Introduction to Cosmology

244 Dark Energy

푤휑≈−1 and dynamical quintessence with푤휑>−1. This means that the slow-roll

approximation is not necessarily applicable to dynamical quintessence, and that the

latter generally requires exact solution of the equation of motion [Equation (7.24)].

Given a potential like Equation (11.15) and fixing the current values of parameters

훺m,훺r,훺휑,푤휑one can solve the equation of motion [Equation (7.24)] by numerical

integration. Finding the functions훤(푎),푤휑(휑),푤휑(푎)or훺휑(푎)is a rather complicated

exercise. Finally, when the potential gets shallow, the relatively fast-rolling dynamical

quintessence solution exits from the scaling regime and becomes static, approaching

the푤휑=−1 regime, and an accelerated expansion commences.

The equation of state of dark energy,푤휑, introduces a new degeneracy with훺mand

ℎwhich cannot be resolved by CMB data alone. Using the full set of available data in

Figure 9.1, one can obtain limits to푤휑.

In this model the lookback time is given by

푡(푧)=^1

퐻 0 ∫

1 ∕( 1 +푧)

1

d푎

[

( 1 −훺 0 )+훺m푎−^1 +훺r푎−^2 +훺휑푎^2 −^3 (^1 +푤휑)

]− 1 ∕ 2

. (11.17)

For 1∕( 1 +푧)=0 this gives us the age [Equation (5.55)] of the Universe,푡 0.

For the case of− 1 <푤휑, the effect on galaxy formation is earlier collapse times

and more rapid collapse of overdensities. The equation of state and the amplitude

of velocity fluctuations휎 8 are correlated in such a way that for constant푤휑,higher

(lower)휎 8 produces earlier (later) collapse times, respectively.

K-Essence. As already mentioned, the weakness of the tracking quintessence model

is that the energy density for which the pressure becomes negative is set by an

adjustable parameter which has to be fine-tuned to explain the cosmic coincidence.

Surely one can do better by adding degrees of freedom, for instance by letting휑(푡)

interplay with the decaying cosmological constant휆(푡), or with the matter field, or

with a second real scalar field휓(푡)which dominates at a different time, or by taking

휑(푡)to be complex, or by choosing a double-exponential potential, or by adding a new

type of matter or a dissipative pressure to the background energy density. Surely the

present acceleration could have been preceded by various periods of deceleration and

acceleration. These are only a few of the alternatives that have been proposed.

One interesting alternative calledK-essence[12] comes at the cost of introducing a

nonlinear kinetic energy density functional of the scalar field. In Equation (5.86) we

introduced a Lagrangian which we now slightly generalize to read

phi,푋=

1

2

푔휇휈휕휇휙휕휈휙−푉(휙)≡푋−푉(휑). (11.18)

푋could be replaced by a a positive semi-definite kinetic function of퐾(푋).
If휑describes a perfect fluid with pressure푝and density휌kthe energy-momentum
tensor is given by푝=and
휌k=[ 2 푋퐾′(푋)−퐾(푋)] +푉(휑) (11.19)
where a prime denotes a partial derivative with respect to푋.Then푤휑≡푝∕휌. A nec-
essary condition that the theory be stable is휌>0 and further that the speed of sound
휕푝∕휕휌be positive.