2.4. The Value of H₀ and the Age of the Universe

The immense effort that has been put towards the goal of determining H₀, 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 H₀. Note that there are 3 different measurements based on SNIa, giving different values of H₀. 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 H₀'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.

Table 1. Some recent determinations of the Hubble constant, based on different methods.

Method	N	H0	<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 H₀.

A crude n-weighted average of the different H₀-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 (H₀ 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 H₀-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 H₀ 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 t₀, 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 t₀ 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 t₀ 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.