Physical Foundations of Cosmology

2.5 Kinematic tests 65

whereq 0 ≡−

(

a ̈/aH^2

)

0 .In turn, the value of the deceleration parameterq^0 is
determined by the equation of state. Using Friedmann equation (1.66), we obtain

q 0 =

1

2

(^0)

(

1 + 3

p

ε

)

0

. (2.80)

Thus, measuring the luminosity–redshift dependence for a set of standard candles,
we can, in principle, determine the effective equation of state for the dominant
matter components.
Measurements using Type IA supernovae as standard candles have produced a
spectacular result. The expansion of the universe has been found to be accelerating,
rather than decelerating. In other words,q 0 is negative. In a matter-dominated
universe, the gravitational self-attraction of matter resists the expansion and slows
it down. According to Friedmann equation (1.66), acceleration is possible only if
a substantial fraction of the total energy density is a “darkenergy” with negative
pressure or, equivalently, negative equation of statew≡p/ε.
One possibility is that the dark energy component is a vacuum energy density
or cosmological constant, which corresponds tow=−1. Alternatively, the dark
energy can be dynamical, such as a slightly time-varying scalar field. The latter
case is referred to as “quintessence.” The discovery of cosmic acceleration raises a
number of new problems in cosmology. At present, there is no convincing explana-
tion as to why dark energy came to dominate so late in the history of the universe
and exactly at the time to be observed. Additionally, because the nature of the dark
energy is uncertain, the long-term future of the universe cannot be determined.
If the dark energy is a cosmological constant, then the acceleration will continue
forever and the universe will become empty. On the other hand, if the dark energy
is a dynamical scalar field, then this field may decay, repopulating the universe
with matter and energy. In summary, dark energy is one of the most enigmatic and
challenging issues in cosmology today.
The supernovae that provide evidence for the dark energy component have red-
shifts of order unity and the expansion in (2.79), valid only forz< 0. 3 ,is not
applicable for them. Therefore, to describe the observations, we have to use the
exact formula (2.78) and choose a particular class of cosmological models in order
to compute (χem(z)). For example, for a flat universe comprising only cold matter

and a cosmological constant, so that
0 = (^) + (^) m= 1 ,we have
(χem(z))=χem(z)=

1

H 0 a 0

∫z

0

dz ̃

√

(^) m(1+z ̃)^3 +(1− (^) m)

. (2.81)

Calculating the integral numerically, we can findmbol(z)for different values of (^) m
(Figure 2.13).The best fit to the data is achieved for (^) m 0 .3.