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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

Equation 1 (1)

Here t is the time variable, and r-theta-phi 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/sqrtk. 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, theta, phi) will remain there. Such coordinates are said to be comoving with the expansion. One swaps between physical (ie actual) and comoving distances via

Equation 2 (2)

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

Equation 3 (3)

Equation 4 (4)

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

Equation 5 (5)

Equation 6 (6)

where overdots are time derivatives and H = adot / a is the Hubble parameter. The terms in the fluid equation contributing to rhodot have a simple interpretation; the term 3Hrho is the reduction in density due to the increase in volume, and the term 3Hp 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

Equation 7 (7)

in which k does not appear explicitly.

1 I follow standard cosmological practice of setting the fundamental constants c and hbar 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 mPl rather than the gravitational constant G; with the convention just mentioned they are related by G ident mPl-2. Back.

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