Certainly the most basic of the cosmological parameters is the present
expansion rate, *H*_{o}, because this sets the scale of the
Universe.
Until a few years ago, there was a factor of two uncertainty in
*H*_{o}; with
two separate groups claiming two distinct values, one near
50 km s^{-1} Mpc^{-1} and the other nearer 100 km
s^{-1} Mpc^{-1}, and the errors quoted
by both groups were much smaller than this factor of two difference. This
points out a problem which is common in observational cosmology (or indeed,
astronomy in general). Often the indicated statistical errors give the
impression of great precision, whereas the true uncertainty is dominated
by poorly understood or unknown systematic effects. That was true in
the Hubble constant controversy, and there is no less reason to think
that this problem is absent in modern results. I will return to this point
several times below.

The great leap forward in determination of *H*_{o} came
with the Hubble Space Telescope
(HST) program on the distance scale. Here a particular kind of
variable stars - Cepheid variables - were observed in twenty nearby spiral
galaxies. Cepheids exhibit periodic variations in luminosity by a factor
of two on timescales of 2-40 days. There is a well-determined empirical
correlation between the period of Cepheids and their mean luminosity -
the longer the period the higher the luminosity. Of course, this period -
luminosity relation must be calibrated by observing Cepheids in some object
with a distance known by other techniques and this remains a source of
systematic uncertainty. But putting this problem aside, the Hubble
Space telescope measured the periods and the apparent magnitudes, without
confusion from adjacent bright stars, of a number of Cephieds in each of
these relatively
nearby galaxies, which yielded a distance determination (eq. 3.9).
These galaxies are generally too close (less than 15 Mpc) to sample
the pure Hubble flow - the Hubble flow on these scales is contaminated by
random motion of the galaxies and systematic cosmic flows -
but these determinations do permit a
calibration of other secondary distance indicators which reach further out,
such as supernovae type Ia (SNIa) and the Tully-Fisher relation (the
observed
tight correlation between the rotation velocities of a spiral galaxies and
their luminosities). After an enormous amount of work by a number of very
competent astronomers
[28],
the answer turned out to be *h* = 0.72 ± .10

As I mentioned there is the known systematic uncertainty of calibrating
the period-luminosity relation, but there are other possible systematic
effects that are less well-understood:
How can we be certain that the period-luminosity relation for Cepheids
is the same in all galaxies? For example, is this relation affected by
the concentration of elements heavier than helium (the
metallicity)? In view of such potential problems, other more direct
physical methods, which by-pass the traditional "distance ladder"
are of interest. Chief among these is the Sunyaev-Zeldovich (S-Z) effect
which is relevant to clusters of galaxies
[29].
The baryonic mass of
clusters of galaxies is primarily in the form of hot gas, which typically
exceeds the mass in the visible galaxies by more than a factor of two.
This gas has a temperature between 10^{7} and 10^{8} K
(i.e., the sound speed
is comparable to the one-dimensional velocity dispersion of the galaxies)
and is detected by satellite X-ray telescopes with detectors in the range
of several KeV. The S-Z effect is a small change in the intensity of
the CMB in the direction of such clusters due to Compton scattering of
CMB photons by thermal electrons (classical electron scattering would,
of course, produce no intensity change). Basically, CMB photons are
moved from the Rayleigh-Jeans part of the black body spectrum to the
Wien part, so the effect is observable as a spectral distortion of
the black body spectrum in the range of 100 to 300 GHz. It is a
small effect (on the order of 0.4 milli Kelvin) but still 5 to 10
times larger than the intrinsic anisotropies in the CMB.

By measuring the amplitude of the S-Z effect one determines an optical depth

(5.1) |

where is the frequency
dependent cross section, *l* is the path
length, and *n*_{e} is the electron density.
Because these same clusters emit X-rays via thermal bremsstrahlung, we may
also determine, from the observed X-ray intensity, an emission measure:

(5.2) |

Here we have two equations for two unknowns, *n*_{e} and
*l*. (This is simplifying the actual calculation because
*n*_{e} is a
function of radius in the cluster.) Knowing *l* and the angular
diameter of the cluster
we can then calculate the angular size
distance to the cluster via eq. 3.8. Hence, the Hubble parameter is given
by *H*_{o} = *v* / *D*_{A} where v is the
observed recession velocity of the cluster.
All of this assumes that the clusters have a spherical shape on average,
so the method needs to be applied to a number of clusters.
Even so biases are possible if clusters have more typically
a prolate shape or an oblate shape, or if the X-ray emitting gas is
clumpy. Overall, for a number of clusters
[30]
the answer turns out to be *h* = 0.6 - somewhat smaller than
the HST distance ladder method, but the systematic uncertainties remain
large.

A second direct method relies on time delays in gravitational lenses
[31].
Occasionally, a distant quasar (the source) is lensed by an intervening
galaxy (the lens) into multiple images; that is to say, we observe two
or more images of the same background object separated typically by
one or two seconds of arc. This means that there are
two or more distinct null geodesics connecting us to the quasar with two
or more different light travel times. Now a number of these quasars are
intrinsically variable over time scales of days or months (not periodic
but irregular variables). Therefore, in two distinct images
we should observe the flux variations track each other with a time
delay. This measured delay is proportional to the ratio
*D*_{l} *D*_{s} / *D*_{ls} where
these are the angular size distances to the lens,
the source, and the lens to the source. Since this ratio is proportional
to *H*_{o}^{-1}, the measured time delay, when
combined with a mass model for
the lens (the main source of uncertainty in the method), provides a
determination of the Hubble parameter. This method, applied to several
lenses [32,
33],
again tends to yield
a value of *h* that is somewhat smaller than the HST value, i.e.,
0.6. In a recent summary
[34]
it is claimed that, for four cases
where the lens is an isolated galaxy, the result is *h* = 0.48
± .03,
if the overall mass distribution in each case can be represented
by a singular isothermal sphere. On the other hand, in a well-observed
lens where the mass distribution is constrained by observations of
stellar velocity dispersion
[35],
the implied value of *h* is
0.75^{+.07}_{-.06}. Such supplementary observations are
important
because the essential uncertainty with this technique is in
the adopted mass model of the lens.

It is probably safe to say that
*h* 0.7, with an
uncertainty of
0.10 and perhaps a slight bias toward lower values, but the story is not
over as S-Z and gravitational lens determinations continue to improve.
This is of considerable interest because the best fit to the CMB
anisotropies observed by WMAP implies that
*h* = 0.72 ± .05 in perfect agreement
with the HST result. With the S-Z effect and lenses, there remains
the possibility of a contradiction.

With *h* = .70, we find a Hubble time of *t*_{H} = 14
Gyr. Now in FRW cosmology, the age of the Universe is
*t*_{o} = *f t*_{H} where *f* is a
number depending upon the cosmological model. For an Einstein-de Sitter
Universe (i.e.,
_{k} = 0,
_{Q} = 0,
_{m} = 1)
*f* = 2/3 which means that *t*_{o} = 9.1 Gyr. For an
empty negatively curved Universe, *f* = 1 which means that the age
is the Hubble time. Generally, models with a dominant vacuum energy density
(_{Q}
1,
*w* -1)
are older (*f* 1) and
for the concordance model, *f* = 0.94.
Therefore, independent determinations of the age of the Universe
are an important consistency test of the cosmology.

It is reasonable to expect that the Universe should be older than the
oldest stars it contains, so if we can measure the ages of the oldest
stars, we have, at least, a lower limit on the age of the Universe.
Globular star clusters are old stellar systems in the halo of our
own galaxy; these systems are distributed in a
roughly spherical region around the galactic disk and have low abundances
of heavy elements suggesting they were formed before most of the stars
in the disk. If one can measure the luminosity, *L*_{u}, of
the most luminous
un-evolved stars in a globular cluster (that is, stars still burning
hydrogen in their cores), then one may estimate the age. That is
because this luminosity is correlated with age: a higher
*L*_{u} means
a younger cluster. Up to five years ago, this method yielded globular
cluster ages of
*t*_{gc}
14 ± 2 Gyr, which, combined with the
Hubble parameter discussed above, would be in direct contradiction with
the Einstein-de Sitter
_{m} = 1
Universe. But about ten years ago
the Hipparchus satellite began to return accurate parallaxes for thousands
of relatively nearby stars which led to a recalibration of the
entire distance scale. Distances outside the solar system
increased by about 10% (in fact, the entire Universe suddenly
grew by this same factor leading to a decrease in the HST value for the
Hubble parameter). This meant that the globular
clusters were further away, that *L*_{u} was 20% larger,
and the clusters were correspondingly younger:
*t*_{gc}
11.5 ± 1.3 Gyr. If we assume that the Universe is
about 1 Gyr older than the globular clusters, then the age of the
Universe becomes 12.5 ± 2 Gyr
[36]
which is almost consistent
with the Einstein-de Sitter Universe. At least there is no longer any
compelling time scale argument for a non-zero vacuum energy density,
_{Q} >
0. The value of accurate basic astronomical
data (and what is more basic than stellar positions?)
should never be underestimated.

A second method for determining the ages of stars is familiar to all
physicists, and that is radioactive dating. This has been done recently
by observations of a U^{238} line in a metal-poor galactic star
(an old
star). Although the iron abundance in this star is only 1/800 that of
the sun, the abundances of a group of rare earth metals known as
r-process elements are
enhanced. The r-process is rapid neutron absorption onto iron nuclei
(rapid compared to the timescale for subsequent
decay) which
contributes to certain abundance peaks in the periodic table and which
occurs in explosive events like supernovae. This means that this old
star was formed from gas contaminated by an even older supernova event;
i.e. the uranium was deposited at a definite time in the past.
Now U^{238} is unstable with a half life of 4.5 Gyr which makes it
an ideal probe on cosmological times scales. All we have to do is
compare the observed abundance of *U*^{238} to that of a stable
r-process element (in this case osmium), with what is expected from
the r-process. The answer for the age of this star (or more accurately,
the SN which contaminated the gas out of which the star formed) is
12.5 ± 3 Gyr, which is completely consistent with the globular cluster
ages [37].

If we take 0.6 < *h* < 0.7, and
9.5 Gyr < *t*_{o} < 15.5 Gyr this implies
that 0.59 < *H*_{o} *t*_{o} < 1.1. This is
consistent with a wide range of
FRW cosmologies from Einstein-de Sitter to the concordance model.
That is to say, independent measurements of *H*_{o} and
*t*_{o} are not yet precise enough to stand as a
confirmation or contradiction to the WMAP result.