Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 13

The sign ofqdetermines whether the expansion is slowing down or speeding up.
Find a general expression forqin terms of and verify thatq= 1 /2inaflat
dust-dominated universe.

To conclude this section we derive an explicit solution for the scale factor in a
flat matter-dominated universe. BecauseE=0, (1.17) can be rewritten as

a·a ̇^2 =

4

9

(

da^3 /^2

dt

) 2

=const, (1.23)

and, hence, its solution is

a∝t^2 /^3. (1.24)

For the Hubble parameter, we obtain

H=

2

3 t

. (1.25)

Thus, the current age of a flat (E=0) dust-dominated universe is

t 0 =

2

3 H 0

, (1.26)

whereH 0 is the present value of the Hubble parameter. We see that the result is not
very different from the rough estimate obtained by neglecting gravity. The energy
density of matter as a function of cosmic time can be found by substituting the
Hubble parameter (1.25) into (1.18):

ε(t)=

1

6 πGt^2

. (1.27)

Problem 1.6Estimate the energy density att= 10 −^43 s,1 s and 1 year after the
big bang.

Problem 1.7Solve (1.18) in the limitt→∞for an open universe and discuss the
properties of the solution.

1.3 From Newtonian to relativistic cosmology

General Relativity leads to a mathematically consistent theory of the universe,
whereas Newtonian theory does not. For example, we pointed out that the
Newtonian picture of an expanding, dust-filled universe relies on Birkhoff’s theo-
rem, which is proven in General Relativity. In addition, General Relativity intro-
duces key changes to the Newtonian description. First, Einstein’s theory proposes
that geometry is dynamical and is determined by the matter composition of the
universe. Second, General Relativity can describe matter moving with relativistic