240 Dark Energy
could do both jobs. Then the inflaton field and quintessence would have to be matched
at some time later than푡GUT. This seems quite feasible since, on the one hand, the
initially dominating inflaton potential푉(휑)must give way to the background energy
density휌r+휌mas the Universe cools, and on the other hand, the dark energy density
must have been much smaller than the background energy density until recently.
Recall that quintessence models are constructed to be quite insensitive to the initial
conditions.
On the other hand, nothing forces the identification of the inflaton and quintessence
fields. The inflationary paradigm in no way needs nor predicts quintessence.
In the previously described models of inflation, the inflaton field휑settled to oscil-
late around the minimum푉(휑= 0 )at the end of inflation. Now we want the inflaton
energy density to continue a monotonic roll-down toward zero, turning ultimately into
a minute but nonvanishing quintessence tail. The global minimum of the potential is
only reached in a distant future,푉(휑→∞)→0. In this process the inflaton does not
decay into a thermal bath of ordinary matter and radiation because it does not inter-
act with particles at all, it is said to be sterile. A sterile inflaton field avoids violation
of the equivalence principle, otherwise the interaction of the ultralight quintessence
field would correspond to a new long-range force. Entropy in the matter fields comes
from gravitational generation at the end of inflation rather than from decay of the
inflaton field.
Thetaskisthentofindapotential푉(휑)such that it has two phases of accelerated
expansion: from푡Pto푡endat the end of inflation, and from a time푡F≈푡GUTwhen the
instanton field freezes to a constant value until now,푡 0. Moreover, the inflaton energy
density must decrease faster than the background energy density, equalling it at some
time푡∗when the field is휑∗, and thereafter remaining subdominant to the energy den-
sity of the particles produced at푡end. Finally it must catch up with a tracking potential
at some time during matter domination,푡>푡eq.
The mathematical form of candidate potentials is of course very complicated, and
it would not be very useful to give many examples here. However, it is instructive to
follow through the physics requirements on휑and푉(휑)from inflation to present.
Kination. Inflation is caused by an essentially constant potential푉(휑)according to
Equation (7.36). The condition푉(휑→∞)→0 requires an end to inflation at some
finite time푡endwhen the field is휑endand the potential is푉end≡푉(휑end). The change
in the potential at푡endfrom a constant to a decreasing roll then implies, by Equa-
tion (7.37), that휑̇end≠0, and furthermore, by Equation (7.24), that also휑̈end≠0. Then
the slow-roll conditions in Equation (7.32) for휖and휂are also violated.
During inflation the kinetic energy density of the inflaton is
휌kin=휖푉=푚^2 Planck
16 휋[
푉′^2 (휑)
푉(휑)
]
. (11.8)
Thus when푉′(휑)starts to grow, so does휌kin. and the total energy density of the Uni-
verse becomes dominated by the inflaton kinetic energy density. This epoch has been