**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*. Many authors
rescale the coordinates 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, 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
/ *a* is the Hubble
parameter. In this equation
is the cosmological constant;
astronomers' convention is to write this as a separate term, though
physicists would typically be happier to consider it as part of the
density .

These can also be combined to give

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}^{-2}
rather than the gravitational
constant *G*; with the convention just mentioned they are related by
*G*
*m*_{pl}^{-2}.
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