Introduction to Cosmology

238 Dark Energy

90

80

H

(z

)/(1

+

z) (km/sec/Mpc) 70

60

50

01

z

2

Figure 11.2 Measurements of 퐻(푧)∕( 1 +푧)versus푧demonstrating the acceleration of
the expansion for푧< 0 .8 and deceleration for푧> 0 .8. The BAO-based measurements from
different groups are the filled circles. The line is the훬CDM prediction for(ℎ, 훺푚,훺휆=
0. 7 , 0. 27 , 0. 73 ). For further details, see Buscaet al.[6]. Reproduced with permission © ESO.

Decaying Cosmological Constant. A dynamical approach to remove or alleviate
the extreme need for fine-tuning휆is to choose it to be a slowly varying function of
time,휆(푡). The initial conditions require휆(푡Planck)≈ 10122 휆 0 , from which it decays to
its present value at time푡 0.
The Universe is then treated as a fluid composed of dust and dark energy in which
thedarkenergydensity,휌휆(푡)=휆(푡)∕ 8 휋퐺, continuously transfers energy to the material
component. Its equation of state is then of the form

푝휆=−휌휆

[

1 +^1

3

dln휌휆(푎)

dln푎

]

. (11.4)

In the classical limit when휌휆(푎)is a very slow function of푎so that the derivative
term can be ignored, one obtains the equation of state of the cosmological constant,
푤휆=−1.
The advantage in removing the need for fine-tuning is, however, only replaced by
another arbitrariness: an ansatz for휆(푡)is required and new parameters characterizing
the timescale of the deflationary period and the transfer of energy from dark energy
to dust must be introduced. Such phenomenological models have been presented in
the literature [7, 8], and they can lead to testable predictions.

11.2 Single Field Models

Instead of arguing about whether휆should be interpreted as a correction to the
geometry or to the stress–energy tensor, we could go the whole way and postulate
the existence of a new kind of energy, described by a slowly evolving scalar field휑(푡)
that contributes to the total energy density together with the background (matter and
radiation) energy density. This scalar field is assumed to interact only with gravity and