The table below ^{(2)}
summarizes the current observational
information about the cosmological parameters, with estimated
1 errors. The quantities
in brackets have been deduced using at least some of the
CDM
assumptions. Is is apparent that there is
impressive agreement between the values of the parameters determined
by various methods, including those based on
CDM. In particular,
(A) several different approaches (some of which are discussed further
below) all suggest that
_{m}
0.3; (B) the location of
the first acoustic peak in the CMB angular anisotropy power spectrum,
now very well determined independently by the BOOMERANG
[17]
and MAXIMA1
[18]
balloon data
[19,
20] and by the
DASI interferometer at the South Pole
[21], implies that
_{m} +
_{}
1; and (C) the data
on supernovae of
Type Ia (SNIa) at redshifts *z* = 0.4 - 1.2 from two independent groups
imply that
_{}
- 4/3
1/3. Any two of
these three results then imply that
_{}
0.7. The
1 errors in these
determinations are about 0.1.

Questions have been raised about the reliability of the high-redshift
SNIa results, especially the possibilities that the SNIa properties at
high redshift might not be sufficiently similar to those nearby to use
them as standard candles, and that there might be "grey" dust (which
would make the SNIa dimmer but not change their colors). Although the
available evidence disfavors these possibilities,
^{(3)} additional observations
are needed on SNIa at high redshift, both to control systematic
effects and to see whether the dark energy is just a cosmological
constant or is perhaps instead changing with redshift as expected in
"quintessence" models
[3].
Such data could be
obtained by the proposed SuperNova Acceleration Probe (SNAP) satellite
[23],
whose Gigapixel optical camera and other instruments
would also produce much other useful data. But it is important to
appreciate that, independently of (C) SNIa, (A) cluster and other
evidence for
_{m}
0.3, together with
(B) ~ 1° CMB evidence for
_{m} +
_{}
1, imply that
_{}
0.7.

All methods for determining the Hubble parameter now give compatible
results, confirming our confidence that this crucial parameter has now
been measured robustly to a
1 accuracy of about 10%. The
final result [24]
from the Hubble Key Project on the
Extragalactic Distance Scale is
72& #177; 8 km s^{-1} Mpc^{-1}, or
*h* = 0.72 ± 0.08,
where the stated error is dominated by one systematic uncertainty, the
distance to the Large Magellanic Cloud (used to calibrate the Cepheid
period-luminosity relationship). The most accurate of the direct
methods for measuring distances *d* to distant objects, giving the
Hubble parameter directly as *H*_{0} = *d* /*v*
where the velocity is
determined by the redshift, are (1) time delays between luminosity
variations in different gravitationally lensed images of distant
quasars, giving
*h* 0.65, and
(2) the Sunyaev-Zel'dovich effect
(Compton scattering of the CMB by the hot electrons in clusters of
galaxies), giving
*h* 0.63
[25,
24].
For the rest of this article, I will take *h* = 0.7 whenever I need
to use an explicit value, and express results in terms of
*h*_{70}
*H*_{0}/70 km s^{-1} Mpc^{-1}.

For a CDM universe
with _{m} =
(0.2)0.3(0.4, 0.5), the expansion age is
*t*_{0} = (15.0)13.47(12.41, 11.61)
*h*_{70}^{-1} Gyr.
Thus for _{m}
0.3 - 0.4 and
*h* 0.7, there is
excellent agreement with the latest estimates of the ages of the
oldest globular cluster stars in the Milky Way, both from their Main
Sequence turnoff luminosities
[26],
giving 12 - 13 ± 2 Gyr,
and using the thorium and uranium radioactive decay
chronometers [27],
giving 14 ± 3 Gyr and 12.5 ± 3 Gyr,
respectively.

The simplest and clearest argument that
_{m}
1/3 comes
from comparing the baryon abundance in clusters
*f*_{b}
*M*_{b}/*M*_{tot} to that in the universe as a
whole _{b} /
_{m},
as emphasized by White et al.
[28].
Since clusters are
evidently formed from the gravitational collapse of a region of radius
~ 10 Mpc, they should represent a fair sample of both baryons and
dark matter. This is confirmed in CDM simulations
[29]. The
fair sample hypothesis implies that

(1) |

We can use this to determine
_{m} using
the baryon abundance
_{b}
*h*^{2} = 0.019 ± 0.0024 (95% C.L.) from the
measurement of the deuterium abundance in high-redshift Lyman limit
systems [30,
31].
Using X-ray data from an X-ray flux
limited sample of clusters to estimate the baryon fraction
*f*_{b} = 0.075*h*^{-3/2} gives
[32]
_{m} =
0.25*h*^{-1/2} = 0.3 ± 0.1 (using
*h* = 0.70 ± 0.08). Estimating the baryon fraction
using Sunyaev-Zel'dovich measurements of a sample of 18 clusters gives
*f*_{b} = 0.077*h*^{-1}
[25],
and implies
_{m} =
0.25*h*^{-1} = 0.36 ± 0.1.

There is another way to use clusters to measure
_{m}, which
takes advantage of the fact that the redshift at which structures form
depends strongly on
_{m}. This
happens because in a low-density
universe the growth rate of fluctuations slows when, on the right hand
side of the Friedmann equation,

(2) |

the first (matter) term becomes smaller than either the second
(curvature) term (for the case of an open universe) or the third
(cosmological constant) term. As I have already discussed, the
term appears to be dominant now; note that if we evaluate
the Friedmann equation at the present epoch and divide both sides by
*H*_{0}^{2}, the resulting equation is just

(3) |

Therefore, if we normalize the fluctuation power spectrum
*P*(*k*) for an
_{m} = 1
(Einstein-de Sitter) cosmology and for a
CDM one by
choosing _{8} so
that each is consistent with COBE and has the
same abundance of clusters today, then at higher redshifts the
low-_{m}
universe will have a higher comoving number density of
clusters. Probably the most reliable way of comparing clusters nearby
with those at higher redshift uses the cluster X-ray temperatures; the
latest results, comparing 14 clusters at an average redshift of 0.38
with 25 nearby clusters, give
_{m} = 0.44
± 0.12
[33,
34]. There is greater
leverage in this test if one can use higher redshift clusters, but the
challenge is to find large samples with well understood cluster
selection and properties. The largest such sample now available is
from the Las Companas Distant Cluster Survey, which goes to redshifts
~ 1, from which the preliminary result is
_{m} = 0.30
± 0.12 (90% CL)
[35].

^{2} Further discussion and references are given
in [16].
Back.

^{3} For
example, SNIa at *z* = 1.2 and ~ 1.7 apparently have the brightness
expected in a CDM
cosmology but are brighter than would be expected
with grey dust, and the infrared brightness of a nearer SNIa is also
inconsistent with grey dust
[22].
Back.