Is it possible to consider Dark Energy and Dark Matter as a same and unique

Dark Fluid?
arXiv:astro-ph/0506732v1 29 Jun 2005

Alexandre Arbey∗
Centre de Recherche Astronomique de Lyon (CRAL),
9 avenue Charles André, 69561 Saint Genis Laval Cedex, France

In the standard model of cosmology, the present evolution of the Universe is determined
by the presence of two components of unknown nature. One of them is referenced as “dark
matter” to justify the fact that it behaves cosmologically like usual baryonic matter, whereas
the other one is called “dark energy”, which is a component with a negative pressure. As
the nature of both dark components remains unknown, it is interesting to consider other
models. In particular, it seems that the cosmological observations can also be understood
for a Universe which does not contain two fluids of unknown nature, but only one fluid with
other properties. To arrive to this conclusion, we will review the observational constraints
from supernovæ of type Ia, cosmic microwave background, large scale structures, and the
theoretical results of big-bang nucleosynthesis. We will try to determine constraints on this
unifying “dark fluid”, and briefly review different possibilities to build models of dark fluid.

In the standard model of cosmology, the total energy density of the Universe is dominated today
by the densities of two components: the first one, called “dark matter”, is generally modeled
as a system of collisionless particles (“cold dark matter”) and consequently has an attractive
gravitational effect like usual matter. The second one, generally refered as “dark energy” or

∗
E–mail: arbey@obs.univ-lyon1.fr
 2

“cosmological constant” can be considered as a vacuum energy with a negative pressure, which
seems constant today. The real nature of these two components remains unknown. The reason of
such a distinction between both dark components is mainly historical. Indeed, the cosmological
constant was introduced by Einstein in order to justify the existence of a static Universe. Because
this constant can appear naturally in the Einstein equations as a new fundamental constant,
introducing it was not problematic. Moreover, no distinction was made between the dark matter
and the usual baryonic matter. Therefore, at this epoch, the standard model of cosmology was
not so different from the present one, with however an interesting conceptual difference: the two
components present in the cosmological models were at that moment not considered of unknown
nature, because, firstly no dark matter was needed, and secondly the cosmological constant could
be considered as a fundamental constant of Nature. Thus, the Universe was finally not so strange
at that time, and the distinction between cosmological constant and matter still holds. Later on
important problems have appeared. First, big–bang nucleosynthesis has shown that the baryons
represent only a very small part of the matter density of the Universe [Burles & Tyler 1998], so
that dark matter has to exist and to represent a large fraction of the total density of the Universe.
Moreover, this dark matter has to be non–baryonic, and yet the standard model of particle physics
does not provide good candidates for such a kind of matter. No particle of dark matter have
been detected yet, and the nature of the dark matter remains uncertain. Usual candidates for
dark matter are based on new particle physics theories, like supersymmetry, but as these new
theories have not been verified yet and as some simulations based on cold dark matter seem to give
problematic results [Kazantzidis et al. 2004], this answer is not definitive yet.
A second problem arises from the cosmological constant: if it has always been constant, it means
that its density was very small in the early Universe in comparison to the density of the other
fluids, and appears to be dominant today only “by chance”. This is the coincidence problem.
To solve this question of coincidence, one prefers to consider that this energy is not constant,
and the fluid which was called “cosmological constant” is now referred as “dark energy”. The
nature of this dark energy is also unknown and even stranger than usual fluids, as its pressure is
negative today. Usual models for such a fluid are the quintessence models, which consider that the
behavior of the fluid can be explained with a real scalar field associated to a potential dominating
today [Peebles & Ratra 1988]. However, this potential is unknown, and recent observations of
the supernovæ of type Ia seem to indicate that the ratio of pressure over density is near to -1,
corresponding to a real cosmological constant, and can even be less than -1, what then rules out
the involvement of usual real scalar fields and calls for even more exotic explanations.
 3

To summarize, the present standard model of cosmology assumes the existence of two components
of completely unknown nature, and the general approach to this assumption is to find constraints
to better understand their behaviors. The usual method to find the best values of the different
parameters of the model is to try to predict observations and to adjust the parameters to improve
the accuracy of the predictions. In this paper, I will consider a model in which the dark matter
and the dark energy are in fact different aspects of a same fluid, that I will call “dark fluid”.
Such an idea is simpler because one replaces two components of unknown nature by only one.
In the following I will consider constraints from the observations of supernovæ of type Ia, cosmic
microwave background (CMB), large scale structures, and the theoretical predictions from big-bang
nucleosynthesis (BBN), and I will show then that they cannot distinguish between a model based
on two components and a model using only one. I will also use the observations to set constraints
on the dark fluid. In a last paragraph, I will provide some ideas on what could be the nature of
the dark fluid.

I. BASIS OF THE MODEL AND DEFINITIONS

If one considers a Friedmann–Lemaı̂tre Universe with different fluids: photons, neutrinos, baryons
and a dark fluid, the Friedmann equations take the form:
 2
ȧ 8πG k ä 4πG
= H2 = ρ− 2 , =− {ρ + 3P } , (1)
a 3 a a 3
where P and ρ denote the total pressure and the total density in the Universe, respectively. It will
be assumed in the whole article that the value of a today is 1. For the dark fluid model, pressure
and density can be expanded as:

P = Pr + PD ,

ρ = ρr + ρb + ρD , (2)

where r denotes the radiation (i.e. photons + neutrinos), b the baryonic matter and D the dark
fluid.
In the standard model of cosmology, the equivalent expressions are:

P = Pr + Pϕ ,

ρ = ρr + ρb + ρdm + ρϕ , (3)

where dm denotes the dark matter and ϕ the dark energy.

For each fluid, one can write the conservation of the energy–momentum tensor in a homogeneous
 4

and isotropic spacetime, which reads:

d d(a3 )
(ρfluid a3 ) = −Pfluid . (4)
dt dt
This equation is equivalent to the second Friedmann equation. As usual, one can define a cosmo-
logical parameter corresponding to each fluid by:
ρ0
Ω0fluid = fluid , (5)
ρc0
where ρc0 is the critical density today. A cosmological parameter containing the curvature term can
also be written:
k
ΩK ≡ − , (6)
a2 3H 2 /(8πG)
so that one gets finally a simple equation:

ΩK + Ωr + Ωb + ΩD = 1 . (7)

For a flat Universe, one has ΩK = 0, and the Friedmann equations are remarkably simplified.
Finally, one can define the equation of state of the fluid as:
Pfluid
ωfluid ≡ , (8)
ρfluid
and one knows that for baryonic matter ωb = 0, for radiation ωr = 1/3 and for a real cosmological
constant ωφ = −1. For simplicity reasons, it is considered in the following that this fluid is per-
fect, i.e. the entropy variations and the shear stress can be ignored. For other assumptions, it is
necessary to specify a dark fluid model.
A basis for the dark fluid model is now defined, and a comparison with the observations of super-
novæ of type Ia is given in the following.

II. CONSTRAINTS FROM THE SUPERNOVAE OF TYPE IA

Cosmological constraints from the supernovæ of type Ia are based on the joint observations of the
redshift z and of the apparent luminosity l of a large number of supernovæ. Supernovæ of type Ia
are often considered as standard candles, i.e. the absolute luminosity L is approximately the same
for every supernova (it is not completely true and recent studies correct the value of the absolute
luminosity to reflect the deviations from the standard candle behavior [Tonry et al. 2003]), so that
it is possible to determine for each supernova the luminosity distance
1/2
L

dL = . (9)
4πl
 5

This luminosity distance depends on the reddening induced by the expansion of the Universe, and
thus can reveal the presence of the cosmological components, through the equation:
(q Z z )
c 1+z 0 dz ′
dL (z) = q S ΩK p , (10)
H0 Ω0 0 F (z ′ )
K

where z = a−1 − 1 denotes the redshift, S is given by





 sin x if k > 0 ,

S(x) ≡ x if k = 0 , (11)



 sinh x if k < 0 ,


and F is defined as
dz
F (z) = −H0−1 (1 + z)−2 . (12)
dt
One can see that F is directly related to the first Friedmann equation through the term dz/dt =
−a−2 da/dt. If there is only one component – replacing the two dark components – it shall have
the same influence on the luminosity distance – and then on the expansion of the Universe – as
dark matter and dark energy would have. Through the Friedmann equations, it seems clear that if

ρD = ρdm + ρϕ ,

PD = Pdm + Pϕ = Pϕ , (13)

the dark fluid would provide the same effect on the expansion of the Universe as the two compo-
nents.
Observations on the supernovæ of type Ia enable to give constraints on the dark component den-
sities and on the dark energy behavior at low redshift [Riess et al. 2004]. From these constraints,
it should be possible to characterize the dark fluid at low redshift. At first, the cosmological pa-
rameter corresponding to the dark fluid can be written in function of those related to dark matter
and to dark energy:

Ω0D = Ω0dm + Ω0ϕ . (14)

At low redshift, one can consider that the equation of state for the dark energy is, at first order
in z:

ωϕ = ωϕ0 + ωϕ1 z , (15)

and that the equation of state for the dark fluid can have the same form:

0 1
ωD = ωD + ωD z . (16)
 6

The observations have given constraints on the values of ωϕ0 and ωϕ1 , and one would like to deduce
0 and ω 1 . The equation of state of the dark fluid writes:
from them constraints on ωD D

PD Pϕ ρϕ
ωD = = = ωϕ . (17)
ρD ρdm + ρϕ ρdm + ρϕ

At first order in z, the dark matter density evolves like ρdm = ρ0dm a−3 = ρ0dm (1 + 3z) . It would be
interesting to know the behavior of ρϕ . Let us assume that, at first order:

ρϕ = ρ0ϕ + ρ1ϕ z . (18)

The equation of conservation of the energy-momentum tensor for each fluid satisfies:

d d(a3 )
(ρϕ a3 ) = −Pϕ . (19)
dt dt

At first order in z, this equation becomes:

d 0 d(1 − 3z)
(ρϕ + z(ρ1ϕ − 3ρ0ϕ )) = −(ωϕ0 ρ0ϕ ) , (20)
dt dt

so that the density of dark energy reads:

ρ1ϕ = 3ρ0ϕ (1 + ωϕ0 ) . (21)

Then, the relation between the ratio pressure/density for the dark fluid becomes:

ρ0ϕ (1 + 3(1 + ωϕ0 )z)

ωD = (ωϕ0 + ωϕ1 z) , (22)
ρ0dm (1 + 3z) + ρ0ϕ (1 + 3(1 + ωϕ0 )z)

and one can determine the value of the two first terms of the expansion:

0 ωϕ0 Ω0ϕ
ωD = ,
Ω0dm + Ω0ϕ
1 ωϕ1 Ω0ϕ 3Ω0dm Ω0ϕ (ωϕ0 )2
ωD = + . (23)
Ω0dm + Ω0ϕ (Ω0dm + Ω0ϕ )2

The favored values for the cosmological parameters of the usual standard model from the
supernovæ of type Ia [Riess et al. 2004] combined with the results of other observations
[Tegmark et al. 2004 I] are:

h = 0.70 ± 0.04

Ω0K = −0.012 ± 0.022 Ω0dm = 0.25 ± 0.04 (24)

Ω0b = 0.049 ± 0.012 Ω0ϕ = 0.712 ± 0.044

ωϕ0 = −1.02 ± 0.19 ωϕ1 = 0.6 ± 0.5 .

 7

From these values, one can calculate the parameters of the dark fluid:

Ω0D = 0.962 ± 0.084

0
ωD = −0.76 ± 0.25 (25)
1
ωD = 1.0 ± 0.6 .

These values are of course not completely representative of the dark fluid model, because they
come from data analyses based on the usual standard model. Nevertheless, one can use them as
test–parameters at low redshift.
Recent supernova observations tend to show that the dark energy has a negative pressure. More-
over, ωϕ < −1 is not at all excluded, and in that case the dark energy cannot be explained
anymore thanks to the usual models (see for example [Caldwell et al. 2003] for a possible answer
to this problem). From the precedent constraints, one can see that this difficulty vanishes with a
dark fluid. Hence, the pressure of the fluid has to be negative today at cosmological scales, and
0 ≥ −1, it seems possible to model the dark
seems to increase strongly with the redshift. As ωD
fluid with a scalar field.
The study of supernovæ provided us properties of the equation of state of our dark fluid at low
redshift independently from the specification of a dark fluid model. We will now try to extract
constraints from the information concerning large scale structures.

III. LARGE SCALE STRUCTURES

We will not consider here a complete scenario of structure formation, which would require the
specification of a precise model of dark fluid. Nevertheless, one can study the necessary conditions
for the fluid parameters to enable the perturbations to grow and to give birth to large scale
structures.
Let us consider the case where the equation of state of our fluid does not change during the growth
of perturbations, and, to simplify, that the entropy perturbations can be ignored and that the
Jeans length is smaller than any other considered scale. One can define the local density contrast
of the dark fluid as:

ρD (~x, t))
δ (~x, t) ≡ −1 , (26)
ρD (t)

where ρD (~x, t) is the local value of the density, and ρD (t) is the mean background density, i.e. the
apparent cosmological density. In the fluid approximation, one can write the evolution equation of
 8

the local density contrast [Gaztañaga & Lobo 2001]:

!
1 d2 δ Ḣ 1 dδ 2
2 2
+ 2+ 2 − (1 + ωD )(1 + 3ωD )ΩD δ
H dt H H dt 3
2 2
4 + 3ωD 1 1 1 dδ 3
 
= + (1 + ωD )(1 + 3ωD )ΩD δ2 . (27)
3(1 + ωD ) 1 + δ 1+δ H2 dt 2

To solve this equation, one can define a new variable reflecting the expansion:

η = ln a , (28)

so that, if one assumes that the dark fluid is completely dominant at the time of growth of pertur-
bations (in that case, the Friedmann equations reveal that Ḣ/H 2 = −3(1 + ωD )/2), equation (27)
becomes:
2  2
d2 δ 1 − 3ωD dδ 2 4 + 3ωD 1 1 dδ 3

2
+ − (1+ωD )(1+3ωD )δ = + (1+ωD )(1+3ωD )δ2 .
dη 2 dη 3 3(1 + ωD ) 1 + δ 1+δ dη 2
(29)
Because the coefficients of the above equation are time–dependant only, one can separate the spatial
and temporal parts so that

δl (~x, t) = δ0 (~x) D(t) , (30)

where D is called the “linear growth factor”. In the linear approximation, where δ is small, equation
(29) becomes:

d2 D 1 − 3ωD dD 2
+ − (1 + ωD )(1 + 3ωD )D = 0 . (31)
dη 2 2 dη 3

Its solutions take the form D = D1 aα1 + D2 aα2 , with D1 and D2 being two integration constants,
and

α1 = 1 + 3ωD , (32)
3
α2 = − (1 + ωD ) . (33)
2

Thus, in the case of a dominant dark fluid, we have only a growing mode if ωD > −1/3 or if
ωD < −1. One can however note that the last inequality seems very difficult to achieve with
standard model for dark matter or dark energy models.
Let us now consider the observations of the cosmic microwave background to get constraints at
earlier times.
 9

IV. COSMIC MICROWAVE BACKGROUND

A power spectrum of temperature fluctuations can be deduced from the observations of the cosmic
microwave background (CMB) [Spergel et al. 2003]. Predicting this power spectrum requires a
hard work, and a program like CMBFAST [Seljak & Zaldarriaga 1996] is able to produce it for
the cosmological standard model. In our case, we will consider only the position of the peaks to
constrain the parameters of the dark fluid, and we will make some assumptions on the dark fluid
properties.
First, one should note that at high redshift, in the standard model the density of dark energy is
nearly negligible in comparison to that of the dark matter. As the usual model seems to be able
to correctly reproduce the fluctuations of the CMB, one can assume that our dark fluid should not
behave very differently from the superposition dark matter/dark energy, and so should behave at
the moment of recombination nearly like matter. Therefore, one can write the density of our fluid
as a sum of a matter–like term (m) and of another term of unknown behavior (o):
−3
a

ρD = ρls
Dm + ρDo . (34)
als

One can note that this equation gives no constraint on the behavior of the dark fluid, as the second
term is not restricted to any behavior yet.
We do not want to specify a model of dark fluid and we would like to be as general as possible.
Nevertheless, we will consider for simplicity only the background properties of the dark fluid, and
we will not try to reproduce the whole power spectrum, but only consider the position of its peaks
without trying to find their amplitude. The conformal time is defined by:
Z
τ= dt a−1 (t) . (35)

The spacing between the peaks is then given, to a good approximation, by [Hu & Sugiyama 1995]:

τ0 − τls
∆l ≈ π , (36)
cs τls

where cs is the average sound speed before last scattering, and τ0 and τls the conformal time today
and at last scattering. This average sound speed reads:

τls −1/2
9ρb (t)
Z 
cs ≡ τls−1 dτ 3 + , (37)
0 4ρr (t)

where ρb is the density of baryonic matter and ρr is the density of relativistic fluids (radiation and
neutrinos).
 10

Let us consider that the Universe is flat so that the Friedmann equations are simplified.
In this case, the first Friedmann equation can be written:
8πG
H2 = (ρb + ρr + ρDm + ρDo ) . (38)
3
Using the evolution equation of the different densities, this becomes:
−3 !
a 8πG

2
H = H02 Ω0b a−3 + Ω0r a−4 + Ωls
Dm + ρDo . (39)
als 3
The precedent equation cannot be solved if the form of ρDo is not given. In our case, it is possible
to assume that the fraction
ρDo (τ )
ΩDo (τ ) ≡ P (40)
ρ(τ )
does not vary too rapidly before the moment of last scattering (denoted ls), so that an effective
average can be defined:
Z τls
ls
ΩDo ≡ τls−1 ΩDo (τ )dτ . (41)
0

In the following, we will only consider an approximate and effective density:

3 ls
ρDo ≈ H 2 ΩDo . (42)
8πG
Let us then replace this density in the Friedmann equation:

ls
 
H 2 (1 − ΩDo ) = H02 (Ω0b + Ωls 3
Dm als )a
−3
+ Ω0r a−4 . (43)

This time, provided one fixes the values of the different cosmological parameters and knowing the
initial conditions, this equation can be solved. While replacing usual time by conformal time, the
Friedmann equation becomes:
2
da

ls
 
= H02 (1 − ΩDo )−1 (Ω0b + Ωls 3 0
Dm als )a(τ ) + Ωr . (44)
dτ
The value of the conformal time at the moment of last scattering is given by:
v
u ls (s s )
u
−1 t 1 − ΩDo Ω0r Ω0r
τls = 2H0 als + 0 − . (45)
Ωb + Ωls
0 3
Dm als Ωb + Ωls 3
Dm als Ωb + Ωls
0 3
Dm als

One can use the same method to evaluate the conformal time today. The Friedmann equation
reads, after last scattering:
2 !
da ρD

= H02 Ω0b a(τ ) + Ω0r + a(τ ) 4
. (46)
dτ ρC
0
 11

Let us make now the further assumption that the dark fluid has an approximate equation of state:

ρD = ρ̃0D a−3(1+ω D ) , (47)

where ρ̃0D is an effective value of the dark fluid density, such that

3(1+ω D ) ls
ρ̃0D = als ρD , (48)

and ω D is the average value of ωD over the conformal time, weighted by:

ρD (τ )
ΩD (τ ) = P (49)
ρ(τ )

to reflect the fact that the equation of state of our fluid should be more significant when its density
contributes more heavily to the total density of the Universe. ω D is then given by:
Z τ0
ΩD (τ )ωD (τ )dτ
0
ωD ≡ Z τ0 . (50)
ΩD (τ )dτ
0

Defining the effective cosmological parameter Ω̃0D = 3ρ̃0D /(8πGH02 ), the Friedmann equation be-
comes:
2
da
  
= H02 Ω0b a(τ ) + Ω0r + Ω̃0D a(τ )(1−3ω D ) . (51)
dτ

One can then integrate the equation, and show that:

τ0 = 2H0−1 F (ω D ) , (52)

with

1
Z 1  −1/2
F (ω D ) ≡ da Ω0b a + Ω0r + Ω̃0D a(1−3ω D ) . (53)
2 0

There is no analytical integration for this function, but in a few cases. In particular, one has for a
flat Universe:

1
q q 
F (0) = Ω0b + Ω0r + Ω̃0D − Ω0r , (54)
Ωb + Ω̃0D
0

and
 q 
1 1 − Ω̃0r + 2 Ω0D
F (−1/3) = (Ω0D )−1/2 ln  q  . (55)
2 Ω0b + 2 Ω0r Ω0D
 12

0
The case ω D = −1/3 looks much more probable than ω D = 0 when one considers the value of ωD
which was obtained from the supernova data. One finally gets the spacing between peaks:
 v (s s )−1 
u 0
uΩ + Ωls 3
Dm als Ω0r Ω0r
∆l = πcs F (ω D )t b
−1 
ls
als + 0 − − 1 . (56)
1 − ΩDo Ωb + Ωls 3
Dm als Ω0b + Ωls 3
Dm als

The sound velocity cs is then given by:

" ! #−1/2
als 9Ω0b
q Z
ls
 
0 ls 3 0
cs = τls−1 H0−1 1− ΩDo da 3+ a (Ω b + Ω Dm a ls )a + Ω r . (57)
0 4Ω0r

This equation can be integrated analytically, and one finally gets:

!1/2 (s s )−1
1 Ω0r Ω0r Ω0r
cs = als + 0 − × (58)
3 Ω0b Ωb + Ωls 3
Dm als Ω0b + Ωls 3
Dm als
 q 
Ω0r (7Ω0b + 4Ωls 3 0 0 ls 3 0 ls 4 0 0 0 0 0 ls 3
Dm als ) + 2 3Ωb (Ωb + ΩDm als )(Ωb als + ΩDm als + Ωr )(3Ωb als + 4Ωr ) + 6als Ωb (Ωb + ΩDm als )
ln  q  .
Ω0r (7ωb0 + 4Ωls 3 0 0 0 ls 3
Dm als ) + 4Ωr 3Ωb (Ωb + ΩDm als )

ls
One can notice that cs does not depend on ΩDo . The approximate value of als can be taken
from [Hu et al. 2001]:

0 2 −0.74
a−1
ls ≈ 1008(1 + 0.00124(Ωb h ) )(1 + c1 (Ωls 3 c2
Dm als ) ) , (59)

where

c1 = 0.0783(Ω0b h2 )−0.24 (1 + 39.5(Ω0b h2 )0.76 )−1 ,

c2 = 0.56(1 + 21.1(Ω0b h2 )1.28 )−1 . (60)

Unfortunately, this is an implicit equation, and we cannot find als analytically.

ls
Finally, one has a direct dependence between ∆l and the parameters Ωls 0
Dm , ΩDo , Ω̃D and ω D . Even

if this is only an approximate formula, it gives the possibility to visualize directly the effect of the
dark fluid on the position of the peaks, provided that its behavior does not differ much from the
ls
requirements of the approximations on ΩDo and ω D .
However, the calculated ∆l cannot be directly related to the observed spacing between peaks, as
shifts of peaks can be induced by other effects. In particular, the location of the i–th peaks can be
approximated by:

li = ∆l(m − φi ) = ∆l(m − φ̄ − δφi ) , (61)

where φ̄ is the shift of the first peak, corresponding to an overall shift, and the δφi is the specific
shift of the i–th peak. We have fortunately access to the fitting formulae of [Doran & Lilley 2002]:
 13

Overall phase shift φ̄ :

ls
h i
a2
φ̄ = (1.466 − 0.466n) a1 rls + 0.291ΩDo , (62)

where a1 and a2 are given by:

 
a1 = 0.286 + 0.626 Ωb h2 ,
 2  3
a2 = 0.1786 − 6.308Ωb h2 + 174.9 Ωb h2 − 1168 Ωb h2 , (63)

n is the spectral index and rls is defined by:

ρr (als ) Ω0
rls = = ls r 4 . (64)
ρDm (als ) ΩDm als

Relative shift of second peak δφ2 :

−c3
δφ2 = c0 − c1 rls − c2 rls + 0.05 (n − 1) , (65)

with

ls ls
  n   o
c0 = −0.1 + 0.213 − 0.123ΩDo exp − 52 − 63.6ΩDo Ωb h2 ,
  2 
c1 = 0.063 exp −3500 Ωb h2 + 0.015 , (66)
 2
c2 = 6 × 10−6 + 0.137 Ωb h2 − 0.07 ,
ls ls
 
c3 = 0.8 + 2.3ΩDo + 70 − 126ΩDo Ωb h2 .

Relative shift of third peak δφ3 :

d2
δφ3 = 10 − d1 rls + 0.08 (n − 1) , (67)

with

ls
 
d1 = 9.97 + 3.3 − 3ΩDo Ωb h2 , (68)
ls
ls

ls
 (2.25 + 2.77ΩDo ) × 10−5
d2 = 0.0016 − 0.0067ΩDo + 0.196 − 0.22ΩDo Ωb h2 + .
Ωb h2

One can now compare our results to the data. The WMAP experiment provides the precise location
of the two first peaks [Spergel et al. 2003]:

lp1 = 220.1 ± 0.8 ,

lp2 = 546 ± 10 , (69)

 14

ls
Ωls
Dm ΩDo Ω̃0D ωD l1 l2 l3
0.600 0.125 0.95 -0.10 220.3 544.8 817.2
0.400 0.120 0.95 -0.15 220.0 546.1 830.6
0.270 0.080 0.95 -0.20 219.4 544.8 835.4
0.200 0.010 0.50 -0.13 220.2 544.5 835.9
0.250 0.035 0.50 -0.10 220.7 544.1 833.5
0.018 0.013 0.25 -0.07 219.6 543.4 837.4
0.017 0.001 0.10 0.17 219.5 543.1 837.6

TABLE I: Position of the peaks for h = 0.7, n = 1, Ω0b = 0.049 and Ω0r = 9.89 × 10−5 , in function of the
parameters of the dark fluid.

and BOOMERanG gives the position of the third peak [Bernardis et al. 2002]:

lp3 = 825 ± 13 . (70)

To evaluate roughly the value of the parameters of the fluid, one can fix the other parameters as
follows:

n = 1 ,

h = 0.70 , (71)

Ω0b = 0.049 ,

Ω0r = 9.89 × 10−5 .

For these values, one finds in Table 1 the resulting positions of the peaks in function of parameters
of the dark fluid.
One can note that a large range of values is possible. It is not so strange because we have many
parameters for our fluid. A more complete analysis is not needed here, because the strongest
constraints would come from the specification of a model. Without specifying a model, Table 1
shows that the values of the parameters of the dark fluid are not stringently constrained. One can
nevertheless see that for large values of Ω̃0D , the permitted values of ω D are negative, and hence
in that case one can assume that our fluid behaves today like a cosmological constant whereas it
could have behaved mainly like matter at last scattering. For small values of Ω̃0D , ω D is positive,
so that the density of dark fluid should decrease more rapidly than a matter density after last
ls
scattering. In this case, ΩDo is very small, and the fluid should have behaved like matter before
and around last scattering. Small values of Ω̃0D look therefore unrealistic, because as ω D is then
 15

ls
positive, unless our dark fluid has an oscillating density, ΩDo should be much larger and certainly
dominant.
ls
The value of ΩDo , which can be as much as 0.1, also shows that before recombination, the fluid
may have behaved differently from matter, and perhaps like radiation. One can also note that low
ls
values of Ω̃0D corresponds to low values of ΩDo , and then in that case the fluid mostly behaves like
matter.

If one combines these results with the results from the CMB, it seems that the constraints
become −1/3 < ωD < 0 to enable the perturbations to grow. This shows in fact that during the
growth of the perturbations the behavior of the dark fluid should not be too different from that
of matter. This result also confirms that the value of the effective Ω̃0D which appears in equation
(51) cannot be too small (it has to be at least larger than 0.2).
If one wants to perform a much more precise study of a specified model, it would be interesting
to simulate the whole process of structure formation, and to compare the results with surveys like
SDSS [Tegmark et al. 2004 II], or 2dF [Percival et al. 2001]. We will not study here further the
CMB power spectrum, as other features seem more model–dependant, and we want to consider
here the general case. We will now consider the results of the big-bang nucleosynthesis and their
influence on the establishment of a dark fluid model.

V. BIG-BANG NUCLEOSYNTHESIS

Recent analyses of the big–bang nucleosynthesis (BBN) [Coc et al. 2004] indicate a discrepancy
between the value of the baryonic density calculated from the observed Li and 4 He abundances, and
the one calculated with the observations of deuterium. Some explanations can be found. Problems
could have appeared in the measurement of Li and 4 He abundances, or the Li on the stellar surface
could be altered during stellar evolution, or we have no accurate knowledge of the reaction rates
related to 7 Be destruction, or the expansion rate during BBN could have been modified through
an accelerated cosmological expansion. Thus, two possibilities can be considered for the equation
of state of the dark fluid. First, if it is correct to consider a Universe dominated by radiation at
BBN time, the main constraint is that the dark fluid density should be small in comparison to the
radiation density; otherwise Friedmann equations indicates that the expansion rate of the Universe
would be different from the one in the standard BBN, changing then the temperature evolution
rate and so the abundance of the elements. It means that, if one assumes that the dark fluid
 16

behavior does not change violently during BBN, the equation of state of this fluid around the time
of BBN has to be ωD (BBN) ≤ 1/3, or that its density was completely negligible before BBN. In
the case of a real radiative behavior ωD (BBN) = 1/3, the dark fluid behaves like extra–families of
neutrinos, and its density can be constrained. The effective extra–neutrinos number at the BBN
time is defined by:
ρD (BBN)
∆Neff (BBN) ≡ , (72)
ρν (BBN)
where ρν is the standard density of a single relativistic neutrino species. The usual bound on
the number of neutrinos is ∆Neff < 1 [Burles & Tyler 1998], which corresponds in our case for a
temperature around 1 MeV (a−1 ≈ 3 − 4 × 109 ) to:
4/3
7 4 π2 4

ρD (BBN) < T ≈ 3 × 10−2 (MeV)4 . (73)
8 11 15 BBN
This limit is only valid in the case ωD (BBN) = 1/3.
If the abundance of the elements is as observed, a modification of the expansion rate could
provide, as presented in [Salati 2003], a correction to the predicted values. If our fluid is the
dominant component at the time of the BBN, it can have a big influence on the expansion rate
of the Universe. Evaluating the density of the dark fluid and its evolution during the time of the
BBN so that the observations are retrieved would require further studies that I will not develop here.

We have seen that a dark fluid may be compatible with the cosmological observations, and
could be an interesting approach to the ambivalence of dark energy and dark matter. Let us now
consider possible paths to model a dark fluid.

VI. MODELS OF DARK FLUID

In the literature, only few fluids behaving like a dark fluid are considered. Different ways to model
the dark fluid are possible, and I will consider here in particular two of them: the generalized
Chaplygin gas, based on D-brane theories, and another one using scalar fields.

A. Generalized Chaplygin Gas

The generalized Chaplygin gas (GCG) is an exotic fluid derived from D-brane theories
[Bento et al. 2002]. It can be described by an equation of state:
A
Pch = − , (74)
ραch
 17

where α is a constant, 0 < α ≤ 1, and A is another positive constant. This equation of state
corresponds to a density evolving like:
1/(1+α)
1 − As

ρch = ρ0ch As + 3(1+α) , (75)
a

where As = A/(ρ0ch )(1+α) and ρ0ch is the Chaplygin gas density today. Such a behavior could be
interesting in order to model the dark fluid, because for high values of a this density is mainly
constant, and for low values of a it evolves like matter. This behavior has to be compared with
the observations.
For the comparison with the data of supernovæ of type Ia, let us derive the equation of state of
the GCG at low redshift:

Pch As
ωch = =− ≈ −As [1 − 3(1 − As )(1 + α)z] , (76)
ρch As + (1 − As )(1 + z)3(1+α)

so that one can deduce:

0
ωch = −As ,
1
ωch = 3As (1 − As )(1 + α) , (77)

to be compared with the constraints (25), and one finally gets:

As = 0.76 ± 0.25 ,

α = 0.8 ± 2.4 . (78)

We have of course no constraint on α. The analysis of CMB has been done in [Bento et al. 2003],
and the results for h = 0.7 and n = 1, combined with the ones from the supernovæ of type Ia are:

As = 0.75 ± 0.13 , (79)

α = 0.4 ± 0.2 .

If one considers now the BBN, at this time the GCG behaves like matter, so that it is compatible
with the standard BBN scenario. The large scale structure formation has been studied in a case
where the Chaplygin gas adds only a background density to a Universe containing cold dark matter
[Multamäki et al. 2004], and such a scenario seems then possible.
So, the GCG seems to be in agreement with the cosmological observations. A further analysis
is still needed, in particular concerning the growth of structures with a dominant Chaplygin gas
density, or the local behavior of such a fluid.
 18

B. Scalar Fields

One can also consider the idea that the dark fluid could be explained thanks to a scalar
field. Indeed, scalar fields are very useful in explaining the behavior of the dark energy today
[Ratra & Peebles 2000, Hebecker & Wetterich 2001], and recent analyses have shown that they
can behave like matter on local scales [Arbey et al. 2003, Kiselev 2005] as well as on cosmological
scales [Arbey et al. 2002, Guzman & Urena–Lopez 2003].
Let us therefore consider a real scalar field associated with a Lagrangian density

L = gµν ∂µ ϕ ∂ν ϕ − V (ϕ) . (80)

Its pressure and its density on cosmological scales are given by

1 2
Pϕ = ϕ̇ − V (ϕ) ,
2
1
ρϕ = ϕ̇2 + V (ϕ) . (81)
2

So, the pressure is negative if the potential dominates, and negligible if the potential equilibrates
the kinetic term. Thus, a scalar field can be a good candidate for the dark fluid if it respects in
particular the following constraints:
– its density at the time of the BBN decreases at least as fast as the density of radiation,
– its density from the time of last scattering to the time of structure formation evolves nearly
like matter, and so 21 ϕ̇2 ≈ V (ϕ),
– after the growth of perturbations, because the scalar field is dominating the Universe, its
potential does not equilibrate the kinetic term anymore and begins to dominate; thus it will
behave like a cosmological constant in the future.

The main parameter of such a model is the same as that of quintessence models: the po-
tential. When one considers quintessence models with real scalar field, one looks for potentials
which provide a cosmological constant–like behavior today, and decreasing potentials seem to be
favored. If one considers now complex scalar fields, it was shown in [Arbey et al. 2002] that such
a field can behave like cosmological matter when its potential has a dominant m2 |φ|2 term, which
corresponds to an increasing term in the potential. Thus, a way to find a “good” potential would
be to consider a superposition of a decreasing potential which would begin to dominate today,
and of the increasing quadratic term which has to dominate at least until structure formation and
can nevertheless lead to an attractive effect on local scales today. A more detailed study of the
 19

possible potentials will be presented in a future publication.

One could also hope that a scalar field might explain the excess of gravity on local scales.
To do that, one can imagine a Universe filled with a scalar field. In the part – and time – of the
Universe where the density of baryonic matter is high, the scalar field would, through gravitational
interaction, have a large kinetic term which could even equilibrate the potential, so that one has
an attractive net force on local scales (easier to achieve with a complex scalar field associated to
an internal rotation, see [Arbey et al. 2003]), whereas in the parts where no baryons are present,
the field would not vary much, and the potential dominates, providing repulsion. Thus, on local
scales, where the baryon density is high, the field behaves like matter. Where the baryonic density
is small, i.e. away from galaxies and clusters, the gravitational interaction is not strong enough
to increase the kinetic term of the scalar field, so that the potential dominates, and one can
then observe the effects of a negative pressure. In that case, the scalar field will have a negative
pressure on cosmological scales, providing a locally negative pressure in average. In the past,
baryons were uniformly dense, so that the kinetic term was large everywhere, and one could have
then a uniform matter behavior under these conditions. In that way, the local behavior can be in
agreement with the cosmological one, and a complete cosmological scenario can be built. Such a
scenario has of course to be studied further.

VII. CONCLUSION AND PERSPECTIVES

Astrophysical and cosmological observations are usually interpreted in terms of dark matter and
dark energy. We have seen here that they can also be analyzed differently. Thus, it is possible
to develop a model of dark fluid, which could advantageously replace a model containing in fact
two dark components. Of course, hard work and studies are required to test completely the dark
fluid hypothesis. Nevertheless today, as it seems difficult to find a model for dark energy and
as problems concerning cold dark matter remain, it is worthwhile to investigate different ideas
such as an unification of dark energy and dark matter – that finally does not seem stranger than
trying to determine the nature of two components at the same time – which could be achieved in
particular thanks to D-brane theories (in particular through the Chaplygin gas), or thanks to the
so–useful scalar fields. Of course, other models may also account for dark fluid.

An important question remains, how to interpret the dark matter problem on local scales
 20

and could the dark fluid account for the excess of gravity inside local structures? I will only pro-
vide here a qualitative analysis of whether a fluid with a negative pressure on cosmological scale can
have an attractive effect on local scale, such as it is observed in galaxies (for example, with the ro-
tation curves of spiral galaxies [Persic et al. 1986, Gentile et al. 2004, Carignan & Purton 1998]).

Let us consider only the quasi–Newtonian limit of general relativity. In that case, devia-
tions from the Minkowski metric ηµν = diag(1, −1, −1, −1) are accounted for by the perturbation
hµν . In the harmonic coordinate gauge, it satisfies the condition:
1
∂α hαµ − ∂µ hαα = 0 . (82)
2
One can show that the perturbation hµν is related to the source tensor:
1
Sµν = Tµν − gµν T αα (83)
2
through the integral
Sµν (~r ′ ) 3 ′
Z
hµν (~r) = − 4 G d ~r . (84)
| ~r ′ − ~r |
If the energy–momentum tensor is written as:

T µν = (P + ρ)U µ U ν − P gµν , (85)

one can show that, at leading order in hµν ,

1
S µν = (P + ρ)U µ U ν − η µν (ρ − P ) . (86)
2
The gravitational potential is in fact Φ = h00 /2, so that:
S00 (~r ′ ) 3 ′
Z
Φ (~r) = − 2 G d ~r . (87)
| ~r ′ − ~r |
For a fluid at rest, U µ = (1, 0, 0, 0), with an equation of state P = ωρ, one has:
1
S00 = ρ(1 + 3ω) , (88)
2
and consequently this fluid has an attractive effect only if ω > −1/3. From the study of super-
novæ it seems that our dark fluid is not in that state today, so that its effects are mainly repulsive.
Nevertheless, it is possible that the dark fluid has a different behavior on cosmological and on local
scales. Indeed, the density and pressure of the fluid on cosmological scale are spacial averages of
the local density and pressure, and one can assume that:

ρ (t, ~r) = ρcosmo (t) + δρ (t, ~r) ,

P (t, ~r) = P cosmo (t) + δP (t, ~r) , (89)

 21

where ρcosmo and P cosmo are the cosmological density and pressure, with the spatial averages:

< δρ (t, ~r) >=< δP (t, ~r) >= 0 . (90)

The density of dark fluid on cosmological scales today is of the order of the critical density, i.e.
ρ0c ≈ 9 × 10−29 g.cm−3 . One can compare it to the estimated matter density in the Milky Way at
the radius of the Sun ρSun ≈ 5 × 10−24 g.cm−3 [Olling & Merrifield 2001]. Hence, even if the dark
fluid’s local density would represent 1% of this total matter local density only, its value would be
much higher than the cosmological densities today. Therefore, on local scales, one can assume that
δρ (t, ~r) ≫ ρcosmo (t) and thus write:

1
S00 ≈ (δρ + 3δP ) . (91)
2

To have a net attraction, we get finally the same kind of constraint as before, δρ > −3δP , but this
time we do not have to use the cosmological constraints, because the local behavior of the dark
fluid can be very different from the cosmological one. Moreover, if no model is specified, one can
still hope that δP could be negligible on local scales, so that the local behavior of the dark fluid is
matter-like, and that the usual Newtonian equation can be retrieved:

δρ (~r ′ ) 3 ′
Z
Φ (~r) = − G d ~r . (92)
| ~r ′ − ~r |

This local behavior will of course have to be verified quantitatively for each dark fluid model, but
nevertheless gives hope for a unified explanation on any scale. In particular, considering scalar
fields, this qualitative analysis tends to show that it would be interesting to try to find a potential
which gives today a negative pressure on cosmological scale, but which also gives a matter behavior
in local structures, i.e. where the density of baryons is high.
To conclude, the dark fluid appears as an interesting possibility to explain the observations. As
the properties of the dark fluid are different from dark matter and dark energy, models of dark
fluid are worth to be studied, and we can now use many precise observations as strong constraints
on such models.
 22

Acknowledgements

I would like to thank Farvah Mahmoudi, Hélène Courtois, Julien Devriendt, Thierry Sousbie and
Wolfgang Hillebrandt for their comments and for useful discussions.

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