and Cosmology

4.5 Achievements and Problems of the Standard Model
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(4.31) we find H(a) ≈ H 0
√
Ωm a −3/2 , and (4.68) yields
r H,com (z) ≈ 2 c H 0
1
√ (1 + z)Ωm
for z eq ≫ z ≫ 0 .
(4.69)
In earlier phases, z ≫ z eq , H is radiation-dominated,
H(a) ≈ H 0
√
Ωr /a 2 , and (4.68) becomes
r H,com (z) ≈
c 1
√
H 0 Ωr (1 + z)
for z ≫ z eq .
(4.70)
The earlier the cosmic epoch, the smaller the comoving
horizon length, as was to be expected. In particular, we
will now consider the recombination epoch, z rec ∼ 1000,
for which (4.69) applies (see Fig. 4.16). The comoving
length r H,com corresponds to a physical proper length
r H,prop = ar H,com , and thus
r H,prop (z rec ) = 2 c H 0
Ω −1/2
m (1 + z rec ) −3/2 (4.71)
is the horizon length at recombination. We can then
calculate the angular size on the sky that this length
corresponds to,
θ H,rec = r H,prop(z rec )
D A (z rec )
,
where D A is the angular-diameter distance (4.45) to the
last scattering surface of the CMB. Using (4.47), we
find that in the case of Ω Λ = 0
D A (z) ≈ c H 0
2
Ω m z
and hence
θ H,rec ≈
for z ≫ 1 ,
√ √
Ω m Ωm
∼
z rec 30 ∼ √ Ω m 2 ◦ for Ω Λ = 0 .
(4.72)
This means that the horizon length at recombination
subtends an angle of about one degree on the sky.
Fig. 4.16. The horizon problem: the region of space which
was in causal contact before recombination has a much
smaller radius than the spatial separation between two regions
from which we receive the CMB photons. Thus the
question arises how these two regions may “know” of each
other’s temperature
The Horizon Problem: Since no signal can travel
faster than light, (4.72) means that CMB radiation
from two directions separated by more than
about one degree originates in regions that were
not in causal contact before recombination, i.e., the
time when the CMB photons interacted with matter
the last time. Therefore, these two regions have
never been able to exchange information, for example
about their temperature. Nevertheless their
temperature is the same, as seen from the high degree
of isotropy of the CMB, which shows relative
fluctuations of only ΔT/T ∼ 10 −5 !
Redshift-Dependent Density Parameter. We have defined
the density parameters Ω m and Ω Λ as the current
density divided by the critical mass density ρ cr today.
These definitions can be generalized. If we existed at
a different time, the densities and the Hubble constant
would have had different values and consequently we
would obtain different values for the density parameters.
Thus we define the total density parameter for an