String Theory Demystified

270 String Theory Demystifi ed
Interestingly, the dilaton fi eld introduces a type of duality into the model. In fact,
this duality is related to T-duality, because it relates large and small distances. Given
a set of Kasner exponents p j and a dilaton fi eld f, there exists a dual solution with
D
∑
p′ =− p φ′ = φ−2p ln t
(16.9)
j j j
j=
1
Notice that since p′ =−p
j j, expanding dimensions in the theory are the contracting
dimensions in the dual theory and vice versa.
Pre-big-bang cosmology can be described in terms of this duality. It allows for
the universe to go through the following stages of evolution:
• It starts out in a large, fl at, and cold state.
• It contracts to a self-dual point. The universe enters a state where it is small,
highly curved, and very hot. This is the “big bang.”
• It enters an expansion phase which is the universe we live in.
This was the fi rst attempt at a cosmological model using string theory. However,
it has since been discarded in favor of brane-based cosmological models. This is
because several problems with the model could not be resolved, and brane models
of the universe are compelling because of how the fi elds of the standard model and
gravity are described. Before going on to brane-world cosmology though, let’s see
how the Kasner metric can describe an accelerating universe.
An interesting effect that can arise when considering some spatial dimensions
contracting and others expanding is that the contracting dimensions actually cause
the expanding dimensions to accelerate. 1 Suppose that we have n > 1 contracting
dimensions with three expanding spatial dimensions. It can be shown that they
cause the three spatial dimensions not only to expand, but to do so in an infl ationary
manner without a cosmological constant.
We write the number of space-time dimensions as D = n + 4, where we
understand that the n dimensions which contract are all spatial and the remaining
dimensions are 3 + 1 dimensional space-time. The metric can be written in a
general form which is split between time, the expanding dimensions, and the
contracting dimensions as
3
D
2 2 2 ⎛ 2⎞
ds =− dt + a t dx b t dx
i
⎝
⎜
⎠
⎟ +
−1
⎛
() ∑ () ∑ ⎝
⎜
2 2
m
i=
1
m=
4
1 Levin, Janna, “Infl ation from Extra Dimensions,” Phys. Lett. vol. B343, 1995, 69–75.
⎞
⎠
⎟
(16.10)