and Cosmology

4.2 An Expanding Universe
physical volume V = a 3 (t)V x will change due to expansion.
Thus, a 3 = V/V x is the volume, and c 2 ρ a 3 the
energy contained in the volume, each divided by V x .
Taken together, (4.17) corresponds to the first law of
thermodynamics in an expanding universe.
The Friedmann–Lemaître Expansion Equations.
Next, we will present equations for the scale factor a(t)
which follow from GR for a homogeneous isotropic universe.
Afterwards, we will derive these equations from
the relations stated above – as we shall see, the modifications
by GR are in fact only minor, as expected from
the argument that a small section of a homogeneous
universe characterizes the cosmos as a whole. The field
equations of GR yield the equations of motion
and
(ȧ
a
) 2
= 8πG
3 ρ − Kc2
a 2 + Λ 3
(
ä
a =−4πG ρ + 3P )
3 c 2 + Λ 3
(4.18)
, (4.19)
where Λ is the aforementioned cosmological constant
introduced by Einstein. Compared to equations (4.13)
and (4.14), these two equations have been changed
in two places. First, the cosmological constant occurs
in both equations, and second, the equation of motion
(4.19) now contains a pressure term. The pair of
equations (4.18) and (4.19) are called the Friedmann
equations.
The Cosmological Constant. When Einstein introduced
the Λ-term into his equations, he did this solely
for the purpose of obtaining a static solution for the
resulting expansion equations. We can easily see that
(4.18) and (4.19), without the Λ-term, have no solution
for ȧ ≡ 0. However, if the Λ-term is included, such
a solution can be found (which is irrelevant, however, as
we now know that the Universe is expanding). Einstein
had no real physical interpretation for this constant, and
after the expansion of the Universe was discovered he
discarded it again. But with the genie out of the bottle,
the cosmological constant remained in the minds of
cosmologists, and their attitude towards Λ has changed
frequently in the past 90 years. Around the turn of the
millennium, observations were made which strongly
suggest a non-vanishing cosmological constant, i.e., we
believe today that Λ ̸= 0.
But the physical interpretation of the cosmological
constant has also been modified. In quantum mechanics
even completely empty space, the so-called vacuum,
may have a finite energy density, the vacuum energy
density. For physical measurements not involving gravity,
the value of this vacuum energy density is of no
relevance since those measurements are only sensitive
to energy differences. For example, the energy of a photon
that is emitted in an atomic transition equals the
energy difference between the two corresponding states
in the atom. Thus the absolute energy of a state is measurable
only up to a constant. Only in gravity does the
absolute energy become important, because E = mc 2
implies that it corresponds to a mass.
It is now found that the cosmological constant is
equivalent to a finite vacuum energy density – the equations
of GR, and thus also the expansion equations, are
not affected by this new interpretation. We will explain
this fact in the following.
4.2.5 The Components of Matter in the Universe
Starting from the equation of energy conservation
(4.14), we will now derive the relativistically correct
expansion equations (4.18) and (4.19). The only change
with respect to the Newtonian approach in Sect. 4.2.3
will be that we introduce other forms of matter. The essential
components of the Universe can be described as
pressure-free matter, radiation, and vacuum energy.
Pressure-Free Matter. The pressure in a gas is determined
by the thermal motion of its constituents. At room
temperature, molecules in the air move at a speed comparable
to the speed of sound, c s ∼ 300 m/s. For such
agas,P ∼ ρ c 2 s ≪ ρc2 , so that its pressure is of course
gravitationally completely insignificant. In cosmology,
a substance with P ≪ ρc 2 is denoted as (pressure-free)
matter, also called cosmological dust. 2 We approximate
P m = 0, where the index “m” stands for matter. The
constituents of the (pressure-free) matter move with
velocities much smaller than c.
2 The notation “dust” should not be confused with the dust that is
responsible for the extinction of light – “dust” in cosmology only
denotes matter with P = 0.
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