Structure Formation with Scalar Field Dark Matter: The Fluid Approach
ABSTRACT The properties of nearby galaxies that can be observed in great detail
suggest that a better theory rather than cold dark matter (CDM) 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
develop and simulate a hydrodynamical approach for the early formation of
structure in the Universe, this approach is based on the fact that dark matter
is on the form of some kind of scalar field (SF) with a potential that goes as
$\mu^2\Phi^2/2+\lambda\Phi^4/4$, we expect that the fluctuations coming from
the SF will give us some information about the matter distribution we observe
these days.
- [Show abstract] [Hide abstract]
ABSTRACT: We consider the growth of cosmological perturbations to the energy density of dark matter during matter domination when dark matter is a scalar field that has undergone Bose-Einstein condensation. We study these inhomogeneities within the framework of both Newtonian gravity, where the calculation and results are more transparent, and General Relativity. The direction we take is to derive analytical expressions, which can be obtained in the small pressure limit. Throughout we compare our results to those of the standard cosmology, where dark matter is assumed pressureless, using our analytical expressions to showcase precise differences. We find, compared to the standard cosmology, that Bose-Einstein condensate dark matter leads to a scale factor, gravitational potential and density contrast that increase at faster rates.Physical review D: Particles and fields 12/2011; 85(2). - SourceAvailable from: Curtis J. Saxton[Show abstract] [Hide abstract]
ABSTRACT: This paper investigates spheroidal galaxies comprising a self-interacting dark matter halo (SIDM) plus de Vaucouleurs stellar distribution. These are coupled only via their shared gravitational field, which is computed consistently from the density profiles. Assuming conservation of mass, momentum and angular momentum, perturbation analyses reveal the galaxy's response to radial disturbance. The modes depend on fundamental dark matter properties, the stellar mass, and the halo's mass and radius. The coupling of stars and dark matter stabilises some haloes that would be unstable as one-fluid models. However the centrally densest haloes are unstable, causing radial flows of SIDM and stars (sometimes in opposite directions). Depending on the dark microphysics, some highly diffuse haloes are also unstable. Unstable galaxies might shed their outskirts or collapse. Observed elliptical galaxies appear to exist in the safe domain. Halo pulsations are possible. The innermost node of SIDM waves may occur within ten half-light radii. Induced stellar ripples may also occur at detectable radii if higher overtones are excited. If any SIDM exists, observational skotoseismology of galaxies could probe DM physics, measure the sizes of specific systems, and perhaps help explain peculiar objects (e.g. some shell galaxies, and the growth of red nuggets).Monthly Notices of the Royal Astronomical Society 04/2013; 430(3):1578-1598. · 5.52 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: A relativistic complex scalar boson field at finite temperature $T$ is examined below its critical Bose-Einstein condensation temperature. It is shown that at the same $T$ the state with antibosons has higher entropy, lower Helmholtz free energy and higher pressure than the state without antibosons, but the same Gibbs free energy as it should. This implies that the configuration without antibosons is metastable. Results are generalized for arbitrary $d$ spatial dimensions.Physics Letters A 08/2012; 376(45). · 1.63 Impact Factor
Page 1
arXiv:1101.4039v2 [gr-qc] 12 May 2011
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 16 May 2011 (MN LATEX style file v2.2)
Structure formation with scalar field dark matter: the fluid
approach
A. Su´ arez1⋆and T. Matos1
1Departamento de F´ ısica, Centro de Investigaci´ on y de Estudios Avanzados del IPN, 07000 M´ exico D.F., M´ exico
16 May 2011
ABSTRACT
The properties of nearby galaxies that can be observed in great detail suggest that a
better theory rather than cold dark matter (CDM) 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 develop and simulate a hydrodynamical
approach for the early formation of structure in the Universe, this approach is based
on the fact that dark matter is on the form of some kind of scalar field (SF) with a
potential that goes as µ2Φ2/2 + λΦ4/4, we expect that the fluctuations coming from
the SF will give us some information about the matter distribution we observe these
days.
Key words: theory – dark matter – large scale structure of Universe.
1 INTRODUCTION
We begin this work remebering the framework of the stan-
dard cosmological model: a homogeneous and isotropic Uni-
verse whose evolution is best described by Friedmann’s equa-
tions that come from general relativity and whose main in-
gridients can be described by fluids whose characteristics
are very similar to those we see in our Universe. Of course,
the Universe is not exactly homogeneous and isotropic but
this standard model does give us a framework within which
we can study the evolution of structure like the observed
galaxies or clusters of galaxies from small fluctuations in
the density of the early Universe. In this model 4 per cent
of the mass in the Universe is in the baryons, 22 per cent
is non-baryonic dark matter and the rest in some form of
cosmological constant. Another idea that has been around
just a bit less than hundred years and in which many of
the cosmological models are based in is that of an homoge-
neous and isotropic Universe, although it has always been
clear that this homogeneity and isotropy are only found un-
til certain level. Now we know that the anisotropies are very
important and can grow as big as the large scale structure
we see today.
Nowadays the most accepted model in cosmology which
explains the evolution of the Universe is known as ΛCDM,
because it has achieved some observations with outstanding
success, like for example the fact that the cosmic microwave
background can be explained in great detail and that it pro-
vides a framework within one can understand the large-scale
isotropy of the Universe and important characteristics on the
⋆E-mail:asuarez@fis.cinvestav.mx ; tmatos@fis.cinvestav.mx
origin, nature and evolution of density fluctuations which are
believed to give rise to galaxies and other cosmic structure.
There remain, however, certain problems at galactic scales,
like the cusp profile of central densities in galactic halos, the
over 500 substructures predicted by numerical simulations
which are not found in observations, etc.. See for example
Moore et al. (1999), Clowe (2006) and Penny et al. (2009).
In the big bang model, gravity plays an essential role,
it collects the dark matter in concentrated regions called
’Dark matter haloes’. In the large dark matter haloes, the
baryons are believed to be so dense as to radiate enough
energy so they will collapse into galaxies and stars. The most
massive haloes, hosts for the brightest galaxies, are formed
in regions with the highest local mass density. Less massive
haloes, hosts for the less bright galaxies, appear in regions
with low local densities, i.e, regions were the local density is
not well defined, Peebles & Nusser (2010). These situations
appear to be the same as in our extragalactic neighborhood,
but still there are problems.
Observations point out to a better understanding of the
theory beginning with the less occupied space called the ’Lo-
cal void’, which contains just a few galaxies which are big-
ger than the expected. 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 mat-
ter in the surroundings, Peebles & Nusser (2010).
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 be-
lieved 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.
Page 2
2 A. Su´ arez and T. Matos
The incorporation of a new kind of dark matter, differ-
ent from the one proposed by the ΛCDM model into the big
bang theory holds out the possibility of resolving some of
these issues.
Recent works have introduced a dynamic scalar field
with a certain potential V (Φ) as a candidate to dark matter,
although there is not yet an agreement for the correct form
of the potential of the field. Lee & Koh (1996), and inde-
pendently Matos & Guzm´ an (2000) suggested bosonic dark
matter (SFDM) as a model for galactic halos. Another inter-
esting work pointing this way was done by Matos & Ure˜ na
(2000) and independently by Sahni & Wang (2000) where
they used a potential of the form cosh to explain the core
density problem for disc galaxy halos in the ΛCDM model.
Matos & Guzm´ an (2000) presented a model for the dark
matter in spiral galaxies, in which they supposed that dark
matter is a scalar field endowed with a scalar potential.
Several recent work have also suggested that SFDM
can be composed of spin-0 bosons which give rise to Bose-
Einstein Condensates (BECs), which at the same time can
make up the galaxies we are observing in our Universe.
Hu, Barkana & Gruzinov (2008) proposed that dark matter
is composed of ultra/light scalar particles who are initially in
the form of a BEC. In their work Woo & Chiueh (2008) used
a bosonic dark matter model to explain the structure for-
mation via high-resolution simulations, finally Ure˜ na (2010)
reviewed the key properties that may arise from the bosonic
nature of SFDM models.
The main objective of this work in difference with
others is to introduce SFDM and assume that dark
matter its itself a scalar field that involves an auto-
interacting potential of the form V (Φ)
λΦ4/4, where µΦ
∼
scalar field, Lee & Koh (1996), Matos & Ure˜ na (2001) and
Hu, Barkana & Gruzinov (2008). With the mass µΦ
10−22eV and only one free parameter when λ is taken equals
to zero, the SFDM model fits the following important fea-
tures:
=µ2Φ2/2 +
10−22
eV is the mass of the
∼
(i) The
rameters
Matos, V´ azquez-Gonz´ alez & Maga˜ na (2009).
(ii) The rotation curves of galaxies, Boehmer & Harko
(2007), and the central density profile of LSB galaxies,
Bernal, Matos & N´ u˜ nez (2008),
(iii) With this mass, the critical mass of collapse for a real
scalar field is just 1012M⊙, i.e., the one observed in galactic
haloes, Alcubierre et al. (2002).
(iv) The central density profile of the dark matter is flat,
Bernal, Matos & N´ u˜ nez (2008).
(v) The scalar field has a natural cut off, thus the sub-
structures in clusters of galaxies is avoided naturally. With a
scalar field mass of µΦ ∼ 10−22eV the amount of substruc-
tures is compatible with the observed one, Matos & Ure˜ na
(2001).
cosmological
of all
evolution
components
ofthe
of
density
the
pa-
the Universe,
In this paper we show that the SFDM predicts galaxy for-
mation earlier than the cold dark matter model, because
they form BEC at a critical temperature Tc >> TeV. So,
if SFDM is right, this would imply that we have to see big
galaxies at high redshifts. In order to do this, we study the
density fluctuations of the scalar field from a hydrodynam-
ical point of view, this will give us some information about
the energy density of dark matter halos necessary to obtain
the observational results of large-scale structure. Here we
will give some tools that might be necessary for the study
of the early formation of structure.
In section 2 we analyse the analytical evolution of the
SF, then in section 3 we treat the SF as a hydrodynamical
fluid in order to study its evolution for the density contrast,
in section 4 we compare our results with those obtained by
the CDM model for the density contrast in the radiation
dominated era just before recombination and finally we give
our conclusions.
2 THE BACKGROUND
In this section we perform a transformation in order to solve
the Friedmann equations analytically with the approxima-
tion H << µΦ. The scalar field (SF) we deal with depends
only on time, Φ = Φ0(t), and of course the background is
only time dependent as well.
We usetheFriedmann-Lemaˆ ıtre-Robertson-Walker
(FLRW) metric with scale factor a(t). The background Uni-
verse is composed only by SFDM (Φ0) endowed with a scalar
potential. We begin by recalling the basic background equa-
tions. From the energy-momentum tensor T for a scalar
field, the scalar energy density T0
Ti
0 and the scalar pressure
jare given by
T0
0= −ρΦ0= −
?1
2
˙Φ2
0+ V
?
, (1)
Ti
j= pΦ0=
?1
2
˙Φ2
0− V
?
δi
j, (2)
where the dots stand for the derivative with respect to the
cosmological time and δi
Equation of State (EoS) for the scalar field is pΦ0= ωΦ0ρΦ0
with
j is Kronecker’s delta. Thus, the
ωΦ0=
1
2˙Φ2
1
2˙Φ2
0− V
0+ V.
(3)
Notice that background scalar quantities have the sub-
script 0. Now the following dimensionless variables are de-
fined
˙Φ0
H,H
x ≡
κ
√6
u ≡
κ
√3
√V
,
being κ2≡ 8πG and H ≡ ˙ a/a the Hubble parameter. Here
we take the scalar potential as V = m2Φ2/2?2+ λΦ4/4,
where, µ = mc/? and m is the mass given in kilograms, and
from now on we will use units where c = 1, then for the
ultra-light boson particle we have that µΦ ∼ 10−22eV.
With these variables, the density parameter ΩΦ for the
background 0 can be written as
ΩΦ0= x2+ u2. (4)
In addition, we may write the EoS of the scalar field as
ωΦ0=x2− u2
ΩΦ0
. (5)
Since ωΦ0is a function of time, if its time average tends
to zero, this would imply that Φ2-dark matter can be able
to mimic the EoS for CDM, see Matos, Maga˜ na & Su´ arez
(2010) and Matos, V´ azquez-Gonz´ alez & Maga˜ na (2009).
Page 3
Structure formation with scalar field dark matter: the fluid approach3
Now we express the SF, Φ0, in terms of the new vari-
ables S and ˆ ρ0, where S is constant in the background and
ˆ ρ0 will be the energy density of the fluid also in the back-
ground. So, our background field is proposed as
Φ0 = (ψ0e−imt/?+ ψ∗
0eimt/?) (6)
where,
ψ0(t) =
?
ˆ ρ0(t)eiS/?
(7)
and with this our SF in the background can be finally ex-
pressed as,
Φ0 = 2?
with this we obtain
ˆ ρ0cos(S − mt/?), (8)
˙Φ2
0
=ˆ ρ0
?˙ˆ ρ0
ˆ ρ0cos(S − mt/?)
−
To simplify, observe that the uncertanty relation implies
that m∆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 << µΦ ∼ 10−22
eV, so with these considerations at hand for the background,
in (9) we have
2(˙S − m/?) sin(S − mt/?)
?2
(9)
˙Φ2
0= 4m2
?2ˆ ρ0sin2(S − mt/?) (10)
Finally, substituting this last equation and equation (8)
into (1) when taking λ = 0, we obtain
ρΦ0= 2m2
?2ˆ ρ0[sin2(S−mt/?)+cos2(S−mt/?)] = 2m2
Comparing this result with (4) we have that the identity
ΩΦ0= 2m2ˆ ρ0/?2holds for the background, so comparing
with (11),
2ˆ ρ0m
?sin(S − mt/?)
2ˆ ρ0m
?cos(S − mt/?).
?2ˆ ρ0.(11)
x =
?
?
We plot the evolution of the energies (12) and (13) 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 (12) and (13) Fig.
1 shows the kinetic and the potential energies of the scalar
field. Observe the excelent accordance with the numerical
results in Matos, V´ azquez-Gonz´ alez & Maga˜ na (2009) for
the kinetic and potential energies of the background respec-
tively.
(12)
u =
(13)
3SCALAR FIELD FLUCTUATIONS
If dark matter is some kind of elemental particle with mass
µ, then it would be about 1068µ 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 function that only depends on
the three spatial coordinates and time.
Now a days it is known that our Universe is not exactly
0
1e-05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0001 0.001 0.01 0.1 1
u2
a
Potential energy
0
1e-05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0001 0.001 0.01 0.1 1
x2
a
Kinetic energy
Figure 1. Analytical evolution of the potential (top panel) and
kinetic (bottom panel) energies of the scalar field dark matter.
isotropic and spatially homogeneous like the 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 µ << 1 eV, the occupation numbers in galactic
haloes are so big that the dark matter behaves as a classical
field that obeys the Klein-Gordon equation (?2+m2/?2)Φ =
0, where ? is the D’Alambertian and we have set c = 1.
By definition, a perturbation done in any quantity, is
the difference between its value in some event in real space-
time, and its corresponding value in the background. So, for
example for the SF we have
Φ = Φ0(t) + δΦ(x,t),(14)
where the background is only time dependent, while the per-
turbations also depend on the space coordinates. Similar
cases apply for the metric;
g00
=−a2(1 + 2φ),
a2B,i,
a2[(1 − 2ψ)δij + 2E,ij].
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 tensor gµν is diagonal,
and so the calculations become much easier. We will only
work with scalar perturbations, vector and tensor perturba-
g0i
=
gij
=(15)
Page 4
4 A. Su´ arez and T. Matos
tions are eliminated from the beginning, so that only scalar
perturbations are taken into account. Another advantage in
using this gauge is that φ will play the role of the gravita-
tional potential, this will help us to have a simpler physical
interpretation, i.e., both potentials φ and ψ are then related.
This metric has already been used in other works, Bardeen
(1980), Ma & Bertschinger (1995) and Malik (2009).
For the perturbed Klein-Gordon where we have used
equation (14) and we have set˙φ = 0, we have:
δ¨Φ + 3Hδ˙Φ −1
The SF Φ has very hard oscillations from the beginning,
this oscillations are transmitted to the fluctuations which
apparently seems to grow very fast and are too big. Never-
theless, 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, Matos, Maga˜ na & Su´ arez (2010). In
order to drop out these oscillations, in what follows we per-
form two transformations. The first one changes the per-
turbed Klein-Gordon equation into a kind of ’Schr¨ odinger’
equation and the second transforms this last equation into
a hydrodynamical system, where we can interpret the phys-
ical quantities easier and the observable quantities become
much clear. Now we express the perturbed SF δΦ in terms
of the field Ψ,
a2ˆ∇2δΦ + V,ΦΦδΦ + 2V,Φφ = 0(16)
δΦ = Ψe−imt/?+ Ψ∗eimt/?, (17)
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 and the expresion for the potential of the scalar
field, (16) transforms into
− i?(˙Ψ +3
where we have defined
2HΨ) +?2
2m(?Ψ + 9λ|Ψ|2Ψ) + mφΨ = 0, (18)
? =d2
dt2+ 3Hd
dt−
1
a2ˆ∇2. (19)
Notice that this last equation could represent a kind of
’Gross-Pitaevskii’ equation in an expanding Universe. The
only modification of equation (18) 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 Laplacianˆ∇2= ∂2xtransforms into the D’ Alam-
bertian ?.
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 con-
densates waves function, the field is proposed as,
Ψ =
?
ˆ ρeiS, (20)
where Ψ will be the condensates wave function with ˆ ρ =
ˆ ρ(x,t) and S = S(x,t), Ginzburg & Landau (1985). Here
we have separated Ψ into a real phase S and a real amplitude
√ˆ ρ and the condition | Ψ |2= ΨΨ∗= ˆ ρ is satisfied. From
(20) we have
˙ˆ ρ+3Hˆ ρ −?
mˆ ρ?S +
?2
2m2(
?
a2m
ˆ?√ˆ ρ
√ˆ ρ
ˆ∇Sˆ∇ˆ ρ −?
) +?2
2a2[ˆ∇(S/m)]2
m
˙ˆ ρ˙S = 0
?˙S/m+ωˆ ρ + φ +
−
?2
2(˙S/m)2= 0 (21)
Now, taking the gradient of (21) then dividing by a and
using the definition
v ≡
we have,
?
ma
ˆ∇S (22)
˙ˆ ρ+3Hˆ ρ −?
mˆ ρ?S +1
1
2aˆ ρ∇p +1
av∇ˆ ρ −?
m
?2
˙ˆ ρ˙S = 0
˙ v
+Hv +
a∇φ +
2m2a∇(?√ˆ ρ
√ˆ ρ
)
+
1
a(v·∇)v − ?(˙ v + Hv)(˙S/m) = 0.
where in (21) ω = 9?2λ/2m2and in (23) we have defined
p = ωˆ ρ2.
It is worth noting that, to this moment this last set of
equations do not involve any approximations with respect
to equation (18) and can be used in linear and non-linear
regimes.
Now, neglecting squared terms, second order time
derivatives and products of time derivatives in this last set
of equations we get,
(23)
∂ˆ ρ
∂t
∂v
∂t
+∇·(ˆ ρv) + 3Hˆ ρ = 0(24)
+Hv + (v·∇)v −
∇φ = 0
4πGˆ ρ
?2
2m2∇(1
2ˆ ρ∇2ˆ ρ) + ω∇ˆ ρ
+ (25)
∇2φ
where the equation for the gravitational field is given by
Poisson’s equation (26). In these equations we have intro-
duced r = a(t)x, such that 1/aˆ∇ = ∇ = ∂r.
Equation (22) 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 there ex-
ists an extra term of third order for the partial derivatives in
the waves amplitude which goes as the gradient of
this term would result in a sort of ’quantum pressure’ that
would act against gravity. We remain that φ represents the
gravitational field. These two sets of equations (24) and (25)
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 =
which can describe a force or a sort of negative quantum
pressure.
For equation (24) we have that ˆ ρ will represent the mass
density or the particles density number of the fluid, where
all the particles would have the same mass. Finally, these
equations describe the dynamics of a great number of non-
interacting identical particles that manifest themselves in
the form of a fluid, also equation (18) can describe a great
number of non-interacting but self-interacting identical par-
ticles 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 equilibrium.
For this let ˆ ρ0 be the mass density of the fluid in equi-
librium, the average velocity v0 will be taken as zero in
=(26)
?2
2m2
?√ˆ ρ
√ˆ ρ,
?2
2m2
?√ˆ ρ
√ˆ ρ
Page 5
Structure formation with scalar field dark matter: the fluid approach5
equilibrium, so we will only have v(x,t) out of equilibrium.
Then, the matter in the Universe will be considered as a
hydrodynamical fluid inside an Universe in expansion. This
system will then evolve in this Universe and later on they
will collapse because of their gravitational attraction.
Then from (24) for the mass density of the fluid in equi-
librium we have,
∂ˆ ρ0
∂t
with solutions of the form
ˆ ρ0 =ρ0i
a3,
where as we know, in general if we have an equation of state
of the form ˆ p = ωˆ ρ and consider CDM or dust as dark matter
such that ˆ p = 0 it holds that ˆ 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 (28).
Now for the system out of equilibrium we have
+ 3Hˆ ρ0 = 0, (27)
(28)
∂δˆ ρ
∂t
∂δv
∂t
∇2δφ
equations that are valid in a Universe in expansion. In order
to solve system (29) we look for solutions in the form of
plane waves, for this the convenient ansatz goes as
+3Hδˆ ρ + ˆ ρ0∇·δv = 0
?2
2m2∇(1
+ Hδv −
2∇2δˆ ρ
ˆ ρ0) + ω∇δˆ ρ + ∇δφ = 0
=4πGδˆ ρ(29)
δˆ ρ=ˆ ρ1(t)exp(ik · x/a),
δv
=
v1(t)exp(ik · x/a),
δφ=φ1(t)exp(ik · x/a).
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 (29), we then have
dˆ ρ1
dt
dv1
dt
+3Hˆ ρ1+ iˆ ρ0
ak · v1 = 0,
?v2
(30)
+Hv1+ iˆ ρ1
a
q
ˆ ρ0− 4πGa2
k2+ ω
?
k = 0, (31)
φ1
+4πGa2
k2ˆ ρ1 = 0. (32)
where we have defined the velocity
v2
q=
?2k2
4a2m2
(33)
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 = λk + v2, where v2 is the vector perpendicular to the
wave propagation vector k · v2 = 0. In terms of v2 for equa-
tions (30)-(32) we have
dˆ ρ1
dt
dλ
dt
+3Hˆ ρ1+ iˆ ρ0
ak2λ = 0
a(v2
(34)
+ Hλ +i
q
ˆ ρ0− 4πGa2
k2+ ω)ˆ ρ1 = 0, (35)
in addition to an equation for v2, dv2/dt + Hv2 = 0, with
solutions v2 = C/a with C a constant of integration, i.e.,
perpendicular modes to the wave vector are eliminated with
the expansion of the Universe. Now, if we use the result (28),
then equation (34) can be written as
d
dt
?ˆ ρ1
ˆ ρ0
?
= −ik2λ
a
(36)
System (34)-(35) can be treated as in the case of a Uni-
verse with no expansion, so combining the two equations
and with the aid of (36), we get
d2δ
dt2+ 2Hdδ
dt+
?
(v2
q+ ωˆ ρ0)k2
a2− 4πGρ0
?
δ = 0,(37)
where δ = ˆ ρ1/ˆ ρ0 = ρ1/ρ0 is defined as the density contrast.
This will be a fundamental equation in the understanding
of the evolution of the primordial fluctuations.
4 RESULTS
First we will give a brief summary of the results for the
ΛCDM model, this will enable us to make a direct compar-
ison with our results.
For CDM the equation for the evolution of the density
contrast is given by,
d2δ
dt2+ 2Hdδ
dt+
?
c2
sk2
a2− 4πGˆ ρ0
?
δ = 0,(38)
where cs is defined as the sound velocity (which in our
case it is not). Now lets analyze equation (38) at the be-
ginning of the the matter dominated era a time just after
the epoch of equality,and just before recombination when
the radiation has cooled down and the photons do not inter-
act with the electrons anymore, for a relativistic treatment
see Gorini et al. (2008). In this era, a ? aeq, practically all
the interesting fluctuation modes are well within the hori-
zon, and the evolution of the perturbations can be well de-
scribed within the newtonian analysis. At this time, matter
behaves like dust with zero pressure. So we have a ∼ t2/3,
c2
equation (38) we have
sk2/a2≈ 0 and ˆ ρ0 ∼ t−2therefore H = (2/3)1/t. For
d2δ
dt2+4
3
1
t
dδ
dt−2
3
1
t2δ = 0. (39)
The solutions to this equation are of the form
δ(t) → t2/3C1+C2
where C1 and C2 are integration constants, from this solu-
tion we can see that we have modes that will disappear as
time goes by, and modes that grow proportionally to the
expansion of the Universe. This is an important result, be-
cause then the density contrast will grow proportionally 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 galaxies and all the large-scale
structure we see now a days.
Now lets see what happens to the SFDM at this epoch
(a ? aeq). The evolution of the perturbations in this case
will be given by equation (37).
In general we have that in equation (37) the term vq
is very small throughout the evolution of the pertubations
t,
(40)
Page 6
6A. Su´ arez and T. Matos
1e-06
1e-05
0.0001
0.001
1e-05 0.0001 0.001 0.01 0.1 1
delta
a
’k=0.001CDM.dat’
’k=0.001SFDM.dat’
Figure 2. Evolution of the perturbations for the CDM model
(dots) and SFDM model (lines) for k = 1 ∗ 10−3hMpc−1. Notice
how after the epoch of equality (aeq ∼ 10−4) the evolution of
both perturbations in nearly identical, a = 1 today. In this case
we have taken λ = 0.
(vq ? 10−3ms−1for small k), so it really does not have a
significant contribution on its evolution.
When the condition λ = 0 is taken we can have a BEC
that might be or might not be stable, if there exists stability
the results of SFDM are consistent with those obtained from
CDM (in this case both equations (37) and (38) are almost
equal), the condition of stability for the BEC in the SFDM
case will come from the study of λ together with Q.
d2δ
dt2+ 2Hdδ
dt+
?
v2
qk2
a2− 4πGρ0
?
δ = 0,(41)
As we can see in Fig. 2 the perturbations used for the
ΛCDM model grow in a similar way for the SFDM model,
when λ = 0, in this case both perturbations can give birth
to structures quite similar in size, and this will happen with
all the fluctuations as long as k is kept small.
When λ ?= 0 the results are quite different, so when
discussing the evolution of the density perturbations, there
are two different cases: i) In the case of λ > 0 the amplitude
of the density contrast tends to decrease as λ grows bigger
and bigger away from zero until the amplitude of the density
takes negative values (around λ ∼ 108), telling us that this
kind of fluctuations can not grow in time, and hence do not
form a BEC. ii) On the other hand if λ < 0 the fluctuations
for the density contrast alway grow despite their size, this
results means either than the fluctuations grow and form a
stable BEC or than the density grows because it is collapsing
into a single point and our BEC might be unstable, the study
of the stability of this fluctuations needs then to be studied
with non-linear perturbation theory. These results are shown
in Fig. 3, in both figures 2 and 3 the initial condition for δ
goes as δ ∼ 1 ∗ 10−5in accordance with the data obtained
from WMAP.
If these fluctuations result stable and because they are
big in size, this means that they can only give birth to large
structures. These fluctuations can then help for the forma-
tion of large clusters or other large-scale structure in the
Universe at its early stages (around a ? aeq). Then, as these
kind of SFDM can only interact with radiation in a gravita-
tional form it is not limited by its interaction with radiation,
and the dark matter halos can then create potential wells
1e-05
0.0001
0.001
0.01
1e-05 0.0001 0.001 0.01 0.1 1
delta
a
’k=0.01CDM.dat’
’k=0.01SFDM.dat’
Figure 3. Evolution of the perturbations for the CDM model
(dots) and SFDM model (lines) for k = 1 ∗ 10−2hMpc−1and
λ ?= 0 and negative. Notice how after the epoch of equality (aeq ∼
10−4) the evolution of both perturbations is now different from
the one in Fig. 2, a = 1 today. In this case we can clearly see that
the SFDM fluctuations grow quicker than those for the CDM
model.
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.
5CONCLUSIONS
The new observational instruments and telescopes until to-
day have perceived objects as far as z = 8.6, Lehnert et al.
(2010). The cosmic background radiation can bring us in-
formation 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 ob-
tained that for the matter dominated era the low-k modes
grow, when CDM decouples from radiation in a time just
before recombination it grows in a milder way than it does
in the matter dominated era (Fig. 2).
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 structure forma-
tion, because locations with a high density of dark matter
can support the formation of galactic structure.
In this work we have assumed that there is only one
component to the mass density, and that this component is
given by the scalar field dark matter. In this case equation
(37) is valid for all sub-horizon sized perturbations in our
non-relativistic specie, so for sub-horizon perturbations the
newtonian treatment worked with in the evolution of the
perturbations suffices.
The 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
when λ = 0, because in general in a matter dominated Uni-
verse for low-k, vq tends to be a very small quantity tending
to zero, so from (37) we can see that on this era we will
have the CDM profile given by (38), i.e., the SFDM density
Page 7
Structure formation with scalar field dark matter: the fluid approach7
contrast profile is very similar to that of the ΛCDM model,
Fig. 2. On the contrary for λ ?= 0 both models have different
behavior as we can see from Fig. 3, results which show that
linear fluctuations on the SFDM can grow in comparison
with those of CDM, even at early times when the large-
scale modes (small k) have entered the horizon just after
aeq ∼ 10−4, when it has decoupled from radiation, so the
amplitudes of the density contrast start to grow faster than
those for CDM around a ∼ 10−2. Here an important point
is that although CDM can grow it does so in a hierarchical
way, while from Fig. 3 we can see that SFDM can have big-
ger fluctuations just before the ΛCDM model does, i.e., it
might be that no hierarchical model of structure formation
is needed for SFDM and is expected that for the non-linear
fluctuations the behaviour will be quite the same as soon as
the scalar field condensates, in a very early epoch when the
energy of the Universe was about ∼ TeV. 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, cor-
responding to z ≈ 7. It is clear from Fig. 3 that at recombi-
nation z ≈ 1300 there already existed defined perturbations
in the energy density for the SFDM model, which can con-
tribute 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 explana-
tion for the characteristic masses that are being observed,
and therefore it could solve some of the problems present in
the standard ΛCDM model.
Although the observational evidence seems to be in fa-
vor 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.
ACKNOWLEDGMENTS
This work was partially supported by CONACyT M´ exico,
Instituto Avanzado de Cosmologia (IAC) collaboration. A.
Su´ arez is supported by a CONACYT scholarships.
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