Physical Foundations of Cosmology

36 Kinematics and dynamics of an expanding universe

whereHis defined in (1.103). Analyze and discuss the behavior of this solution
in the limitst→0 andt→∞.Derive the corresponding solutions fork=± 1.
(HintUse (1.71), replacing the conformal time with the physical time.)

Problem 1.21Show that the solution of (1.67) and (1.68) for a flat universe with
cold matter (dust) and cosmological constant is

a(t)=a 0

(

sinh

3

2

Ht

) 2 / 3

. (1.108)

Verify that in this case the age of the universe is given by

t 0 =

2

3 H 0

1

√

1 − (^) m
ln

1 +

√

1 − (^) m
√
(^) m

, (1.109)

whereH 0 is the current value of the Hubble constant and (^) mis the cold matter
contribution to the cosmological parameter today.
Problem 1.22Given a nonvanishing cosmological constant, find the static solution
for a closed universe filled with cold matter (Einsteins’s universe). Why is this
solution unstable?
Problem 1.23Find the solutions for an energy component with equation of state
p=−ε/3 in the presence of a cosmological term. Discuss the properties of these
solutions.