and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
156
Local Hubble Law. The Hubble law applies for nearby
sources: with (4.8) and v ≈ zc it follows that
z = H 0
c D ≈ hD
3000 Mpc
for z ≪ 1 , (4.39)
where D is the distance of a source with redshift z.
This corresponds to a light travel time of Δt = D/c.
On the other hand, due to the definition of the Hubble
parameter, we have Δa = (1 − a) ≈ H 0 Δt, where a is
the scale factor at time t 0 − Δt, and we used a(t 0 ) = 1
and H(t 0 ) = H 0 . This implies D = (1 − a)c/H 0 . Utilizing
(4.39), we then find z = 1 − a, or a = 1 − z,
which agrees with (4.38) in linear approximation since
(1 + z) −1 = 1 − z + O(z 2 ). Hence we conclude that the
general relation (4.38) contains the local Hubble law as
a special case.
Energy Density in Radiation. A further consequence
of (4.38) is the dependence of the energy density of radiation
on the scale parameter. As mentioned previously,
the number density of photons is ∝ a −3 if we assume
that photons are neither created nor destroyed. In other
words, the number of photons in a comoving volume element
is conserved. According to (4.38), the frequency ν
of a photon changes due to cosmic expansion. Since
the energy of a photon is ∝ ν, E γ = h P ν ∝ 1/a, the
energy density of photons decreases, ρ r ∝ nE γ ∝ a −4 .
Therefore (4.38) implies (4.24).
Cosmic Microwave Background. Assuming that, at
some time t 1 , the Universe contained a blackbody radiation
of temperature T 1 , we can examine the evolution
of this photon population in time by means of relation
(4.38). We should recall that the Planck function B ν
(A.13) specifies the radiation energy of blackbody radiation
that passes through a unit area per unit time, per
unit area, per unit frequency interval, and per unit solid
angle. Using this definition, the number density dN ν of
photons in the frequency interval between ν and ν + dν
is obtained as
dN ν
dν = 4π B ν
ch P ν = 8πν2 1
( )
c 3 . (4.40)
exp hP ν
k B T 1
− 1
At a later time t 2 > t 1 , the Universe has expanded by
a factor a(t 2 )/a(t 1 ). An observer at t 2 therefore observes
the photons redshifted by a factor (1 + z) = a(t 2 )/a(t 1 ),
i.e., a photon with frequency ν at t 1 will then be
measured to have frequency ν ′ = ν/(1 + z). The original
frequency interval is transformed accordingly as
dν ′ = dν/(1 + z). The number density of photons decreases
with the third power of the scale factor, so
that dN
ν ′ ′ = dN ν /(1 + z) 3 . Combining these relations,
we obtain for the number density dN
ν ′ ′ of photons in the
frequency interval between ν ′ and ν ′ + dν ′
dN
ν ′ ′
dν ′ = dN ν/(1 + z) 3
dν/(1 + z)
1 8π(1 + z) 2 ν ′ 2
1
=
( )
(1 + z) 2 c 3 exp hP (1+z)ν ′
k B T 1
− 1
= 8πν′ 2
1
( )
c 3 , (4.41)
exp hP ν ′
k B T 2
− 1
where we used T 2 = T 1 /(1 + z) in the last step. The
distribution (4.41) has the same form as (4.40) except
that the temperature is reduced by a factor (1 + z) −1 .If
a Planck distribution of photons had been established at
an earlier time, it will persist during cosmic expansion.
As we have seen above, the CMB is such a blackbody
radiation, with a current temperature of T 0 = T CMB ≈
2.73 K. We will show in Sect. 4.4 that this radiation
originates in the early phase of the cosmos. Thus it is
meaningful to consider the temperature of the CMB as
the “temperature of the Universe” which is a function
of redshift,
T(z) = T 0 (1 + z) = T 0 a −1 , (4.42)
i.e., the Universe was hotter in the past than it is today.
The energy density of the Planck spectrum is
( π
ρ r = a SB T 4 2 k 4 )
B
≡ T 4
15 3 c 3 , (4.43)
so ρ r behaves like (1 + z) 4 = a −4 in accordance with
(4.24).
Generally, it can be shown that the specific intensity
I ν changes due to redshift according to
I ν
ν = I ν ′ ′
3 (ν ′ ) . (4.44)
3
Here, I ν is the specific intensity today at frequency ν
and I
ν ′ ′ is the specific intensity at redshift z at frequency
ν ′ = (1 + z)ν.