3.2. Scalar fields and their potentials

In order to obtain the required equation of state, a suitable material must come to dominate the density of the Universe. Such a material is a scalar field, which in particle physics is used to represent spin-zero particles and which we represent by throughout. It transforms as a scalar (that is, it is unchanged) under coordinate transformations. In a homogeneous Universe, the scalar field is a function of time alone.

At present no fundamental scalar field has been observed, but they proliferate in modern particle physics theories. In particular, supersymmetry associates a boson with every fermion (and vice versa), giving a multitude of scalar fields in any supersymmetric theory containing the Standard Model of particle physics.

The traditional starting point for particle physics models is the action, which is an integral of the Lagrange density over space and time and from which the equations of motion can be obtained. As an intermediate step, one might write down the energy-momentum tensor, which sits on the right-hand side of Einstein's equations. Rather than begin there, I will take as my starting point expressions for the effective energy density and pressure of a homogeneous scalar field. These are obtained by comparison of the energy-momentum tensor of the scalar field with that of a perfect fluid, and are

(21)

(22)

One can think of the first term in each as a kinetic energy, and the second as a potential energy. The potential energy V() can be thought of as a form of `configurational' or `binding' energy; it measures how much internal energy is associated with a particular field value (including the mass-energy of the particle number density it represents). Normally, like all systems, scalar fields try to minimize this energy; however, a crucial ingredient which allows inflation is that scalar fields are not always very efficient at reaching this minimum energy state.

Note in passing that a scalar field cannot in general be described by an equation of state; there is no unique value of p that can be associated with a given as the energy density can be divided between potential and kinetic energy in different ways. This is not particularly significant for early Universe inflation, but will be later when we discuss the present Universe.