**2.4. The Value of H_{0} and the Age of the
Universe**

The immense effort
that has been put towards the goal of determining *H*_{0},
with an
accuracy of a few %, has recently come to fruition with different
methods providing consistent (within < 10%) values. In
Table 1, I
present a list of some of the most recent determinations of
*H*_{0}. Note
that there are 3 different measurements based on SNIa, giving
different values of *H*_{0}. Although many of the SNIa they
use are
common, the difference is most probably attributable to the different
local calibrations that they employ. Thus, the differences in their
derived *H*_{0}'s should reflect the *systematic*
uncertainty introduced
by the different local calibrations and it is indeed comparable
to the systematic uncertainty that the individual studies have
estimated.

Method | N | H_{0} |
<z> | reference |

Cepheid | 23 | 75±10 | 0.006 | Freedman et al 2001 |

2-ary methods | 77 | 72±8.0 | 0.1 | Freedman et al 2001 |

IR SBF | 16 | 76±6.2 | 0.020 | Jensen et al. 2000 |

SN Ia | 36 | 73±7.3 | 0.1 | Gibson & Brook 2001 |

SN Ia | 35 | 59±6 | 0.1 | Parodi et al 2000 |

SN Ia | 46 | 64±7 | 0.1 | Jua et al 1999 |

CO-line T-F | 36 | 60±10 | < 0.11 | Tutui et al 2001 |

S-Z | 7 | 66±15 | < 0.1 | Mason et al 2001 |

Grav. Lens | 5 | 68±13 | Koopmans & Fassnacht 2000 | |

The method in the last row of Table 1 is based on
*gravitational lensing*
(cf. [20]).
The basic principle behind this method is that there is a
difference in the light travel time along two distinct rays from a source,
which has been gravitationally lensed by some intervening mass. The
relative time delay
(*t*), between
two images of the source, can be
measured if the source is variable. Then it can be shown that the Hubble
constant is just:

(58) |

where
is the image separation
and
is a constant that
depends on
the lens model. Although, this method has well understood physical
principles, still the details of the lensing model provide quite large
uncertainties in the derived *H*_{0}.

A crude *n*-weighted average of the different
*H*_{0}-determinations in Table 1 gives:

where the uncertainty reflects that of the weighted mean (the individual
uncertainties have not been taken into account). However, there seems to
be some clustering around two preferred values (*H*_{0}
60 and
72 km
s^{-1} Mpc^{-1}) and thus the above averaging provides
biased results. More
appropriate is to quote the median value and the 95% confidence limits:

(59) |

The anisotropic errors reflect the non-Gaussian nature of the distribution
of the derived *H*_{0}-values. Note that the largest part
of the individual
uncertainties, presented in Table 1, of all methods except the last two,
are *systematic* because they rely on local calibrators (like the
distance to
the LMC), which then implies that a systematic offset of the local
zero-point will "perpetuate" to the secondary indicators although
internally they may be self-consistent. A further source of systematic
errors is the peculiar velocity model, used to correct the derived
distances, which can easily introduce ~ 7% shifts in the derived
*H*_{0}
values [146],
[190].

With the value (59), we obtain a *Hubble* time,
*t*_{H}, equal to:

(60) |

the uncertainties reflecting the 95% confidence interval.

It is trivial to state that the present age of the Universe
*t*_{0}, should
always be larger than the age of any extragalactic object. A well known
problem that has troubled cosmologists, is the fact that the predicted age
of the Universe, in the classical Einstein de-Sitter Cosmological
Model is smaller than the measured age of the oldest globular
clusters of the Galaxy. This can be clearly seen from (60) and

(61) |

and although the latest estimates of the globular cluster ages have been drastically decreased to [84]:

One should then add the age of the formation of the globular clusters
and assuming a redshift of formation *z*
5 then this age is ~
0.6 -0.8 Gyr's which brings the lower 95% limit of *t*_{gc}
to ~ 11.6 Gyr's (see however
[30]
for possible formation at *z*
10). It is
evident that there is
a discrepancy between *t*_{0} and
*t*_{gc}. This discrepancy could however be
bridged if one is willing to push in the right direction the 95% limit of
both *t*_{H} and *t*_{gc}.

However, other lines of research point towards the age-problem. For
example, if at some large redshift we observe galaxies with old stellar
populations, for which we know the necessary time for evolution to
their locally "present"state, then we can deduce again the age
of the Universe. In an EdS we have *R*
*t*^{2/3} and thus we have:

(62) |

Galaxies have been found at *z*
3 with spectra that
correspond to a
stellar component as old as ~ 1.5 Gyr's, in their local rest-frame. From
(62) we then have that *t*_{0}
12 Gyr's, in
disagreement with the EdS age (61).

This controversy could be solved in a number of ways, some of which are:

- invoking an open ( < 1) cosmological model,
- assuming that we live in a local underdensity of an EdS
Universe,
- invoking a flat model with
_{}> 0.

The first possibility is in contradiction with many observational data and most importantly with the recent CMB experiments (BOOMERANG, MAXIMA and DASI), which show that = 1 (see [44], [43], [90], [164] [130]).

The second possibility [173] can solve the age-problem by assuming that we live in a local underdense region that extends to quite a large distance, which would then imply that the measured local Hubble constant is an overestimate of the global one by a factor:

where the bias factor *b*, is the ratio of the fluctuations in
galaxies and
mass. To reduce the Hubble constant to a comfortable value to solve the
age problem, say from 72 to 50 km sec^{-1} Mpc^{-1}, one
then needs
*N / N*
-0.9 / *b*. Values
of *b* are highly uncertain and model dependent,
but most recent studies point to *b* ~ 1 (cf.
[88])
which would then mean
that we need to live in a local very underdense region, something
that is not supported by the linearity of the Hubble relation out
to *z*
0.03 (cf. [64])
or out to *z*
0.1 (cf.
[171]).
This is not to
say that we are not possibly located in an underdense region,
but rather that this cannot be the sole cause of the *age*-problem
[193].

Thus we are left with the last possibility of a Universe dominated by
vacuum energy (a Universe with
_{} > 0 - see
section 1.3). If we live in
the accelerated phase (see Fig.1)
we will measure:

ie.,
a larger Hubble constant, and thus smaller Hubble time as we progress in
time, resolving the age-problem. In fact we have strong indications (see
next section), from the SNIa results, which trace the Hubble relation
at very large distances (see section 3.2),
and from the combined
analysis of CMB anisotropy and galaxy clustering measurements
in the 2dF galaxy redshift survey
[54],
for a flat Universe with
_{}
0.7. Then from (23) we obtain the age of the Universe in such a model:

Indeed the resolution of the *age*-problem gives further support to the
_{} > 0 paradigm.