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

(1)

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

(2)

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

(3)

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

(4)

(5)

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

(6)

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 c² 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. Back.