Environmental Engineering Reference
In-Depth Information
We conclude that the origin of about 96% of the energy content of the Universe is
unknown to us. In Sect.
4
we will consider the possibility that the whole dark sector
is composed by just one dark fluid.
We finally rewrite the Friedmann equation (Eq.
2
) for a flat space Universe as
8
ˀ
G
3
1
a
4
+
ˁ
b
0
1
a
3
+
ˁ
dm
0
1
a
3
+
ˁ
ʛ
H
2
=
ˁ
r
0
,
(8)
where possible contribution of massive neutrinos were omitted. Note that the first
two terms on the right hand side of the above equation correspond to the “light”
sector, while the last two to the dark sector.
3 Small Perturbations in Newtonian Cosmology
In the study of the Universe at small scales, the homogeneous and isotropic descrip-
tion is no longer valid. Strictly, this situation can only be completely confronted into
the framework of GR. The problem to study the Universe at these scales is that all the
symmetries present in the homogeneous and isotropic description are not present.
A possible solution, the one we adopt, is to treat only with small departures to the
background evolution.
Thus, we want to study the evolution of a fluid with energy density
ˁ
=
ˁ(
r
,
t
)
and velocity field
r
˙
=
u
=
u
(
r
,
t
)
in the presence of a gravitational field
ʦ(
r
,
t
)
.
The continuity, Euler and Poisson equations are
D
Dt
+∇
r
·
(ˁ
u
)
=
0
,
(9)
D
u
Dt
=−
∇
r
P
−∇
r
ʦ
=
0
,
(10)
ˁ
and
2
∇
r
ʦ
=
4
ˀ
G
ˁ,
(11)
respectively. We have used the convective derivative
D
·∇
r
which
describes the time derivative of a quantity at rest in the comoving fluid frame. Adding
an equation of state
P
/
Dt
≡
∂/∂
t
+
u
the problem is solvable in principle, but in practice
such a situation is intractable. The alternative studied here is to treat only small
departures from the background description introduced in the previous section. To
this end, let us first consider coordinates
x
which are comoving with the background
evolution, these are defined by
=
P
(ˁ,
S
)
1
x
≡
r
,
(12)
a
(
t
)
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