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5.1. Scalar fields and their potentials

In particle physics, a scalar field is used to represent spin zero particles. 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.

In particle theories, scalar fields are a crucial ingredient for spontaneous symmetry breaking. The most famous example is the Higgs field which breaks the electro-weak symmetry, whose existence is hoped to be verified at the Large Hadron Collider at CERN when it commences experiments next millennium. Scalar fields are also expected to be associated with the breaking of other symmetries, such as those of Grand Unified Theories, supersymmetry etc.

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, which I'll call phi. These are obtained by comparison of the energy-momentum tensor of the scalar field with that of a perfect fluid, and are

Equation 33 (33)
Equation 34 (34)

One can think of the first term in each as a kinetic energy, and the second as a potential energy. The potential energy V(phi) 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. 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 rho as the energy density can be divided between potential and kinetic energy in different ways.

In a given theory, there would be a specific form for the potential V(phi), at least up to some parameters which one could hope to measure (such as the effective mass and interaction strength of the scalar field). However, we are not presently in a position where there is a well established fundamental theory that one can use, so, in the absence of such a theory, inflation workers tend to regard V(phi) as a function to be chosen arbitrarily, with different choices corresponding to different models of inflation (of which there are many). Some example potentials are

Equation 35 (35)
Equation 36 (36)
Equation 37 (37)

The strength of this approach is that it seems possible to capture many of the crucial properties of inflation by looking at some simple potentials; one is looking for results which will still hold when more `realistic' potentials are chosen. Figure 3 shows such a generic potential, with the scalar field displaced from the minimum and trying to reach it.

Figure 3

Figure 3. A generic inflationary potential.

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