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the dark sector can be described by just one dark fluid which can be characterized
with very simple assumptions, and that there is no observation relying on zero and
first order cosmological perturbation theory that can distinguish it from the
CDM
model, concluding that the standard decomposition of the dark sector is arbitrary
(Kunz
2009
).
The paper is organized as follows, in Sect.
2
we develop the background evolution
of the Universe in Newtonian theory, where arguments are supplemented to under-
stand the results in a curved relativistic framework. In Sect.
3
we study the theory of
small perturbations to the background evolution, which thereafter are generalized to
curved spacetimes. In Sect.
4
we introduce the dark fluid and show explicitly that it
is degenerated with the
ʛ
ʛ
CDM model. Finally in Sect.
5
we summarize our results.
2 Homogeneous and Isotropic Cosmology
in Newtonian Gravity
One of the cornerstones of Modern Cosmology is that the Universe is homogeneous
and isotropic at very large scales (from above about 150Mpc), and this paradigm
is called the Cosmological Principle. We observe essentially the same structures on
the sky, a uniformly random field of distribution, type and composition of galaxies;
moreover, as we look in any direction we detect the same background of cosmic
microwave background radiation with a blackbody spectrum at a temperature of
2
725 K with slight differences of the order of 10
−
5
K. Assuming that we do not
live in a privileged position in the Universe, the foundations of the Cosmological
Principle relies on firm grounds.
To properly discuss this large scale scenario in a Newtonian framework, consider
a spherical region of the space and the total mass
M
contained in it, and denote the
radius of that sphere by
R
.
, which is in general a function of time. Take a small
region over the sphere with mass
m
. Ignoring all other forces except for gravity, the
homogeneity of the Universe allows us to write the conservation of energy
E
as
(
t
)
2
R
R
8
ˀ
G
3
ˁ
M
+
2
E
mR
2
=
(1)
GR
3
where we defined the mass density
ˁ
M
≡
3
M
/(
4
ˀ
)
and a dot means derivative
with respect to time
t
. We can write
R
(
t
)
=
c
˄
0
a
(
t
)
where
c
is the speed of light,
˄
0
an arbitrary time scale and
a
(
t
)
a dimensionless function of time called the scale
mc
2
R
0
, and we choose
factor, then we define
K
≡−
2
E
/
˄
0
such that
K
can take
one of the three values
−
1, 0 or 1, thus we obtain the Friedmann equation
˙
2
Kc
2
a
2
a
a
8
ˀ
G
3
ˁ
−
H
2
≡
=
,
(2)
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