**2.2. Equations of motion**

The hot big bang theory is based on the *cosmological principle*, which
states that the Universe should look the same to all observers. That tells
us that the Universe must be homogeneous and isotropic, which in turn tells
us which metric must be used to describe it. It is the Robertson-Walker
metric

Here *t* is the time variable, and *r*-- are (polar)
coordinates. The constant *k* measures the spatial curvature, with
*k*
negative, zero and positive corresponding to open, flat and closed
Universes respectively. If *k* is zero or negative, then the range of
*r* is from zero to infinity and the Universe is infinite, while
if *k*
is positive then *r* goes from zero to
1/*k*. Usually the
coordinates are rescaled to make *k* equal to -1, 0 or +1. The
quantity *a*(*t*) is the scale-factor of the Universe, which
measures its
physical size. The form of *a*(*t*) depends on the properties of the
material within the Universe, as we'll see.

If no external forces are acting, then a particle at rest at a given set of
coordinates
(*r*, ,
) will remain there. Such
coordinates are said
to be *comoving* with the expansion. One swaps between physical (ie
actual) and comoving distances via

The expansion of the Universe is governed by the properties of material
within it. This can be specified
^{(1)} by the energy density
(*t*) and the pressure *p*(*t*). These are often related by an equation of
state, which gives *p* as a function of
; the classic examples are

In general though there need not be a simple equation of state; for example there may be more than one type of material, such as a combination of radiation and non-relativistic matter, and certain types of material, such as a scalar field (a type of material we'll encounter later which is crucial for modelling inflation), cannot be described by an equation of state at all.

The crucial equations describing the expansion of the Universe are

where overdots are time derivatives and
*H* = / *a* is the Hubble
parameter. The terms in the fluid equation contributing to
have
a simple interpretation; the term
3*H* is the
reduction in density due
to the increase in volume, and the term 3*Hp* is the reduction in energy
caused by the thermodynamic work done by the pressure when this expansion
occurs.

These can also be combined to form a new equation

in which *k* does not appear explicitly.

^{1} I follow standard cosmological
practice of setting the fundamental constants *c* and
equal to one.
This makes the energy density and mass density interchangeable (since the
former is *c*^{2} times the latter). I shall also normally
use the Planck mass
*m*_{Pl} rather than the gravitational constant *G*;
with the convention just
mentioned they are related by
*G*
*m*_{Pl}^{-2}. Back.