String Theory Demystified

CHAPTER 16 String Theory and Cosmology 273
where b is a constant of integration and
n n n
q =
nn
+ +
2
3 6
( + 3)
This solution gives a Kasner type metric. Explicitly, we have
−2
p
3
2 2 ⎛ t ⎞ ⎛ ⎞ 2 ⎛ t ⎞
ds =− dt + a 1−dx b 1
i
⎝
⎜
t ⎠
⎟
⎝
⎜∑
⎠
⎟ + −
⎝
⎜
t ⎠
⎟
i=
1
2q
⎛
⎝
⎜
D−1
∑
m=
4
dx
2
m
⎞
⎠
⎟
(16.19)
(16.20)
The Randall-Sundrum Model
The approaches described in the previous section are no longer considered tenable.
The current line of research into cosmology from a string/M-theory perspective was
launched with a brane-based approach called the Randall-Sundrum model. 2 This
model is not a string/M-theory approach per se. Instead, it is simply a model which
invokes the existence of extra dimensions and the existence of branes. Moreover, the
model was not developed for the purposes of cosmology. The model was put forward
as a possible solution to the hierarchy problem of particle physics. To review, the
hierarchy problem is the fact that there is an enormous energy gap between the natural
or fundamental energy scales of gravity and the electroweak theory. The electroweak
scale is on the order of just 100 GeV, while the gravitational scale is on the order of
a whopping 10 18 GeV. The beauty of the Randall-Sundrum model is that it solves the
hierarchy problem with a simple model based on branes and higher-dimensional
space-time. We discuss the Randall-Sundrum model because the basic idea, two 3branes
connected along an extra spatial dimension, was the starting point for an idea
of how to approach big-bang cosmology in string theory.
Now let’s describe the basics of the model, which will form the basis of
cosmological models more directly connected to string theory. We consider a fi vedimensional
space-time with two branes called the visible brane (our universe) and
the hidden brane. The branes form boundaries to a fi ve-dimensional region called
the bulk. The branes have the usual 3 + 1 dimensional space-time. Gauge interactions
are restricted to the brane, while gravity can propagate along the extra dimension
and hence into the bulk, as well as in the branes.
We denote the extra spatial dimension by y and refer to the other space-time
coordinates as x µ . The fi ve-dimensional metric is denoted by g AB. The two branes
2 First proposed by Lisa Randall and Raman Sundrum in “A Large Mass Hierarchy from a Small Extra
Dimension”, Phys.Rev.Lett. 83 (1999):3370–3373. Available on the arXiv at http://lanl.arxiv.org/abs/
hep-ph/9905221.