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The Hubble parameter H0 ident 100h km s-1 Mpc-1 remains uncertain, although no longer by the traditional factor of two. The range of h determinations has been shrinking with time [64]. De Vaucouleurs long contended that h approx 1. Sandage has long contended that h approx 0.5, although a recent reanalysis of the Type Ia supernovae (SNe Ia) data coauthored by Sandage and Tammann [111] concludes that the latest data are consistent with h = 0.6±0.04.

The Hubble parameter has been measured in two basic ways: (1) Measuring the distance to some nearby galaxies, typically by measuring the periods and luminosities of Cepheid variables in them; and then using these ``calibrator galaxies'' to set the zero point in any of the several methods of measuring the relative distances to galaxies. (2) Using fundamental physics to measure the distance to some distant object(s) directly, thereby avoiding at least some of the uncertainties of the cosmic distance ladder [109]. The difficulty with method (1) was that there was only a handful of calibrator galaxies close enough for Cepheids to be resolved in them. However, the HST Key Project on the Extragalactic Distance Scale has significantly increased the set of calibrator galaxies. The difficulty with method (2) is that in every case studied so far, some aspect of the observed system or the underlying physics remains somewhat uncertain. It is nevertheless remarkable that the results of several different methods of type (2) are rather similar, and indeed not very far from those of method (1). This gives reason to hope for convergence.

3.1. Relative Distance Methods

One piece of good news is that the several methods of measuring the relative distances to galaxies now mostly seem to be consistent with each other. These methods use either ``standard candles'' or empirical relations between two measurable properties of a galaxy, one distance-independent and the other distance-dependent. The favorite standard candle is SNe Ia, and observers are now in good agreement. Taking account of an empirical relationship between the SNe Ia light curve shape and maximum luminosity leads to h = 0.65±0.06 [103], h = 0.64+0.08-0.06 [61], or h = 0.63±0.03 [52, 93], and the slightly lower value mentioned above from the latest analysis coauthored by Sandage and Tammann agrees within the errors. The HST Key Project result using SNe Ia is h = 0.65±0.02±0.05, where the first error quoted is statistical and the second is systematic [50], and their Cepheid metallicity-dependent luminosity-period relationship [65] has been used (this lowers h by 4%). Some of the other relative distance methods are based on old stellar populations: the tip of the red giant branch (TRGB), the planetary nebula luminosity function (PNLF), the globular cluster luminosity function (GCLF), and the surface brightness fluctuation method (SBF). The HST Key Project result using these old star standard candles is [43] h = 0.66±0.04±0.06, including the Cepheid metallicity correction. The old favorite empirical relation used as a relative distance indicator is the Tully-Fisher relation between the rotation velocity and luminosity of spiral galaxies. The ``final'' value of the Hubble constant from the HST Key Project taking all of these into account, including the metallicity dependence of the Cepheid period-luminosity relation, is [45] h = 0.74±0.04±0.07, where the first error is statistical and the second is systematic. The largest source of systematic uncertainty is the distance to the LMC, which is here assumed to have a distance modulus of 18.45. This is a significantly higher h than their previous [85] h = 0.71±0.06, or h = 0.68±0.06 including the Cepheid metallicity dependence, using a LMC distance modulus of 18.5.

3.2. Fundamental Physics Approaches

The fundamental physics approaches involve either Type Ia or Type II supernovae, the Sunyaev-Zel'dovich (S-Z) effect, or gravitational lensing of quasars. All are promising, but in each case the relevant physics remains somewhat uncertain.

The 56Ni radioactivity method for determining H0 using Type Ia SNe avoids the uncertainties of the distance ladder by calculating the absolute luminosity of Type Ia supernovae from first principles using plausible but as yet unproved physical models for 56Ni production. The first result obtained was that h = 0.61±0.10 [3, 17]; however, another study [77] (cf. [126]) found that uncertainties in extinction (i.e., light absorption) toward each supernova increases the range of allowed h. Demanding that the 56Ni radioactivity method agree with an expanding photosphere approach leads to h = 0.60+0.14-0.11 [86]. The expanding photosphere method compares the expansion rate of the SN envelope measured by redshift with its size increase inferred from its temperature and magnitude. This approach was first applied to Type II SNe; the 1992 result h = 0.6±0.1 [114] was subsequently revised upward by the same authors to h = 0.73±0.06±0.07 [115]. However, there are various complications with the physics of the expanding envelope [110, 35].

The S-Z effect is the Compton scattering of microwave background photons from the hot electrons in a foreground galaxy cluster. This can be used to measure H0 since properties of the cluster gas measured via the S-Z effect and from X-ray observations have different dependences on H0. The result from the first cluster for which sufficiently detailed data was available, A665 (at z = 0.182), was h = (0.4 - 0.5)±0.12 [13]; combining this with data on A2218 (z = 0.171) raised this somewhat to h = 0.55±0.17 [12]. The history and more recent data have been reviewed by Birkinshaw [14], who concludes that the available data give a Hubble parameter h approx 0.6 with a scatter of about 0.2. But since the available measurements are not independent, it does not follow that h = 0.6±0.1; for example, there is a selection effect that biases low the h determined this way.

Several quasars have been observed to have multiple images separated by theta ~ a few arc seconds; this phenomenon is interpreted as arising from gravitational lensing of the source quasar by a galaxy along the line of sight (first suggested by [100]; reviewed in [129]). In the first such system discovered, QSO 0957+561 (z = 1.41), the time delay Deltat between arrival at the earth of variations in the quasar's luminosity in the two images has been measured to be, e.g., 409±23 days [89], although other authors found a value of 540±12 days [94]. The shorter Deltat has now been confirmed [72, 117]. Since Deltat approx theta2 H0-1, this observation allows an estimate of the Hubble parameter. The latest results for h from 0957+561, using all available data, are h = 0.64±0.13 (95% C.L.) [72], and h = 0.62±0.07 [39], where the error does not include systematic errors in the assumed form of the lensing mass distribution.

The first quadruple-image quasar system discovered was PG1115+080. Using a recent series of observations [113], the time delay between images B and C has been determined to be about 24±3 days. A simple model for the lensing galaxy and the nearby galaxies then leads to h = 0.42±0.06 [113], although higher values for h are obtained by more sophisticated analyses: h = 0.60±0.17 [63], h = 0.52±0.14 [73]. The results depend on how the lensing galaxy and those in the compact group of which it is a part are modelled.

Another quadruple-lens system, B1606+656, leads to h = 0.59±0.08±0.15, where the first error is the 95% C.L. statistical error, and the second is the estimated systematic uncertainty [41]. Time delays have also recently been determined for the Einstein ring system B0218+357, giving h = 0.69+0.13-0.19 (95% C.L.) [11].

Mainly because of the systematic uncertainties in modelling the mass distribution in the lensing systems, the uncertainty in the h determination by gravitational lens time delays remains rather large. But it is reassuring that this completely independent method gives results consistent with the other determinations.

3.3. Conclusions on H0

To summarize, relative distance methods favor a value h approx 0.6 - 0.8. Meanwhile the fundamental physics methods typically lead to h approx 0.4 - 0.7. Among fundamental physics approaches, there has been important recent progress in measuring h via the Sunyev-Zel'dovich effect and time delays between different images of gravitationally lensed quasars, although the uncertainties remain larger than via relative distance methods. For the rest of this review, we will adopt a value of h = 0.65±0.08. This corresponds to t0 = 6.52h-1 Gyr = 10±2 Gyr for Omegam = 1 - probably too low compared to the ages of the oldest globular clusters. But for Omegam = 0.2 and OmegaLambda = 0, or alternatively for Omegam = 0.4 and OmegaLambda = 0.6, t0 = 13±2 Gyr, in agreement with the globular cluster estimate of t0. This is one of the weakest of the several arguments for low Omegam, a non-zero cosmological constant, or both.

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