String Theory Demystified

CHAPTER 16 String Theory and Cosmology 271
Here, a(t) is a scale factor associated with the three expanding spatial dimensions
and b(t) is a scale factor associated with the contracting spatial dimensions. Solving
the Einstein equations in vacuum gives the following:
a
3 n 0
a
3
2 2 0 2
b
+ =
b
( )
a
k
+ ( H + nH ) H + =
a b a
a
a
b
+ + − +
b
− n 1
( 3H ( n 1)
H ) H k
a b b 2
b
Here we have introduced two Hubble constants. One is the usual Hubble constant
associated with our expanding universe H a :
( n)
= 0
a
Ha
a
= (16.11)
The second Hubble constant is associated with the contracting extra dimensions:
b
Hb
b
= (16.12)
The constants k (3) and k (n) are related to the local curvature and so can be +1, 0, −1.
We choose the locally fl at case and so set k (3) = k (n) = 0. This allows the equations to be
simplifi ed somewhat, giving three relations for the Hubble constants:
The ratio H b /H a can then be written as
2 nn ( −1)
2
H + nH H + H = 0
a a b b
6
H + ( 3H+ nH ) H = 0
a a b a
H
+ ( 3H + nH ) H = 0
H
H
b
a
b
a b b
⎛
2
3n± 3n + 6n⎞
=−⎜
⎟
⎝ nn ( −1)
⎠
(16.13)
The choice of sign corresponds to an accelerating or decelerating universe (for
the expanding extra dimensions). Of course, the choice of sign here is arbitrary, the
model doesn’t dictate why we would pick one sign or the other—it only describes
that an accelerating universe is possible. We take the + sign for the accelerating