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4.4. Inflation from scalar fields

We have already seen that a cosmological constant due to nonzero vacuum energy results in accelerating cosmological expansion. While this is a good candidate for explaining the observations of Type Ia supernovae, it does not work for explaining inflation at early times for the simple reason that any period of accelerated expansion in the very early universe must end. Therefore the vacuum energy driving inflation must be dynamic. To implement a time-dependent "cosmological constant," we require a field with the same quantum numbers as vacuum, i.e. a scalar. We will consider a scalar field minimally coupled to gravity, with potential V(phi) and Lagrangian

Equation 79 (79)

where we have modified a familiar Minkowski-space field theory by replacing the Minkowski metric etaµnu with the FRW metric gµnu. The equation of motion for the field is

Equation 80 (80)

This is the familiar equation for a free scalar field with an extra piece, 3Hphi dot, that comes from the use of the FRW metric in the Lagrangian. We will be interested in the ground state of the field p = 0. This is of interest because the zero mode of the field forms a perfect fluid, with energy density

Equation 81 (81)

and pressure

Equation 82 (82)

Note in particular that in the limit phi dot rightarrow 0 we recover a cosmological constant, p = - rho, as long as the potential V(phi) is nonzero. The Friedmann equation for the dynamics of the cosmology is

Equation 83 (83)

(Note that we have written Newtons' constant G in terms of the Planck mass, so that G = mPl-2 in units where hbar = c = 1.) In the phi dot rightarrow 0 limit, we have

Equation 84 (84)

so that the universe expands exponentially,

Equation 85 (85)

This can be generalized to a time-dependent field and a quasi-exponential expansion in a straightforward way. If we have a slowly varying field (1/2)phi dot2 << V(phi), we can write the equation of motion of the field as

Equation 86 (86)

and the Friedmann equation as

Equation 87 (87)

so that the scale factor evolves as

Equation 88 (88)

This is known as the slow roll approximation, and corresponds physically to the field evolution being dominated by the "friction" term 3Hphi dot in the equation of motion. This will be the case if the potential is sufficiently flat, V'(phi) << V(phi). It is possible to write the equation of state of the field in the slow roll approximation as

Equation 89 (89)

where the slow roll parameter epsilon is given by

Equation 90 (90)

This parameterization is convenient because the condition for accelerating expansion is epsilon < 1:

Equation 91 (91)

Specifying a model for inflation is then as simple as selecting a potential V(phi) and evaluating its behavior as a source of cosmological energy density. Many models have been proposed: Refs. [27, 28] contain extensive reviews of inflationary model building. We will discuss the observational predictions of various models in Section 4.8 below. In the next section, we will discuss one of the central observational predictions of inflation: the generation of primordial density fluctuations.

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