Introduction to Cosmology

The Cosmological Constant 237

Explaining simultaneously all these and still other data, all consistent with an infla-
tionary cold dark matter model with a cosmological constant, requires an accelerat-
ing universe. Recent supernova data plotted in Figure 11.1 as퐻(푧)∕( 1 +푧)versus푧[5]
show that an early deceleration followed by a recent acceleration is favored, determin-
ing the time of transition from deceleration to acceleration to푧= 0. 74 ± 0 .05. Observa-
tions of baryonic acoustic oscillations [6] plotted in Figure 11.2 as퐻(푧)∕( 1 +푧)versus
푧determine this moment to have occurred at푧≈ 0 .8.

Dynamical Models. The cosmological constant corresponds to static gravity with
the equation of state푤=−1 as in Equation (5.28). If푤were a function of the scale푎
the expansion history would be different, but unfortunately all functions aread hoc.
Some simple one-parameter formulas are

푤(푧)=푤 0 +푤 1 푧, (11.1)

and
푤(푎)=푤 0 +푤 1 ( 1 −푎). (11.2)

One relatively successful two-parameter formula is

푤(푎)=푤 0 [ 1 +푏ln( 1 +푧)]−^2. (11.3)

The advantage of this formula is that it covers the whole observed range of푧.

90

80

70

60

50

40

30

0.0 0.5 1.0

Simon9

Stern2

Moresco8

Buscal

Zhang4

Blake3

Chuang1

z

H

(z

)/(1

+

z)

1.5 2.0

Figure 11.1Evidence for transition from deceleration in the past to acceleration today. From
reference [6]. From Farook, O. and Ratra, B., Hubble parameter measurement constraints
on the cosmological deceleration-acceleration transition redshift,Astrophys. J. Lett., 766 ,L7,
published 4 March 2013. © AAS. Reproduced with permission. (See plate section for color
version.)