and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
154
Fig. 4.8. The scale factor a(t) as a function of cosmic time t for
three models with a vanishing cosmological constant, Ω Λ = 0.
Closed models (K > 0) attain a maximum expansion and then
recollapse. In contrast, open models (K ≤ 0) expand forever,
and the Einstein–de Sitter model of K = 0 separates these two
cases. In all models, the scale factor tends towards zero in the
past; this time is called the Big Bang and defines the origin of
the time axis
If Ω Λ = 0 then q 0 > 0, ä < 0, i.e., the expansion decelerates,
as expected due to gravity. However, if Ω Λ
is sufficiently large the deceleration parameter may
become negative, corresponding to an accelerated expansion
of the Universe. The reason for this behavior,
which certainly contradicts intuition, is seen in the vacuum
energy. Only a negative pressure can cause an
accelerated expansion – more precisely, as seen from
(4.22), P < −ρc 2 /3 is needed for ä > 0. Indeed, we believe
today that the Universe is currently undergoing an
accelerated expansion and thus that the cosmological
constant differs significantly from zero.
Age of the Universe. The age of the Universe at
a given scale factor a follows from dt = da(da/dt) −1 =
da/(aH). This relation can be integrated,
t(a) = 1 ∫a
da [ a −2 Ω r + a −1 Ω m + (1 − Ω m − Ω Λ )
H 0
0
+ a 2 ] −1/2
Ω Λ , (4.34)
where the contribution from radiation for a ≫ a eq can
be neglected because it is relevant only for very small a
and thus only for a very small fraction of cosmic time.
To obtain the current age t 0 of the Universe, (4.34)
is calculated for a = 1. For models of vanishing spa-
Fig. 4.9. Top panel: scale factor a(t) as a function of cosmic
time, here scaled as (t −t 0 )H 0 , for an Einstein–de Sitter model
(Ω m = 1, Ω Λ = 0; dotted curve), an open universe (Ω m = 0.3,
Ω Λ = 0; dashed curve), and a flat universe of low density
(Ω m = 0.3, Ω Λ = 0.7; solid curve). At the current epoch,
t = t 0 and a = 1. Bottom panel: age of the universe in units of
the Hubble time t H = H0 −1 for flat world models with K = 0
(Ω m + Ω Λ = 1; solid curve) and models with a vanishing
cosmological constant (dashed curve). We see that for a flat
universe with small Ω m (thus large Ω Λ = 1 − Ω m ), t 0 may be
considerably larger than H0
−1
tial curvature K = 0 and for those with Λ = 0, Fig. 4.9
displays t 0 as a function of Ω m .
The qualitative behavior of the cosmological models
is characterized by the density parameters Ω m and Ω Λ ,
whereas the Hubble constant H 0 determines “only” the
overall length- or time-scale. Today, mainly two families
of models are considered:
• Models without a cosmological constant, Λ = 0. The
difficulties in deriving a “sensible” value for Λ from
particle physics is often taken as an argument for neg-