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which can be integrated yielding
t
2
/
3
(
1
+
w
)
,
a
(
t
)
∝
(6)
for
w
1 corresponds to non-evolving dark energy and the growth
of the scale factor as a function of time
t
becomes exponential while its energy
density remains constant. In general, the situation ismore complicated and an analytic
expression for the scale factor cannot be found. There are several fluids whichmust be
considered, namely matter, incoherent electromagnetic radiation, massive neutrinos
and possibly dark energy, and therefore, the energy density of each one of themmust
contribute to the Friedmann equation. At this point it is convenient to introduce the
redshift
z
through
a
=−
1.
w
=−
)
−
1
, which is commonly used instead of the scale
factor. Consider a Universe filled with matter (
m
), radiation (
r
), dark energy with
EoS parameter
w
=
(
1
+
z
=−
1(
ʛ
) and with a possible non-zero curvature, the Friedmann
equation can be written as
4
1
/
2
2
3
H
(
z
)
=
ʩ
ʛ
+
ʩ
K
(
1
+
z
)
+
ʩ
M
(
1
+
z
)
+
ʩ
r
(
1
+
z
)
,
(7)
3
H
0
where
ʩ
i
=
8
ˀ
G
ˁ
i
0
/
for matter, radiation and dark energy and
ʩ
K
=
H
0
, are the energy contents parameters at present time—Note that
ʩ
j
Kc
2
−
1.
Several independent probes of the expansion history of theUniversewhich include
redshift-distance measurements of Supernovae type Ia (Riess et al.
1998
,
1999
;
Perlmutter et al.
1999
) and observations of Baryon Acoustic Oscillations (Percival
et al.
2010
) agree in the fact that nowadays the Universe is spatially very flat (
/
=
ʩ
K
0) and the dominant components to the energy content are dark energy
ʩ
de
0
0), and additional tiny
contributions of radiation are also present. The question whether all this matter can
be provided by the standard model of particles arises, it turns out that the answer is no
for several reasons: The theory of Big Bang Nucleosynthesis (Gamow
1948
;Olive
et al.
2000
) is very accurate in predicting the relative abundances of light nuclei of
atoms, these results are very dependent in the quantity of baryons
b
present at that
time, and to obtain the observed abundances it is necessary that
.
7 (with
w
de
−
1) and matter
ʩ
M
0
.
3 (with
w
M
=
04; other
constrictions to this parameter arise when one consider observations of the perturbed
Universe, for example, measurements of the anisotropies in the temperature of the
Cosmic Microwave Background Radiation (Ade et al.
2013
) and large scale structure
observations (Reid et al.
2015
), both agreeing on similar values to the above quoted
for
ʩ
b
0
.
ʩ
b
. Moreover, analysis of virialized cosmic structures as clusters of galaxies and
rotation curves in spiral galaxies show that there is a lot of missing matter that we
do not observe; for a review see Roos and Mod (
2012
). Therefore, the matter sector
that fills the Universe must be split into two components,
ʩ
M
=
b
+
ʩ
dm
, one is
the contribution of the standard model of particles, and the other is the dark matter,
which comprises about 80% of the total matter and whose fundamental nature is still
unknown.
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