and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
150
Radiation. If this condition is no longer satisfied, thus
if the thermal velocities are no longer negligible compared
to the speed of light, then the pressure will also
no longer be small compared to ρc 2 . In the limiting
case that the thermal velocity equals the speed of light,
we denote this component as “radiation”. One example
of course is electromagnetic radiation, in particular the
CMB photons. Another example would be other particles
of vanishing rest mass. Even particles of finite mass
can have a thermal velocity very close to c if the thermal
energy of the particles is much larger than the rest
mass energy, i.e., k B T ≫ mc 2 . In these cases, the pressure
is related to the density via the equation of state for
radiation,
P r = 1 3 ρ r c 2 . (4.20)
Vacuum Energy. The equation of state for vacuum energy
takes a very unusual form which results from the
first law of thermodynamics. Because the energy density
ρ v of the vacuum is constant in space and time,
(4.17) immediately yields the relation
P v =−ρ v c 2 . (4.21)
Thus the vacuum energy has a negative pressure. This
unusual form of an equation of state can also be made
plausible as follows: consider the change of a volume V
that contains only vacuum. Since the internal energy is
U ∝ V, and thus a growth by dV implies an increase
in U, the first law dU =−P dV demands that P be
negative.
4.2.6 “Derivation” of the Expansion Equation
Beginning with the equation of energy conservation
(4.14), we are now able to derive the expansion equations
(4.18) and (4.19). To achieve this, we differentiate
both sides of (4.14) with respect to t and obtain
2 ȧ ä = 8πG (˙ρ a 2 + 2 a ȧ ρ ) .
3
Next, we carry out the differentiation in (4.17), thereby
obtaining ˙ρa 3 + 3ρa 2 ȧ =−3Pa 2 ȧ/c 2 . This relation is
then used to replace the term containing ˙ρ in the
previous equation, yielding
(
ä
a =−4πG ρ + 3P )
. (4.22)
3 c 2
This derivation therefore reveals that the pressure term
in the equation of motion results from the combination
of energy conservation and the first law of thermodynamics.
However, we point out that the first law in
the form (4.17) is based explicitely on the equivalence
of mass and energy, resulting from Special Relativity.
When assuming this equivalence, we indeed obtain
the Friedmann equations from Newtonian cosmology,
as expected from the discussion at the beginning of
Sect. 4.2.1.
Next we consider the three aforementioned components
of the cosmos and write the density and pressure
as the sum of dust, radiation, and vacuum energy,
ρ = ρ m + ρ r + ρ v = ρ m+r + ρ v , P = P r + P v ,
where ρ m+r combines the density in matter and radiation.
In the second equation, the pressureless nature of
matter, P m = 0, was used so that P m+r = P r . By inserting
the first of these equations into (4.14), we indeed obtain
the first Friedmann equation (4.18) if the density ρ there
is identified with ρ m+r (the density in “normal matter”),
and if
ρ v =
Λ
8πG . (4.23)
Furthermore, we insert the above decomposition of density
and pressure into the equation of motion (4.22) and
immediately obtain (4.19) if we identify ρ and P with
ρ m+r and P m+r = P r , respectively. Hence, this approach
yields both Friedmann equations; the density and the
pressure in the Friedmann equations refer to normal
matter, i.e., all matter except the contribution by Λ.
Alternatively, the Λ-terms in the Friedmann equations
may be discarded if instead the vacuum energy density
and its pressure are explicitly included in P and ρ.
4.2.7 Discussion of the Expansion Equations
Following the “derivation” of the expansion equations,
we will now discuss their consequences. First
we consider the density evolution of the various cosmic
components resulting from (4.17). For pressure-free
matter, we immediately obtain ρ m ∝ a −3 which is in
agreement with (4.11). Inserting the equation of state
(4.20) for radiation into (4.17) yields the behavior
ρ r ∝ a −4 ; the vacuum energy density is a constant in