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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.

PACS numbers:

INTRODUCTION

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 [21]. 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

some problems.

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-

ings [21].

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) [18]. 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), [19] 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 [18] 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 [19], [11]. Several

authors have studied numerically the collapse and virial-

ization of SFDM/BEC. [1] found that the critical mass

for collapse in the SFDM/BEC model is of the order of a

Milky Way-sized halo mass [25], [24], [27]. 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 [18].

In a recent paper, [26] 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

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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

Cluster [8].

Summarizing, with the mass mΦ∼ 10−22eV and only

one free parameter, the SFDM model fits the following

important features:

1. The cosmological evolution of the density parame-

ters of all the components of the Universe [14].

2. The rotation curves of galaxies [3] and the central

density profile of LSB galaxies [2],

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 [1].

4. The central density profile of the dark matter is flat

[2].

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 [15].

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-

clusions.

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

0and the

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

jis the Kronecker delta. Thus,

the Equation of State (EoS) for the scalar field is pΦ0=

ωΦ0ρΦ0with

ωΦ0=

1

2˙Φ2

1

2˙Φ2

0− V

0+ V.

(3)

Notice that background scalar quantities at zero order

have the subscript 0. Now the following dimensionless

variables are defined

x ≡

κ

√6

˙Φ0

H,

u ≡

κ

√3

√V

H,

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

ωΦ0=x2− u2

ΩΦ0

.(5)

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

?˙ˆ ρ0

ˆ ρ0

Φ0= 2ˆ ρ0cos(S − mt), (6)

˙Φ2

0= ˆ ρ0

cos(S − mt)

− 2(˙S − m) sin(S − mt)

?2

(7)

To simplify, observe that the uncertinty relation im-

plies that µc2∆t ∼ ?, and for the background in the

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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,

˙Φ2

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

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

’OxP.dat’

0

1e-05

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1

x2

a

Kinetic energy

’OxP.dat’

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 [14] 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 [13].

For the perturbed Klein-Gordon where we have used

equation (10) an we have set˙φ = 0, we have:

(11)

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

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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 [17]. 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

−i?(˙Ψ+3

2HΨ)+?2

2m(?Ψ+9λ|Ψ|Ψ)+mφΨ = 0, (14)

where we have defined

? =d2

dt2+ 3Hd

dt−1

a2∇2.(15)

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) [10]. Here we have separated Ψ

into a real phase S and a real amplitude√ˆ ρ. From (16)

we have

Ψ =

ˆ ρeiS/?,(16)

˙ˆ ρ + 3Hˆ ρ +1

mˆ ρ2S +

1

a2m∇S∇ˆ ρ(17)

−

1

m

˙ˆ ρ˙S = 0

˙S/m + ωˆ ρ + φ +

?2

2m2(2√ˆ ρ

√ˆ ρ)(18)

+

1

2a2[∇(S/m)]2−1

2(˙S/m)2= 0

Now, taking the gradient of (19) then dividing by a and

using the definition

? v ≡∇S

ma

(19)

we have,

˙ˆ ρ + 3Hˆ ρ +1

mˆ ρ2S +

a∇ˆ ρ +1

1

am? v∇ˆ ρ −1

2m2a∇(2√ˆ ρ

m

˙ˆ ρ˙S = 0

˙? v + H? v +ω

a∇φ +

?2

√ˆ ρ)(20)

+

1

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,

∂ˆ ρ

∂t+ ∇ · (ˆ ρ? v) + 3Hˆ ρ = 0

?2

2m2∇(1

∇2φ = 4πGˆ ρ (22)

∂? v

∂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

2m

“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 =

2m

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-

rium.

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

attraction.

?2

2√ˆ ρ

√ˆ ρ, this term would result in a sort of

?2

?√ˆ ρ

√ˆ ρ

which can describe a

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Then from (21) for the mass density of the fluid in

equilibrium we have,

∂ˆ ρ0

∂t

+ 3Hˆ ρ0= 0,

with solutions of the form

ˆ ρ0=ρ0i

a3, (23)

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

∂δˆ ρ

∂t

?2

2m2∇(1

+ 3Hδˆ ρ + ˆ ρ0∇ · δ? v = 0

2∇2δˆ ρ

∇2δφ = 4πGδˆ ρ.

∂δ? v

∂t

+ Hδ? v −

ˆ ρ0) + ∇δφ = 0

(24)

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

dρ1

dt

+ 3Hρ1+ iˆ ρ0

?

ˆ ρ0

φ1+ 4πGa2

a

?k ·? v1 = 0,

?

d? v1

dt

+ H? v1+ iρ1

a

v2

q

− 4πGa2

k2

?k = 0,

k2ρ1 = 0.(26)

where we have defined the velocity

v2

q=

?2k2

4a2m2

(27)

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

dρ1

dt

+ 3Hρ1+ iˆ ρ0

ak2λ = 0

dλ

dt+ Hλ +i

a(v2

ˆ ρ0

q

− 4πGa2

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

?ρ1

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.

d

dtˆ ρ0

?

= −ik2λ

a

.(29)

d2δ

dt2+ 2Hdδ

dt+v2

q

k2

a2− 4πGˆ ρ0

?

δ = 0, (30)

3. RESULTS

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

d2δ

dt2+ 2Hdδ

dt+c2

s

k2

a2− 4πGˆ ρ0

?

δ = 0, (31)

qk2/a2≈ 0 and ˆ ρ0 ∼ t−2therefore H =

d2δ

dt2+4

3

1

t

dδ

dt−2

3

1

t2δ = 0.

The solutions to this equation are of the form

δ(t) → t2/3C1+C2

t,

(32)

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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

d2δ

dt2+1

t

dδ

dt+ (A

tk2−3

8

1

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.

1e-05

1.02e-05

1.04e-05

1.06e-05

1.08e-05

1.1e-05

1.12e-05

1.14e-05

0 1e-05 2e-05 3e-05 4e-05 5e-05

a

6e-05 7e-05 8e-05 9e-05 0.0001

amplitude

Density Contrast

’delta.dat’

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.

-1e-05

-8e-06

-6e-06

-4e-06

-2e-06

0

2e-06

4e-06

6e-06

8e-06

1e-05

0 1e-05 2e-05 3e-05 4e-05 5e-05

a

6e-05 7e-05 8e-05 9e-05 0.0001

amplitude

Density Contrast

’delta.dat’

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

by radiation.

1e-05

1.2e-05

1.4e-05

1.6e-05

1.8e-05

2e-05

2.2e-05

2.4e-05

2.6e-05

2.8e-05

3e-05

0 1e-05 2e-05 3e-05 4e-05 5e-05

a

6e-05 7e-05 8e-05 9e-05 0.0001

amplitude

Density Contrast

’delta.dat’

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.

4. CONCLUSIONS

The new observational instruments and telescopes un-

til today have perceived objects as far as z = 8.6 [12].

Page 7

7

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

CONACYT scholarships.

∗Electronic address: abril@fis.cinvestav.mx

†Electronic address: tmatos@fis.cinvestav.mx

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