Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 27

Problem 1.17Verify that, for a nonflat universe with a mixture of matter and
radiation, one has

a(η)=am·

{

(ηsinhη+coshη−1), k=−1;

(ηsinη+ 1 −cosη), k=+ 1.

(1.83)

Problem 1.18Consider a closed universe filled with matter whose equation of
state isw=p/ε, wherewis constant. Verify that the scale factor is then

a(η)=am

(

sin

(

1 + 3 w

2

η+C

)) 2 /(1+ 3 w)

, (1.84)

whereCis a constant of integration. Analyze the behavior of the scale factor for
w=− 1 ,− 1 / 2 ,− 1 / 3 ,0 and+ 1 /3. Find the corresponding solutions for flat and
open universes.

1.3.5 Milne universe

Let us consider an open universe withk=−1 in the limit of vanishing energy
density,ε→ 0 .In this case, (1.67) simplifies to

a ̇^2 = 1

and has a solution,a=t.The metric then takes the form

ds^2 =dt^2 −t^2 (dχ^2 +sinh^2 χd 

2 ), (1.85)

and describes a spacetime known as a Milne universe. One might naturally expect
that the solution of the Einstein equations for an isotropic space without matter
must be Minkowski spacetime. Indeed, the Milne universe is simply a piece of
Minkowski spacetime described in expanding coordinates. To prove this, we begin
with the Minkowski metric,

ds^2 =dτ^2 −dr^2 −r^2 d 

2. (1.86)

Replacing the Minkowski coordinatesτandrby the new coordinatestandχ,
defined via

τ=tcoshχ, r=tsinhχ, (1.87)

we find that

dτ^2 −dr^2 =dt^2 −t^2 dχ^2 ,