**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*() and
Lagrangian

(79) |

where we have modified a familiar Minkowski-space field theory by
replacing the Minkowski metric
^{µ} with the FRW metric
*g*^{µ}. The
equation of motion for the field is

(80) |

This is the familiar equation for a free scalar field with an extra piece,
3*H*,
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

(81) |

and pressure

(82) |

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

(83) |

(Note that we have written Newtons' constant *G* in terms of the
Planck mass, so that
*G* = *m*_{Pl}^{-2} in units where
= *c* = 1.)
In the
0 limit, we have

(84) |

so that the universe expands exponentially,

(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)^{2} <<
*V*(),
we can write the equation of motion of the field as

(86) |

and the Friedmann equation as

(87) |

so that the scale factor evolves as

(88) |

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

(89) |

where the *slow roll parameter*
is given by

(90) |

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

(91) |

Specifying a model for inflation is then as simple as selecting a
potential
*V*()
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.