**4.1. H_{0}, q_{0} and the Age of the
Universe**

The quest for understanding the geometry of our universe has been one
of the central aims of cosmology since the 1960s
and Alan Sandage in 1970 even described the whole of observational
cosmology as being a "search for two numbers". The first of these
numbers - the Hubble parameter
*H*_{0} = ( /
*a*)_{0}, provides us with measure of the observable
size of the universe and its age. The second *q*_{0} =
-*H*_{0}^{-2}( / *a*)_{0}
is called the deceleration parameter and
probes the equation of state of matter and the cosmological density
parameter. In the presence of a cosmological constant,

(13) |

In a critical density universe with
_{m} +
_{} = 1, the
deceleration parameter

(14) |

consequently a critical density universe will accelerate if
_{m} < 2/3.
The observational quest for *q*_{0} showed that
evolutionary effects play
a dominant role in this important quantity and for a while it was felt that
it may be virtually impossible to disentangle the true cosmological
`signal' for *q*_{0} from evolutionary `noise'. Recent
years however have witnessed an important turnaround with the
development of new and more powerful techniques
which are either less sensitive to evolutionary effects
or for which evolutionary effects are better understood.

In the next section, we shall consider several promising cosmological tests which could shed light on the composition of the universe and its geometrical properties. These tests include gravitational lensing, the use of high redshift supernovae as calibrated standard candles, and the angular size-redshift relation. Before we do that however, we shall turn our attention to another fundamental quantity which has traditionally played an important role in constraining cosmological models - the age-redshift relation.

The presence of a
-term leads to an
increase in the age of the universe
with far-reaching observational consequences. To appreciate this let us
first consider the critical density Einstein-de Sitter universe with
*a*
*t*^{2/3}, so that

(14) |

The value of *H*_{0} therefore serves to determine the age
uniquely in a spatially
flat matter dominated universe. Moderately high values
*H*_{0} 75 km
s^{-1} Mpc^{-1}
result in an age for the universe which is smaller than the ages of the
oldest globular clusters making an Einstein-de Sitter universe with a high
value of *H*_{0} difficult to reconcile with
observations. The situation can
be remedied if we live in an open universe. Assuming for simplicity that
the universe is empty (a good approximation if
_{m}
0.2) we get
*a* *t* so
that

(16) |

Combining (15) and (16) we get
2/3 *H*_{0}^{-1}
*t*_{0}
*H*_{0}^{-1}
for matter dominated cosmological models
with _{m}
1.
(A longer age *t*_{0} > *H*_{0}^{-1}
can be achieved in the presence of a cosmological constant.)
An open universe though older, nevertheless has two difficulties
associated with it: the first is related to the growth of density
perturbations which slow down considerably in an open universe leading
to large primordial fluctuations in
the Cosmic Microwave Background which may be difficult to reconcile
with observations (assuming standard adiabatic fluctuations with
scale-invariant initial spectra).
The second is related to the Omega problem:
a low Omega universe requires extreme fine tuning of
initial conditions, which some find to be an unattractive feature of
open/closed models.

Let us now consider a more general situation in which the universe has a -term in addition to normal matter. A closed form expression for the age of the universe in spatially flat models is given by [117]

(17) |

where _{}
= /
3*H*_{0}^{2} = 1 -
_{m}.

In Figure 3
we show the present age of a universe consisting of matter and
a cosmological constant and parametrized in terms of the variables
_{} and
_{m}.
We find that the age of a flat universe with
_{} = 1 -
_{m} is
always greater than that of an open universe for identical values of
1 - _{m}.
Additionally *t*_{0} can exceed
*H*_{0}^{-1} if
_{}
0.74.

No exact forms for *t*(*H*) are available for a time dependent
-term.
To study this and other cases, it is useful to express the Hubble parameter
as a function of the cosmological redshift *z*. This can easily be
done for a general
multicomponent universe consisting of several non-interacting matter
species characterized by equations of state
*P*_{}
= *w*_{}
_{}, for
which the Hubble parameter can be written as

(18) |

where _{total} =
_{},
_{} = 1 +
3*w*_{}
and 1 + *z* = *a*_{0} / *a*(*t*)
is the cosmological redshift parameter.

Let us assume that the universe, in addition to matter and radiation,
consists of a decaying
-term modelled by
a fluid with equation of state *P*_{X} = (*m*/3 -
1)_{X}
so that =
_{0}(*a*_{0}/*a*)^{m},
*m* 2.
The dimensionless Hubble parameter *h*(*z*) then becomes

(19) |

where *m* = 0 corresponds to a cosmological *constant*,
and we neglect the presence of radiation. In a spatially flat universe
_{total}
= _{} +
_{m} = 1
(the present value of
is therefore given by
_{0} =
3*H*_{0}^{2}[1 -
_{m}]).
A useful relationship between the cosmological time parameter *t*
and the cosmological redshift *z* can be obtained by differentiating
1 + *z* = *a*_{0} / *a*(*t*) with respect to
time, so that *dz* / *dt* = - *H*(*z*)(1 + *z*).
This leads to the following completely general
expression for the age of the universe at a redshift *z*

(20) |

with *h*(*z*) supplied by either (18) or (19).

A running debate over the previous decade or so has centered around whether
or not the universe has an `age problem', *i.e.* on
whether matter dominated cosmological models
are substantially younger than their oldest constituents (which happen
to be metal poor old globular cluster stars).
A key role in this controversy is played by
the Hubble parameter, whose present value is known to within an
uncertainty of
about two. Higher values of *H*_{0} clearly give rise to a
younger universe whereas lower values lead to an older one.

At the time of writing lower values *H*_{0}
65 km
s^{-1} Mpc^{-1} are
strongly supported by observations, especially in the light of
new parallax measurements made by the *Hipparcos* satellite for
Cepheid
stars, which has led to a reanalysis of distances to globular clusters and
consequently of their
age estimates ^{(3)}
which have dropped to 11.5 ± 1.5 Gyr
[30,
120].
(Low values of *H*_{0} are also suggested from an analysis
of the Sunyaev-Zeldovich effect from X-ray emitting clusters
[99],
from Type 1a supernovae
[165,
158]
and from Cepheids observed by the HST.)
Lower values of *H*_{0} reconcile
matter dominated flat models with the revised ages of globular clusters
[120]
and with limits from nucleochronology which indicate
*t*_{0} 7.8 Gyr
[30].
Low values of *H*_{0} combined with an
absence of stellar systems with ages greatly exceeding 20 Gyr also argue
against large values of
,
since from Fig 3 we see that
_{}
0.85 suggests an
age *t*_{0}
24 Gyr
(if *H*_{0} = 50km/sec/Mpc).
Finally the recent supernovae based measurements of Perlmutter et
al. (1998b) suggest a best-fit age of the universe
*t*_{0}
14.9 (63/*H*_{0}) Gyr for a spatially flat universe with
_{m}
0.28 and
_{}
0.72.

The above arguments were largely limited to the *present*
age of the universe. Ages of high redshift objects at *z* > 1
provide crucial information about the age of the universe at that redshift
[110].
The existence of at least two high redshift galaxies having
an evolved stellar population and hence an old age sets very severe
constraints on a flat matter dominated universe
[49,
119,
50,
152]. For
instance the radio
galaxy 53W091 at *z* = 1.55 discovered by Dunlop et al. (1996) is
reported to
be at least 3.5 Gyrs old. The age of a spatially flat matter dominated
universe at a redshift *z* is easily obtained from (20) to be

(21) |

Consequently the discovery of 53W091 can be accommodated within an
_{m} = 1,
CDM model only if the Hubble parameter is uncomfortably small
[119]
*H*_{0}
45 km s^{-1} Mpc^{-1}. However both open and flat
-dominated models
alleviate the age problem for 53W091.

At even higher redshifts, recent work
[212]
aimed at age-dating a high redshift QSO at
*z* = 3.62 using delayed iron enrichment by Type Ia supernovae as a
cosmic clock, sets a lower bound of 1.3 Gyr on the age of the universe
at that redshift. This discovery can be accommodated within
a spatially flat cosmology only if
_{m} +
_{} = 1 (low density
open models with
_{m}
0.2 are also permitted).
However, the age dating of stellar populations requires complex modelling
and although both open and flat
-dominated models
are clearly
favoured by current observations, more work needs to be done before
matter dominated flat models are excluded on the basis of age arguments
alone. ^{(4)}

^{3} An important indicator of the absolute
age of a globular cluster star is its luminosity when it leaves the main
sequence. Since luminosity is related to
distance (to the star) ages of globular clusters are very sensitive to
distance callibrators.
Back.

^{4} It may be
appropriate to mention that models with a cosmological constant
may *never be singular* and therefore could possess an infinite
age as demonstrated by the `bouncing models' in
Fig. 1.
However the value of the cosmological constant in such models
is several orders of magnitude larger than permitted by current
observations. The relevance of such models is therefore likely to be
limited to the very early universe and will not affect the
age problem discussed here.
Back.