the Hubble parameter and
the commas stand for the derivative with respect to scalar
field. Here we take the scalar potential as V = m2/2Φ2+
λ/4Φ4, where, m = µc/? and µ is the mass given in
kilograms, then for the ultra-light boson particle we have
that m ∼ 10−22eV.
With these variables, the density parameter ΩΦfor the
background 0 can be written as
ΩΦ0= x2+ u2.(4)
In addition, we may write the EoS of the scalar field as
Since ωΦ0is a function of time, if its temporal average
tends to zero, this would imply that Φ2-dark matter can
be able to mimic the EoS for CDM.
Now we express the SF Φ0for the background in terms
of the new variables S and ˆ ρ0,
with this we obtain
Φ0= 2ˆ ρ0cos(S − mt),(6)
0= ˆ ρ0
cos(S − mt)
− 2(˙S − m) sin(S − mt)
To simplify, observe that the uncertinty relation im-
plies that µc2∆t ∼ ?, and for the background in the
non-relativistic case the relation˙S/m ∼ 0 is satisfied.
Notice also that for the background we have that the
density goes as (ln ˆ ρ0)˙ = −3H, but we also have that
H ∼ 10−33eV<< m ∼ 10−22eV, so with these considera-
tions at hand for the background in (7) we have,
0= 4m2ˆ ρ0sin2(S − mt)
Finally, substituting this last equation and equation
(6) into (1), we obtain that
ρΦ0= 2m2ˆ ρ0[sin2(S−mt)+cos2(S−mt)] = 2m2ˆ ρ0. (8)
Comparing this result with (4) we have that the iden-
tity ΩΦ0= 2m2ˆ ρ0holds for the background, so compar-
ing with (8),
We plot the evolution of the potentials (9) in Fig.1,
where for the evolution we used the e-folding number N
defined as N = ln(a) and the fact that a ∼ tn→ t ∼
eN/n. In terms of the two analytic results (9) Fig.1 shows
the kinetic and the potential energies of the scalar field.
x =2 ˆ ρ0msin(S − mt)
u = 2 ˆ ρ0mcos(S − mt).(9)
0.0001 0.001 0.01 0.1 1
0.0001 0.001 0.01 0.1 1
FIG. 1: Analytical evolution of the kinetic (top panel) and the
potential (low panel) energy of the scalar field dark matter.
Observe the excelent accordance with the numerical
results in  for the kinetic and potential energy of the
background respectively. In what follows we will use the
analytic expresion (9) instead of the numerical ones.
2. SCALAR FIELD FLUCTUATIONS
If dark matter is some kind of elemental particle with
mass m, then it would be about 1068m GeV−1particles
to follow in a single galaxy.
Here we describe a model for the non-interacting mat-
ter such that: i) It can describe it more as a field than
as particles and ii) We find a fuction that only depends
on the three spatial coordinates and time.
Now a days it is known that our Universe is not exactly
isotropic and spatially homogeneous like the Friedmann-
Lemaitre-Robertson-Walker (FLRW) metric describes.
There exist small deviations from this model, and if we
believe these deviations are small enough, they can be
treated by the linear perturbation theory.
Then, if dark matter is composed of scalar particles
with masses m << 1eV, the occupation numbers in
galactic halos are so big that the dark matter behaves
as a classical field that obeys the Klein-Gordon equation
(?2+ m2)Φ = 0, where 2 is the D’Alambertian and
c = 1.
By definition, a perturbation done in any quantity,
is the difference between its values in some event in
real space-time, and its corresponding value in the back-
ground. So, for example for our scalar field (SF) we have
Φ = Φ0(t) + δΦ(? x,t),(10)
where the background is only time dependent, while the
perturbations also depend on the space coordinates. Sim-
ilar cases apply for the metric;
g00 = −a2(1 + 2φ),
g0i = a2Bi,
gij = a2[(1 − 2ψ)δij+ 2Eij].
Here the scale factor a depends on the conformal time,
ψ is a perturbation associated to the curvature and E
is asociated to the expansion. We will work under the
Newtonian gauge, which is defined when B = E = 0. An
advantage of using this gauge is that here the metric ten-
sor gµνis diagonal, and so the calculations become much
easier. We will only work with scalar perturbations, vec-
tor and tensor perturbations are eliminated from the be-
ginning, so that only scalar perturbations are taken into
account. Another advantage in using this gauge is that φ
will play the role of the gravitational potential and then
this will help us to have a simpler physical interpreta-
tion, i.e., both potentials φ and ψ are then related. This
metric has already been used in other works .
For the perturbed Klein-Gordon where we have used
equation (10) an we have set˙φ = 0, we have:
δ¨Φ + 3Hδ˙Φ −1
a2∇2δΦ + V,ΦΦδΦ + 2V,Φφ = 0(12)
The SF Φ has very hard oscillations from the begin-
ning, this oscillations are transmitted to the fluctuations
which apparently seems to grow very fast and are too
big. Nevertheless, this behavior is not physical, because
we only see the oscillations of the fields, but we can not
see clearly the evolution of its density . In order to
drop out these oscillations, in what follows we perform
two transformations. The first one changes the Klein-
Gordon equation into a kind of “Schr¨ odinger” equation
and the second transforms this last equation into a hydro-
dynamical system, where we can interpret the physical
quantities easier and the observable quantities become
much clear. Now we express the perturbed SF δΦ in
terms of the field Ψ,
δΦ = Ψe−imt/?+ Ψ∗eimt/?,(13)
term which oscillates with a frequency proportional to m
and Ψ = Ψ(? x,t) which would be proportional to a wave
function of an ensamble of particles in the condensate.
With this equation (12) transforms into
2m(?Ψ+9λ|Ψ|Ψ)+mφΨ = 0, (14)
where we have defined
Notice that this last equation could represent a kind of
“Gross-Pitaevskii” equation in an expanding Universe.
The only modification of equation (14) in comparison
to the Schr¨ odinger or the Gross-Pitaevskii equation is
the scale factor a−1associated to the co-moving spatial
gradient and that the Laplacian ∇ transforms into the
D’ Alambertian 2.
To explore the hydrodynamical nature of bosonic dark
matter, we will use a modified fluid approach. Then, to
make the connection between the theory of the field and
the condensates waves function, the field is proposed as,
where Ψ will be the condensates wave function with ˆ ρ =
ˆ ρ(? x,t) and S = S(? x,t) . Here we have separated Ψ
into a real phase S and a real amplitude√ˆ ρ. From (16)
˙ˆ ρ + 3Hˆ ρ +1
mˆ ρ2S +
˙ˆ ρ˙S = 0
˙S/m + ωˆ ρ + φ +
Now, taking the gradient of (19) then dividing by a and
using the definition
? v ≡∇S
˙ˆ ρ + 3Hˆ ρ +1
mˆ ρ2S +
a∇ˆ ρ +1
am? v∇ˆ ρ −1
˙ˆ ρ˙S = 0
˙? v + H? v +ω
a(? v · ∇)? v − (˙? v + H? v)(˙S/m) = 0
finally neglecting squared terms, second order time
derivatives, products of time derivatives and setting ω =
0 in this last set of equations we get,
∂t+ ∇ · (ˆ ρ? v) + 3Hˆ ρ = 0
∇2φ = 4πGˆ ρ (22)
∂t+ H? v + (? v · ∇)? v −
2ˆ ρ∇2ˆ ρ) + ∇φ = 0 (21)
where the equation for the gravitational field is given
by Poisson’s equation (22). In these equations we have
introduced ? r = a(t)? x.
Equation (19) shows the proportionality between the
gradient of the phase and the velocity of the fluid. Note
that ? v can represent the velocity field for the fluid and ˆ ρ
will be the particles density number within the fluid. Also
note that there exists an extra term of third order for the
partial derivatives in the waves amplitude which goes as
the gradient of
“quantum pressure” that would act against gravity. We
remain that φ represents the gravitational field. These
two sets of equations (21) would be analogous to Euler’s
equations of classical “fluids”, with the main difference
that there exists a “quantum part”, which we will call Q
and will be given by Q =
force or a sort of negative quantum pressure.
For equation (21) we have that ˆ ρ will represent the
mass density or the particles density number of the fluid,
where all the particles wolud have the same mass. Fi-
nally, these equations describe the dynamics of a great
number of non-interacting identical particles that mani-
fest themselves in the form of a fluid, also equation (14)
can describe a great number of non-interacting identical
particles in the way of a Bose gas, when the probability
density is interpreted as the density number.
Now, these hydrodynamical equations are a set of com-
plicated non-linear differential equations. To solve them
we will restrict ourselves to a vecinity of total equilib-
For this let ˆ ρ0be the mass density of the fluid in equi-
librium, the average velocity ? v0will be taken as cero in
equilibrium, so we will only have ? v(? x,t) out of equilib-
rium. Then, the matter in the Universe will be considered
as a hydrodynamical fluid inside an Universe in expan-
sion. This system will then evolve in this Universe and
later on they will colapse because of their gravitational
√ˆ ρ, this term would result in a sort of
which can describe a
Then from (21) for the mass density of the fluid in
equilibrium we have,
+ 3Hˆ ρ0= 0,
with solutions of the form
where as we know, in general if we have an equation of
state of the form p = ωρ and consider CDM or dust
as dark matter such that p = 0 it holds that ρ ∝ a−3.
Then, when the scale factor was small, the densities were
necessarily bigger. Now, the particles density number
are inversely proportional to the volume, and must be
proportional to a−3, therefore the matter energy density
will also be proportional to a−3, result that is consistent
with our expression (23).
Now for the system out of equilibrium we have
+ 3Hδˆ ρ + ˆ ρ0∇ · δ? v = 0
∇2δφ = 4πGδˆ ρ.
+ Hδ? v −
ˆ ρ0) + ∇δφ = 0
Then, these hydrodynamical equations can describe
the behavior of the fluid if this has moved by a small
amount out of total equilibrium and can also describe
the behavior for a fluid in equilibrium.
Now lets look to a Universe in expansion. In order
to solve system (24) we look for solutions in the form of
plane waves, for this the convenient ansatz goes as
δˆ ρ = ρ1(t)exp(i?k · ? x/a),
δ? v = ? v1(t)exp(i?k · ? x/a)
δφ = φ1(t)exp(i?k · ? x/a).(25)
where ? x is the position vector and?k is a real wavevector
which corresponds to a wavelength λ. If we substitute
these ansatz in the set of equations (24), we then have
+ 3Hρ1+ iˆ ρ0
?k ·? v1 = 0,
+ H? v1+ iρ1
?k = 0,
k2ρ1 = 0.(26)
where we have defined the velocity
To solve the system is convenient to rotate the coordinate
system so that the propagation of the waves will be along
the direction of one of the axes. For this we know that the
velocity vector can be divided into longitudinal (parallel
to?k) and transverse (perpendicular to?k) parts, so we
have ? v1= λ?k +? v2, where ? v2is the vector perpendicular
to the wave propagation vector?k ·? v2= 0. In terms of ? v2
for equations (25) we have
+ 3Hρ1+ iˆ ρ0
ak2λ = 0
dt+ Hλ +i
k2)ρ1 = 0, (28)
in addition to an equation for v2, d? v2/dt = 0+H? v2= 0,
with solutions ? v2 = C/a with C a constant of integra-
tion, i.e., perpendicular modes to the wave vector are
eliminated with the expansion of the Universe. Now, if
we use the result (23), the first equation of (28) can also
be written as
System (28) can be treated as in the case of a Universe
with no expansion, so combining the two equations in
(28) and with the aid of (29), we get
where δ = ρ1/ˆ ρ0is defined as the density contrast. This
will be a fundamental equation in the understanding of
the evolution of the primordial fluctuations.
a2− 4πGˆ ρ0
δ = 0, (30)
First we will give a brief summary of the results for
the CDM model, this will enable us to make a direct
comparison with our results.
For CDM the equation for the evolution of the density
contrast is given by,
where csis defined as the sound velocity (which in our
case it is not). Now lets analyze equation (31) beginning
in the matter dominated era, a time after recombination
when the radiation has cooled down and the photons do
not interact with the electrons anymore. At this time,
matter behaves like dust with zero pressure. So we have
a ∼ t2/3, v2
(2/3)1/t. For equation (31) we have
a2− 4πGˆ ρ0
δ = 0,(31)
qk2/a2≈ 0 and ˆ ρ0 ∼ t−2therefore H =
t2δ = 0.
The solutions to this equation are of the form
δ(t) → t2/3C1+C2
where C1 and C2 are integration constants, from this
solution we can see that we have modes that will disap-
pear as time goes by, and modes that grow proportionally
to the expansion of the Universe. This is an important
result, because then the density contrast will grow pro-
portionally to the expansion of the Universe when this
is dominated by matter. Then, these fluctuations can
maybe grow and give life to the galaxies, clusters of galax-
ies and all the large-scale structure we see now a days.
In an analogous way, we can see what happens with
equation (31) in an era dominated by radiation. In this
case a ∼ t1/2, ˆ ρ0∼ t−2and v2
equation of the fluctuations we have
qk2/a2? 1, then for the
t2)δ = 0,(33)
where A is a constant. This equation has oscillating so-
lutions only, and as we will see these fluctuations will not
grow, they just oscillate and then fade out as time goes
by in this scenario.
0 1e-05 2e-05 3e-05 4e-05 5e-05
6e-05 7e-05 8e-05 9e-05 0.0001
FIG. 2: Evolution of the perturbations for the CDM model
in the radiation dominated era for k = 0.02hMpc−1.
Big fluctuations do not have the time to interact with
the radiation and these survive to the impact with the
photons, and as a result they can grow as shown in Fig.2.
In this model all fluctuation with a size less than 200Mpc
are then fade out.
0 1e-05 2e-05 3e-05 4e-05 5e-05
6e-05 7e-05 8e-05 9e-05 0.0001
FIG. 3: Evolution of the density contrast for the CDM model
in the radiation dominated era for k = 0.7hMpc−1
The results of Fig.3 are no surprise for the CDM model,
since it is believed that in any radiation field any concen-
tration of matter will be dispersed by the shock between
the radiation and the particles. What this would tell us is
that the fluctuations that make up the large-scale struc-
ture we see today, can only grow during the matter dom-
inated era as seen by equation (32). In the epoch where
the radiation dominated the Universe, there was no way
at all for these primordial fluctuations to grow and make
up the structure we see. This result is shown on Fig.3 and
in a time before recombination (z=1300). Another result
obtained shows that some fluctuations can grow during
the radiation dominated era for the CDM model. These
fluctuations are those for which their size is much greater
than the horizon of events in which they are contained.
Now lets analyze the results obtained from equation
(30) for the case of our SFDM. As before we show the
early evolution of the density contrast for a time just
before recombination, when the Universe was dominated
0 1e-05 2e-05 3e-05 4e-05 5e-05
6e-05 7e-05 8e-05 9e-05 0.0001
FIG. 4: Evolution of the density contrast for the SFDM model
in the radiation dominated era for k = 0.7hMpc−1
As we can see in Fig.4 the same perturbations that we
use before with the CDM model can now grow in our
SFDM model, and this will happen with all the other
fluctuations despite their size. Because these fluctuations
are big in size, these means that they can only give birth
to large structures. These fluctuations can then help for
the formation of large clusters or other large-scale struc-
ture in the Universe at its early stages. Then, as these
kind of SFDM can only interact with radiation in a grav-
itational form it is not limited by its interaction with
radiation, and the dark matter halos will create poten-
tial wells that will collapse early in time giving enough
time for the structures to form. Then if DM is some
kind of SFDM the luminous matter will follow the DM
potentials giving birth to large-scale structure.
The new observational instruments and telescopes un-
til today have perceived objects as far as z = 8.6 .
The cosmic background radiation can bring us informa-
tion from z = 1000 to z = 2000. But jet we can not see
anything from the intermediate region, now we know of
a possible galaxy that might be found at a distance of
z = 10.56 but it has jet to be confirmed.
As seen earlier, as expected for the CDM model we
obtained that for the matter dominated era the low-k
modes grow, while in the radiation dominated era the
high-k modes are suppressed not giving enough time for
the perturbations to grow and form the large structures.
Although in general a scalar field is not a fluid, it can
be treated as if it behaved like one.The evolution of its
density can be the appropriate for the purpose of struc-
ture formation. In this case our SFDM has provided to
be an alternative model for the dark matter nature of
the Universe. We have shown that the scalar field with
an ultralight mass of 10−22eV simulates the behavior of
CDM in a Universe dominated by matter, i.e., its den-
sity contrast profile is very similar to that of the CDM
model, on the contrary for the radiation dominated era
both models have different behavior. These facts can be
the crucial difference between both models.
As mentioned before, recent observations have taken
us to very early epochs in the origin of the Universe,
and have made us think that structure had already been
formed, corresponding to z ≈ 7. It is clear from Fig.4
that just before recombination z ≈ 1300 there already
existed well defined perturbations in the energy density
for the SFDM model, which can contribute to the early
formation of structure. Then, if clusters could be formed
as early as these z’s, this would imply that Φ2+ λΦ4as
a model for dark matter could give an explanation for
the characteristic masses that are being observed, and
therefore it could solve some of the problems present in
the standard CDM model.
Although the observational evidence seems to be in
favor of some kind of cold dark matter, the last word has
not been said. Astronomers hope to send satellites that
will detect the finest details of the cosmic background
radiation, which will help us to get information of
structure at the time of recombination, from which it
will be possible to deduce its evolution until now a days.
ACKNOWLEDMENTS This work was partially
supported by CONACyT M´ exico, Instituto Avanzado de
Cosmologia (IAC) collaboration. AS is supported by a
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