Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 25

This last equation, which corresponds to the trace of the Einstein equations, is useful
for finding analytic solutions for a universe filled by dust and radiation.
In the case of radiation,p=ε/3, the expression on the right hand side of (1.71)
vanishes and the equation reduces to

a′′+ka= 0. (1.72)

This is easily integrated and the result is

a(η)=am·

⎧

⎨

⎩

sinhη, k=−1;

η, k=0;

sinη, k=+ 1.

(1.73)

Hereamis one constant of integration and the other has been fixed by requiring
a(η=0)=0. The physical timetis expressed in terms ofηby integrating the
relationdt=adη:

t=am·

⎧

⎨

⎩

(coshη−1), k=−1;

η^2 / 2 , k=0;

(1−cosη), k=+ 1.

(1.74)

It follows that in the most interesting case of a flat radiation-dominated universe,
the scale factor is proportional to the square root of the physical time,a∝

√

t, and
henceH= 1 / 2 t.Substituting this into (1.67), we obtain

εr=

3

32 πGt^2

∝a−^4. (1.75)

Alternatively, the energy conservation equation (1.64) for radiation takes the form

dεr=− 4 εrdlna, (1.76)

also implying thatεr∝a−^4.

Problem 1.15FindH(η) and (η) in open and closed radiation-dominated uni-
verses and express the current age of the universet 0 in terms ofH 0 and
0 .Analyze
the result for
0 1 and give its physical interpretation.

Problem 1.16For dust,p=0, the expression on the right hand side of (1.71) is
constant and solutions of this equation can easily be found. Verify that

a(η)=am·

⎧

⎨

⎩

(coshη−1), k=−1;

η^2 , k=0;

(1−cosη), k=+ 1.

(1.77)

For each case, computeH(η) and (η) and express the age of the universe in terms
ofH 0 and
0 .Show that in the limit
0 →0, we havet 0 = 1 /H 0 , in agreement