**1.2. Overview of Relevant Cosmological Equations**

**1.2.1. The Robertson-Walker Metric**

To place the Hot Big Bang model into a physical context necessitates a sensible mathematical formulation. To assist with this formulation we assume that the universe on a large scale is both homogeneous and isotropic. This assumption is known as the Cosmological Principle and the observed isotropy of the expansion and the CMB are strong testaments to its validity. If we accept this principle to be valid, then our task is to construct a geometrical model of the Universe that explicitly incorporates large scale homogeneity and isotropy. Ideally, this model should be described by a relatively small number of parameters, all of which can be observationally determined. Much of this book is devoted to a modern discussion of attempts to determine these parameters from observations. However, before doing that we must describe the framework that allows observations to be directly connected to our cosmological model.

To begin with, we note that General Relativity is a geometrical
theory concerning the overall curvature of space-time. Within
that context we seek to specify the coordinate properties of
a homogeneous, isotropic, expanding Universe.
If we are to fully describe the Universe in geometrical
terms, we must derive a metric which describes the coordinate paths
that objects are allowed to take. In deriving this metric
we must introduce the concept of an **event**.
An event is something which occurs at a certain place
at a certain time. Hence all events in the universe
can be thought of as occurring in a four-dimensional
**spacetime** continuum, with three spatial dimensions
and one dimension of time.
To compute the separation between any two events in spacetime,
it is necessary to specify the **metric**.
As a simple example, consider the surface of a sphere, which can be
thought of as a two-dimensional
analogue to the four-dimensional spacetime.
Using simple spherical trigonometry, the metric of a sphere
can be written as

(10) |

where *ds* denotes the distance between two points
on the surface of the sphere,
*R* is the radius of the sphere, and
*d* and
*d* are the difference in
latitude and longitude between the two
points (measured in radians).
With this expression, it is possible to compute
the separation between any two points along the surface
of the sphere.
Hence the geometry of the sphere and the physical specification
of events is completely
described by its metric.

The geometry of four-dimensional spacetime
is described by an analogous metric.
However, instead of computing the distance between two points
on the surface of a sphere, we wish to compute the
separation between two events, which involves
both space and time.
Special relativity allows one to show that
the spacetime interval, *ds*, between two events
which occur near each other in flat space
is given by

(11) |

where *dt* is the time interval between the two events
(as determined by an inertial observer),
*c* is the speed of light,
and *dx, dy, dz* correspond to the separation
between the two events in each of the three spatial
dimensions.
Note that, unlike the metric for an ordinary sphere,
the spacetime metric need not always be positive.
The geometry of spacetime is completely specified by
equation 11.
A **geodesic** is the shortest interval between
any two points in spacetime.

Equation 11 assumes a flat Euclidean geometry, in which initially parallel lines always remain parallel. However, according to Einstein's theory of General Relativity, spacetime is curved by gravity, which is a manifestation of the energy density of matter. The separation between two events will therefore depend on the curvature of spacetime. This is schematically shown in Figure 1-6 for three specific curvatures.

For a homogeneous and isotropic universe,
the most general metric in curved spacetime is the
**Robertson-Walker metric**, which was
first proposed in 1934.
Expressed in spherical polar
coordinates (*r*, ,
), this
metric takes the form

(12) |

where *R(t)* is the **universal scale factor**
which describes the time evolution of the
universal expansion,
*k* denotes the curvature of spacetime
(*k* = -1, 0, 1 for negative, zero,
or positive curvature),
and the coordinate *r* is **comoving** with the
universal expansion. If we imagine a particle
at rest with a given set of coordinates *r*,
,
, then as long as no external
forces operate on this particle. then the particle remains at those
coordinates. These coordinates are said to be
**comoving** coordinates. They are related to
physical coordinates through the scale factor,

(13) |

The Robertson-walker metric is independent of any
particular gravitational theory.
Gravity enters through the scale factor *R(t)*
and the curvature constant *k*;
the distance between any two spacetime events
therefore depends on what specific cosmological
model is adopted. This specific cosmological model is
determined by the values of *R(t)* and *k* which can be
observationally determined.