Introduction to Cosmology

Single Field Models 239

with itself. In Section 7.2 we have already discussed this situation in the context of
single-field inflation.
Since a scalar field is mathematically equivalent to a fluid with a time-dependent
speed of sound, one can find potentials푉(휑)for which the dynamical vacuum acts like
a fluid with negative pressure, and with an energy density behaving like a decaying
cosmological constant. In comparison with plain휆(푡)models, scalar field cosmologies
have one extra degree of freedom, since both a potential푉(휑)and a kinetic term^12 휑̇^2
need to be determined.
The simplest example of a minimally coupled and spatially homogeneous scalar
field has the Lagrangian density 5.86 and the equation of motion is then given by the
Klein–GordonEquation 7.24, where the prime indicates derivation with respect to휑.
The energy density and pressure enter in the diagonal elements of푇휇휈,andtheyare

휌휑푐^2 =

1

2

휑̇^2 +푉(휑) and 푝휑=

1

2

휑̇^2 −푉(휑), (11.5)

respectively. Clearly the pressure is always negative if the evolution is so slow that
the kinetic energy density^12 휑̇^2 is less than the potential energy density. Note that in
Equations (7.24) and (11.5) we have ignored terms describing spatial inhomogeneity
which could also have been present.
The conservation of energy-momentum for the scalar field is as in Equation (5.24),

휌̇휑+ 3 퐻휌휑( 1 +푤휑)= 0. (11.6)

As in Equation (5.29), the energy density of the scalar field decreases as푎−^3 (^1 +푤휑).
Inserting Equations (11.5) into Equation (11.6), one indeed obtains Equation (7.24).
The equation of state of the휑field is then a function of the cosmological scale푎(or
time푡or redshift푧),

푤휑=

휑̇^2 + 2 푉(휑)

휑̇^2 − 2 푉(휑)

. (11.7)

Starting early from a wide range of arbitrary initial conditions푤휑oscillates between
≈1and≈−1, until an epoch when휌휑freezes to a small value.
The conditions for acceleration are

푤휑<− 1 ∕ 3 ,푎(푡)∝푡푑with d> 1

so that

푝휑<0or휌휑∝푎−^2.

However, dark energy defined this way and calledquintessenceturns out to be
anotherDeus ex machinawhich not only depends on the parametrization of an arbi-
trary function푉(휑), but also has to be fine-tuned initially in a way similar to the cos-
mological constant.

The Inflaton as Quintessence. Now we have met two cases of scalar fields causing
expansion: the inflaton field acting before푡GUTand the quintessence field describing
present-day dark energy. It would seem economical if one and the same scalar field