Single Field Models 245Solving the equations one finds that the solution is tracking, just like quintessence.
It is also quite insensitive to the initial conditions. The K-field tracks the radiation
energy density until푡eq, when a sharp transition from positive to negative pressure
occurs, with푤휑=−1 as a consequence. The K-essence density휌kthen drops below
휌mand, thereafter, in the matter-dominated epoch the K-field does not track the back-
ground at all, it just stays constant. Thus the time of K-essence domination and accel-
erated expansion simply depends on푡eq. However, this is another case of fine-tuning:
휌kmust drop precisely to the magnitude of the present-day휌휆.
Phantom Fields. Could the scalar field obey an equation of state with negative
kinetic energy violating the weak energy condition휌+푝⩾0?. This can occur in many
models but one is trying to avoid it at all costs. The reason for this is serious problems
with quantum instabilities. Since푤휑<−1, such a universe evolves within a finite time
to aBig Rip singularitywhen the curvature grows toward∞.
However, the singularity can be avoided if the potential has a maximum, for
instance
푉(휑)=푉 0
cosh(휑∕휑 0 ). (11.20)
In contrast to ordinary kinematics, the phantom field evolves toward the top of the
potential and crosses over on the other side. It then turns around and again returns to
the top, executing damped oscillations across the top until it finally settles at푤휑=−1.
This model is strongly disfavored by the data in comparison with genuine푤휑=− 1
models.
Tachyon Fields. Somewhat related are tachyon fields which move with superlu-
minal velocity. Since special relativity in four-dimensional spacetime forbids this,
tachyon models require rather drastic revisions of general relativity or of spacetime.
Speculations have also appeared in the literature that the Universe might have under-
gone shorter periods of this type. It is well to remember that nothing is known about
whether the cosmological constant is indeed constant or whether it will remain so,
nor about the future behavior of a quintessence field and its equation of state.
In higher-dimensional spacetimes tachyons may move in the bulk between our
four-dimensionalbraneand other branes. The potential may be of the form 11.20
so that the field has a ground state at푤휑=∞.
In a flat FRW background one has
휌휑=푉(휑)
√
1 −(휑̇^2 )
and 푝휑=−푉(휑)^2
휌휑
. (11.21)
From Friedmann’s equations one then obtains
푎̈
푎=^8 휋퐺
3
푉(휑)
√
1 −휑̇^2
( 1 − 2 휑̇^2 ). (11.22)
Hence the accelerated expansion occurs for휑̇^2 < 2 ∕3. Note that푤휑=휑̇^2 −1isinthe
range− 1 <푤휑<0.