Introduction to Cosmology

242 Dark Energy

V

T^4 reh

Vend

• 4/3


• 2

inflation kination HBB

ρkin

ργ

φend φ* φF

ρ

In ρ

–√2/3 λ

φ ≡ √3/2 φ/mp

Figure 11.3Schematic view of the scalar potential from inflation to quintessence. The poten-
tial푉(solid line) features two flat regions, the inflationary plateau and the quintessential tail.
The inflation is terminated at휑endby a drastic reduction of푉, leading to a rapid roll-down
of the scalar field from the inflationary plateau towards the quintessential tail. At the end of
inflation the kinetic energy density of the scalar field,휌kin(dash-dotted line), dominates for a
brief period the energy density of the Universe. During this time the radiation energy density
휌훾(dashed line) reduces less rapidly and catches up with휌kinat time푡∗when the field is휑∗,
and the explosive generation of entropy commences. After that the kinetic energy of the scalar
field reduces rapidly to zero and the field freezes asymptotically to a value휑F, while the overall
energy density of the Universe (dash-dot-dotted line) continues to decrease due to the Hubble
expansion. Assuming a quasi-exponential tail given by Equation (11.12), the potential beyond
휑Fis seen departing logarithmically from a pure exponential case (dotted line). Reprinted from
K.DimopoulosandJ.W.F.Valle,Modeling quintessential inflation[9], copyright 2002, with
permission from Elsevier.

where푘⩾1 is an integer,휆>0 is a parameter characterizing the exponential scale,
and푚<푚Pis a mass scale characteristic of the adopted inverse power-law scale. For
more details, see Dimopoulos and Valle [9].

Tracking Quintessence. In a somewhat less arbitrary model [10, 11], one constructs
quintessence in such a way that its energy density is smaller than the background
component for most of the history of the Universe, somehow tracking it with the same
time dependence. This field converges onto the tracker solution with nearly constant
slope−dln휌휑∕dln푎at a value<dln휌푚∕dln푎, and regardless of the value of푤휑.inthe
radiation-domination epoch,푤휑automatically decreases to a negative value at time
푡eqwhen the Universe transforms from radiation domination to matter domination.