Physical Foundations of Cosmology

228 Inflation I: homogeneous limit

causally disconnected regions further complicates the horizon problem. Assuming
that it has, nevertheless, been achieved, we can ask how accurately the initial Hubble
velocities have to be chosen for a given matter distribution.
Let us consider a large spherically symmetric cloud of matter and compare its
total energy with the kinetic energy due to Hubble expansion,Ek. The total energy
is the sum of the positive kinetic energy and the negative potential energy of the
gravitational self-interaction,Ep. It is conserved:

Etot=Eik+Eip=E 0 k+E 0 p.

Because the kinetic energy is proportional to the velocity squared,

Eik=Ek 0 (a ̇i/a ̇ 0 )^2

and we have

Eitot

Eik

=

Eki+Eip

Eki

=

E 0 k+E 0 p

Ek 0

(

a ̇ 0

a ̇i

) 2

. (5.5)

SinceEk 0 ∼

∣∣

E 0 p

∣∣

anda ̇ 0 /a ̇i≤ 10 −^28 ,wefind

Eitot

Eik

≤ 10 −^56. (5.6)

This means that for a given energy density distribution the initial Hubble velocities
must be adjusted so that the huge negative gravitational energy of the matter is
compensated by a huge positive kinetic energy to an unprecedented accuracy of
10 −^54 %. An error in the initial velocities exceeding 10−^54 % has a dramatic conse-
quence: the universe either recollapses or becomes “empty” too early. To stress the
unnaturalness of this requirement one speaks of the initial velocities problem.

Problem 5.1How can the above consideration be made rigorous using the Birkhoff
theorem?

In General Relativity the problem described can be reformulated in terms of the
cosmological parameter (t)introduced in (1.21). Using the definition of (t)we
can rewrite Friedmann equation (1.67) as

(t)− 1 =

k

(Ha)^2

, (5.7)

and hence

(^) i− 1 =(
0 − 1 )
(Ha)^20
(Ha)^2 i

=(

0 − 1 )

(

a ̇ 0

a ̇i

) 2

≤ 10 −^56. (5.8)