Single Field Models 243We saw in Equation (5.33) that radiation energy density evolves as푎−^4 —faster than
matter energy density,푎−^3. Consequently,휌ris now much smaller than휌m.
But once푤휑is negative,휌휑decreases at a slower rate than휌mso that it eventu-
ally overtakes it. At that moment,휑(푡)slows to a near stop, causing푤휑to decrease
toward−1, and tracking stops. Judging from the observed large value of the cosmo-
logical constant density parameter today,훺휆= 0 .714, this happened in the recent past
when the redshift was푧∼2–4. Quintessence is already dominating the total energy
density, driving the Universe into a period of de Sitter-like accelerated expansion.
The tracker field should be an attractor in the sense that a very wide range of ini-
tial conditions for휑and휑̇ rapidly approach a common evolutionary track, so that
the cosmology is insensitive to the initial conditions. Such cosmological solutions,
calledscaling solutions, satisfy휌휑(푡)∕휌m(푡)=constant>0. Scaling solutions define the
borderline between deceleration and acceleration.
Thus the need for fine-tuning is entirely removed, the only arbitrariness remains in
the choice of a function푉(휑). With a judicious choice of parameters, the coincidence
problem can also be considered solved, albeit by tuning the parameters ad hoc.
In Chapter 7 we already discussed the inflationary de Sitter expansion following
theBigBang,whichmayalsobecausedbyascalarinflaton field.Herewejustnote
that the initial conditions for the quintessence field can be chosen, if one so desires,
to match the inflaton field.
Tracking behavior with푤휑<푤boccurs [10, 11] for any potential obeying
훤≡푉′′푉∕(푉′)^2 > 1 , (11.13)
and which is nearly constant over the range of plausible initial휑,
d(훤− 1 )
퐻d푡≪|훤− 1 |, (11.14)
or if−푉′∕푉is a slowly decreasing function of휑. Many potentials satisfy these criteria,
for instance power law, exponential times power law, hyperbolic, and Jacobian elliptic
functions. For a potential of the generic form,
푉(휑)=푉 0 (휑 0 ∕휑)−훽 (11.15)with훽constant, one has a good example of a tracker field for which the kinetic and
potential terms remain in a constant proportion.
The values of푤휑and훺휑depend both on푉(휑)and on the background. The effect
of the background is through the 3퐻̇휑term in the scalar field equation of motion
[Equation (7.24)] when푤changes,퐻also changes, which, in turn, changes the rate
at which the tracker field evolves down the potential.
The tracking potential is characterized asslow rollingwhen the slow-roll parame-
ters [already defined in Equation (7.34)]
휂(휑)≡
푚^2 Planck
16 휋(
푉′′
푉
)
≪ 1 ,휖≡
푚^2 Planck
16 휋(
푉′
푉
) 2
≪ 1 , (11.16)
meaning that휑̈in Equation (7.24) and휑̇^2 in Equation (11.5) are both negligible. At
very early times, however,−푉′∕푉is slowly changing, but is itself not small. This estab-
lishes the important distinction between static and quasi-static quintessence with