String Theory Demystified

266 String Theory Demystifi ed
Einstein’s Equations
Infl ation
In the previous chapter we introduced the Einstein fi eld equations, which give a
classical description of gravity. In this chapter, we discuss the application of the
Einstein fi eld equations to cosmology. For a detailed description of the study of
cosmology in the context of general relativity, please see Relativity Demystifi ed and
any of the references contained therein.
Cosmology is the study of the evolution of the universe as a whole. The starting
point is the Robertson-Walker metric:
2 2 2 2
ds =− dt + a () t dΣ
(16.1)
Here, dΣ 2 represents the spatial part of the metric. The function at () is called the
scale factor. It characterizes the spatial size of the universe and how it changes with
time. The Hubble constant is given by
H a
=
a
(16.2)
We can characterize the spatial structure of the universe by a curvature constant
K. If the space is fl at, has negative curvature (a saddle) or has positive curvature
(a sphere), then K = 0, − 1, + 1,
respectively. Observational evidence indicates that
our universe is fl at.
The behavior of the universe with time is determined by starting with a given
metric believed to describe the overall structure of the universe, and then using it to
work out the components of the curvature tensor. Then we can solve the Einstein
fi eld equations either with or without matter present. This can also be done with or
without a cosmological constant.
In standard cosmology treatments, space is assumed to be isotropic, meaning
that it is the same in all directions. We may not want to make that assumption in
string theory where some spatial dimensions are treated differently.
There are two cosmological models that come up rather repeatedly. A de Sitter
universe is one without matter (a vacuum solution of Einstein’s fi eld equations),
with fl at space, and a positive cosmological constant. An anti-de Sitter universe
(sometimes denoted AdS) is a vacuum solution to the Einstein fi eld equations with
positive cosmological constant and negative scalar curvature.
The cosmological models studied in relativity theory are only a part of modern
cosmology. The second piece which is needed to explain known data is infl ation.
The standard big-bang model begins the universe with a singularity and it expands