Single Field Models 241calledkinationordeflation. Equation (11.7) then dictates that the equation of state is
푤휑=휑̇^2 + 2 푉(휑)
휑̇^2 − 2 푉(휑)
≈ 1 , (11.9)
so that the kinetic energy density decreases as
휌(푎)∝푎−^3 (^1 +푤)=푎−^6 (11.10)from Equation (5.29). This is much faster than the푎−^4 decrease of the radiation
energy density휌r,andthe푎−^3 decrease of the initially much smaller matter energy
density휌m. Consequently, kination ends at the moment when휌rovertakes휌kinat time
푡∗. When constructing phenomenological models for this scenario, one constraint is
of course that휌r(푡end)≪푉end, or equivalently,푡end<푡∗. This behavior is well illustrated
in Figure 11.3, taken from the work of Dimopoulos and Valle [9].
Since matter and radiation are gravitationally generated at푡end, the reheating tem-
perature of radiation is given by
푇reh=훼푇H, (11.11)
where푇His the Hawking temperature [Equation (3.35)], and훼is some reheating effi-
ciency factor less than unity. In Figure 11.3 the radiation energy density휌훾≡휌rstarts
at푇reh^4 ≪푉end, and then catches up with푉(휑)at휑∗. Now the Universe becomes radia-
tion dominated and the hot Big Bang commences. Note that the term ‘hot Big Bang’
has a different meaning here: it does not refer to a time zero with infinite temperature,
but to a moment of explosive entropy generation. This mimics the Big Bang so that
all of its associated successful predictions ensue.
Quintessence. The properties of the inflaton field plays no role any more during the
Big Bang, they are fixed by requirements at Planck time when the quintessence field
is completely negligible. The properties of quintessence, on the other hand, are fixed
by present observations. This makes their identification rather artificial.
Let us continue the argument of the previous paragraph. The kinetic energy den-
sity reduces rapidly to negligible,values by its푎−^6 dependence and the field freezes
ultimately to a nonzero value휑F. The residual inflaton potential푉(휑)again starts to
dominate over the kinetic energy density, however, staying far below the radiation
energy density and, after푡eq, also below the matter energy density.
As we approach푡 0 , the task of phenomenology is to devise a quintessence potential
having a suitable tracker. The nature of the tracker potential is decided by the form of
the quintessence potential. To arrive at the present-day dark energy which causes the
evolution to accelerate, the field휑must be unfrozen again, so휙Fshould not be very
different from휙 0. Many studies have concluded that only exponential trackers are
admissible, and that quintessence potentials can be constructed by functions which
behave as exponentials in휑early on, but which behave more like inverse power poten-
tials in the quintessential tail. A simple example of such a potential is
푉(휑≫휑end)≈푉endexp(−휆휑∕푚P)
(휑∕푚)푘