Environmental Engineering Reference
In-Depth Information
ac
s
√
ˀ/
ak
−
1
ʻ
J
=
ˁ
∼
>ʻ
J
G
, for which perturbations with comoving size
L
<ʻ
J
oscillate and decay.
From Eq. (
26
) we can infer the behavior of dark matter perturbations in the differ-
ent epochs of the cosmic evolution. Let us first consider a matter dominated Universe,
from the Friedmann equation and since
grow while those with
L
a
−
3
and
a
t
2
/
3
, it follows that
ˁ
M
∝
∝
3
t
2
. In this case, Eq. (
26
) has two independent solutions
H
=
2
/
3
t
and 4
ˀ
G
ˁ
=
2
/
t
−
1
and
t
2
/
3
ʴ
dm
∝
a
. The growing mode of the density contrast grows lin-
early with the scale factor and from Eq. (
23
) it follows that the gravitational potential
ˆ
ʴ
dm
∝
∝
is constant in this case. Similar analysis for the growth of the density contrast of the
dark matter in the epochs dominated by radiation and dark energy show that in the
first case the growing mode is logarithmic while for the second case does not exist,
but it remains constant, suppressing the formation of structure. Therefore, in order
for matter perturbations to grow enough to form the structures we observe today, it
must have elapsed a sufficiently long epoch in which the expansion of the universe
was driven by matter (either baryonic or dark).
The equations for the perturbed variables developed so far are valid for non-
relativistic matter fields and for scales which are smaller than the curvature length
scale
cH
−
1
, as discussed in the Introduction. To obtain the complete equations it
is mandatory to use the theory of General Relativity and hydrodynamics in curved
spacetimes. For completeness we present here the equations for a collection of fluids
with equation of state
P
=
w
(ˁ)ˁ
and that do not posses anisotropic stresses. These
are the Poisson equation
G
k
2
ˆ
=−
4
ˀ
ˁ
i
ʔ
i
,
(27)
where
k
2
ʔ
i
=
ʴ
i
+
3
H
(
1
+
w
i
)ʸ/
,
(28)
the continuity equation
3
H
ʴ
w
d
dt
+
(
3
d
dt
P
ʴˁ
1
+
w
)
ʸ
−
+
−
ʴ
=
0
,
(29)
and the Euler equation
k
2
c
s
k
2
a
2
ˆ
=
d
dt
+
w
˙
2
H
(
1
−
3
w
)ʸ
+
w
ʸ
−
)
ʴ
−
0
.
(30)
1
+
a
2
(
1
+
w
First we want to make note that for perturbations with wavelengths well bellow
the Hubble scale, i.e.
k
H
, the Poisson equation reduces to the one found in
the non relativistic treatment. Moreover, for non relativistic matter
w
=˙
w
=
0 and
d
ˆ/
0, recovering the Newtonian Euler and continuity equations.
Now, consider the case of dark energy with EoS parameter
w
dt
=
1. At any epoch
of the cosmic evolution from the continuity equation follows that the density contrast
is a constant. This feature and the fact that its energy density remains also constant,
=−
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