**B. The tests**

The literature on the cosmological tests is enormous compared to what it was just a decade ago, and growing. Our references to this literature are much sparser than in Sec. III, on the principle that no matter how complete the list it will be out of date by the time this review is published. For the same reason, we do not attempt to present the best values of the cosmological parameters based on their joint fit to the full suite of present measurements. The situation will continue to evolve as the measurements improve, and the state of the art is best followed on astro-ph. We do take it to be our assignment to consider what the tests are testing, and to assess the directions the results seem to be leading us. The latter causes us to return many times to two results that seem secure because they are so well checked by independent lines of evidence, as follows.

First, at the present state of the tests, optically selected galaxies are useful mass tracers. By that we mean the assumption that visible galaxies trace mass does not seriously degrade the accuracy of analyses of the observations. This will change as the measurements improve, of course, but the case is good enough now that we suspect the evidence will continue to be that optically selected galaxies are good indicators of where most of the mass is at the present epoch. Second, the mass density in matter is significantly less than the critical Einstein-de Sitter value. The case is compelling because it is supported by so many different lines of evidence (as summarized in Sec. IV.C). Each could be compromised by systematic error, to be sure, but it seems quite unlikely the evidence could be so consistent yet misleading. A judgement of the range of likely values of the mass density is more difficult. Our estimate, based on the measurements we most trust, is

(59) |

and we would put the central value at
_{M0}
0.25. The
spread is meant in the sense of two standard deviations: we would
be surprised to find
_{M0} is
outside this range.

Several other policy decisions should be noted. The first is that we do not comment on tests that have been considered but not yet applied in a substantial campaign of measurements. A widely discussed example is the Alcock and Paczynski (1979) comparison of the apparent depth and width of a system from its angular size and depth in redshift.

In analyses of the tests of models for evolving dark energy
density, simplicity recommends the XCDM parametrization with a
single constant parameter *w*_{X}, as is demonstrated by
the large number of recent papers on this approach. But the more
complete physics recommends the scalar field model with an
inverse power-law potential. This includes the response of the
spatial distribution of the dark energy to the peculiar
gravitational field. Thus our comments on variable dark energy
density are more heavily weighted to the scalar field model than
is the case in the recent literature.

The gravitational deflection of light appears not only as a
tool in cosmological tests, as gravitational lensing, but also as
a source of systematic error. The gravitational
deflections caused by mass concentrations magnify the image of a
galaxy along a line of sight where the mass density is larger than
the average, and reduce the solid angle of the image when the mass
density along the line of sight is low.
The observed energy flux density is proportional to the solid
angle (because the surface brightness, erg cm^{-2} s^{-1}
ster^{-1} Hz^{-1}, is conserved at fixed redshift).
Selection can be biased either way, by the magnification effect
or by obscuration by the dust that tends to accompany
mass. ^{70}
When the tests are more precise we will have to correct them for
these biases, through models for the mass distribution (as in
Premadi et al., 2001),
and the measurements of the associated gravitational shear of
the shapes of the galaxy images. But the biases seem to be
small and will not be discussed here.

And finally, as the cosmological tests improve a satisfactory
application will require a joint fit of all of the parameters to
all of the relevant measurements and constraints.
Until recently it made sense to
impose prior conditions, most famously the hope that if the
universe is not well described by the Einstein-de Sitter model
then surely it is the case either that
is negligibly
small or else that space curvature may be neglected. We
suspect the majority in the community still expect this is true,
on the basis of the coincidences argument in
Sec. II.B.2, but it
will be important to see what comes out of joint fits of both
_{M0} and
_{0},
as well as all the other parameters, as is
becoming the current practice. Our test-by-test discussion is
useful for sorting out the physics and astronomy, we believe; it
is not the prototype for the coming generations of precision
application of the tests.

Our remarks are ordered by our estimates of the model dependence.

**1. The thermal cosmic microwave background
radiation**

We are in a sea of radiation with spectrum
very close to Planck at *T* = 2.73 K, and isotropic to one part in
10^{5} (after correction for a dipole term that usually is
interpreted as the result of our motion relative to the rest
frame defined by the radiation).
^{71}
The thermal spectrum indicates thermal relaxation, for which the
optical depth has to be large on the scale of the Hubble length
*H*_{0}^{-1}. We know space now is close to
transparent at the wavelengths of this
radiation, because radio galaxies are observed at high redshift.
Thus the universe has to have expanded from a state quite
different from now, when it was hotter, denser, and optically
thick. This is strong evidence our universe is evolving.

This interpretation depends on, and checks, conventional local
physics with a single metric description of spacetime. Under
these assumptions the expansion of the universe preserves the
thermal spectrum and cools the temperature as
^{72}

(60) |

Bahcall and Wolf (1968) point out that one can test this temperature-redshift relation by measurements of the excitation temperatures of fine-structure absorption line systems in gas clouds along quasar lines of sight. The corrections for excitations by collisions and the local radiation field are subtle, however, and perhaps not yet fully sorted out (as discussed by Molaro et al., 2002, and references therein).

The 3 K thermal cosmic background radiation is a centerpiece of modern cosmology, but its existence does not test general relativity.

The best evidence that the expansion and cooling of the universe
traces back to high redshift is the success of the standard model
for the origin of deuterium and isotopes of helium and lithium, by
reactions among radiation, leptons, and atomic nuclei as the
universe expands and cools through temperature *T* ~ 1 MeV at
redshift *z* ~ 10^{10}. The free parameter in the
standard model is the present baryon number density. The model
assumes the baryons are uniformly distributed at high redshift,
so this parameter with the known present radiation temperature
fixes the baryon number density as a function of temperature and
the temperature as a function of time. The latter follows from
the expansion rate Eq. (11), which at the epoch of
light element formation may be written as

(61) |

where the mass density
_{r}
counts radiation, which is now at *T* = 2.73 K, the associated
neutrinos, and *e*^{±} pairs. The curvature and
terms
are unimportant, unless the dark energy mass density varies
quite rapidly.

Independent analyses of the fit to the measured element abundances, corrected for synthesis and destruction in stars, by Burles, Nollett, and Turner (2001), and Cyburt, Fields, and Olive (2001), indicate

(62) |

both at 95% confidence limits. Other analyses by
Coc et al. (2002) and
Thuan and Izotov (2002)
result in ranges
that lie between the two of Eq. (62). The difference in
values may be a useful indication of remaining uncertainties; it
is mostly a consequence of the choice of isotopes used to constrain
_{B0}
*h*^{2}.
Burles et al. (2001)
use the deuterium abundance,
Cyburt et al. (2001)
favor the helium and lithium measurements,
and the other two groups use other combinations of abundances.
Equation. (62) is consistent with the summary range,
0.0095
_{B0}
*h*^{2} 0.023
at 95% confidence, of
Fields and Sarkar
(2002).

The baryons observed at low redshift, in stars and gas, amount to (Fukugita, Hogan, and Peebles, 1998)

(63) |

It is plausible that the difference between Eqs. (62)
and (63) is in cool plasma, with temperature *T* ~ 100 eV,
in groups of galaxies. It is difficult to observationally constrain the
idea that there is a good deal more cool plasma in the large voids
between the concentrations of galaxies. A more indirect but eventually
more precise constraint on
_{B0}, from
the anisotropy of the 3 K thermal
cosmic microwave background radiation, is discussed in test (11).

It is easy to imagine complications, such as inhomogeneous entropy per baryon, or in the physics of neutrinos; examples may be traced back through Abazajian, Fuller, and Patel (2001) and Giovannini, Keihänen, and Kurki-Suonio (2002). It seems difficult to imagine that a more complicated theory would reproduce the successful predictions of the simple model, but Nature fools us on occasion. Thus before concluding that the theory of the pre-stellar light element abundances is known, apart from the addition of decimal places to the cross sections, it is best to wait and see what advances in the physics of baryogenesis and of neutrinos teach us.

How is general relativity probed? The only part of the
computation that depends specifically on this theory is
the pressure term in the active gravitational mass density, in
the expansion rate equation (8). If we did not have general
relativity, a simple Newtonian picture might have led us to write
down / *a* =
- 4 *G*
_{r} /
3 instead of Eq. (8). With
_{r} ~
1 / *a*^{4}, as appropriate since most of the mass
is fully relativistic at the redshifts of light element production,
this would predict the expansion time *a* /
is 2^{1/2}
times the standard expression (that from Eq. [61]). The larger
expansion time would hold the neutron to proton number density
ratio close to that at thermal equilibrium,
*n*/*p* = *e*^{-Q/kT}, where *Q*
is the difference between the neutron and proton masses, to lower
temperature. It would also allow more time for free decay of the
neutrons after thermal equilibrium is broken.
Both effects decrease the final ^{4}He abundance. The factor
2^{1/2} increase in expansion time would reduce the helium
abundance by mass to *Y* ~ 0.20. This is significantly less
than what is observed in objects with the lowest heavy element
abundances, and so seems to be ruled out
(Steigman, 2002).
^{73} That is, we have
positive evidence for the
relativistic expression for the active gravitational mass density
at redshift *z* ~ 10^{10}, a striking result.

The predicted time of expansion from the very early universe to
redshift *z* is

(64) |

where *E*(*z*) is defined in Eq. (11).
If = 0 the
present age is *t*_{0} <
*H*_{0}^{-1}. In the
Einstein-de Sitter model the present age is
*t*_{0} = 2 / (3*H*_{0}).
If the dark energy density is significant and evolving, we may
write _{} =
_{0} *f* (*z*), where the function
of redshift is normalized to *f* (0) = 1. Then *E*(*z*)
generalizes to

(65) |

In the XCDM parametrization with constant *w*_{X}
(Eq. [45]),
*f* (*z*) = (1 + *z*)^{3(1+wX)}.
Olson and Jordan (1987)
present the earliest discussion we have found of *H*_{0}
*t*_{0} in this picture (before it got the name).
In scalar field models, *f* (*z*) generally must be evaluated
numerically; examples are in
Peebles and Ratra
(1988).

The relativistic correction to the active gravitational mass density (Eq. [8]) is not important at the redshifts at which galaxies can be observed and the ages of their star populations estimated. At moderately high redshift, where the nonrelativistic matter term dominates, Eq. (64) is approximately

(66) |

That is, the ages of star populations at high redshift are an
interesting probe of
_{M0} but
they are not very sensitive to space curvature or to a near constant dark
energy density. ^{74}

Recent analyses of the ages of old stars
^{75} indicate the expansion
time is in the range

(67) |

at 95% confidence, with central value
*t*_{0} 13
Gyr. Following
Krauss and Chaboyer
(2001)
these numbers add
0.8 Gyr to the star ages, under the assumption star formation
commenced no earlier than *z* = 6 (Eq. [66]). A naive addition in
quadrature to the uncertainty in *H*_{0} (Eq. [6])
indicates the dimensionless age parameter is in the range

(68) |

at 95% confidence, with central value
*H*_{0} *t*_{0}
0.89.
The uncertainty here is dominated by that in *t*_{0}. In the
spatially-flat
CDM model
(_{K0} = 0),
Eq. (68) translates to
0.15
_{M0}
0.8, with central
value _{M0}
0.4. In the open model
with _{0} =
0, the constraint is
_{M0}
0.6 with
the central value
_{M0}
0.1. In the inverse
power-law scalar field dark energy case
(Sec. II.C) with power-law index
= 4, the constraint is
0.05
_{M0}
0.8.

We should pause to admire the unification of the theory and
measurements of
stellar evolution in our galaxy, which yield the estimate of
*t*_{0}, and the measurements of the extragalactic distance
scale, which yield *H*_{0}, in the product in Eq. (68) that
agrees with the relativistic cosmology with dimensionless
parameters in the range now under discussion. As we indicated in
Sec. III, there is a long history of
discussion of the expansion
time as a constraint on cosmological models. The measurements now
are tantalizingly close to a check of consistency with the values
of _{M0} and
_{0}
indicated by other cosmological tests.

**4. The redshift-angular size and redshift-magnitude
relations**

An object at redshift *z* with physical length *l* perpendicular
to the line of sight subtends angle
such that

(69) |

where *a*_{0} = *a*(*t*_{0}).
The angular size distance *r*(*z*) is the coordinate position of
the object in the first line element in Eq. (15), with the
observer placed at the origin. The condition that light moves from
source to observer on a radial null geodesic is

(70) |

which gives

(71) |

where *E*(*z*) is defined in Eqs. (11) and (65).

In the Einstein-de Sitter model, the angular-size-redshift relation is

(72) |

At *z* << 1, =
*H*_{0} *l* / *z*, consistent with the Hubble
redshift-distance relation. At *z* >> 1 the image is
magnified, ^{76}
1 + *z*.

The relation between the luminosity of a galaxy and the energy
flux density received by an observer follows from Liouville's
theorem: the observed energy flux
*i*_{0} per unit time, area, solid angle, and
frequency satisfies

(73) |

with
*i*_{e}
the emitted energy flux (surface brightness) at the source and
_{e} =
_{0}(1 + *z*) the
bandwidth at the source at redshift *z*. The redshift factor (1 +
*z*)^{4}
appears for the same reason as in the 3 K cosmic microwave
background radiation energy density. With Eq. (69) to
fix the solid angle, Eq. (73) says
the observed energy flux per unit area, time, and
frequency from a galaxy at redshift *z* that has luminosity
*L*_{e}
per frequency interval measured at the source is

(74) |

In conventional local physics with a single metric theory the
redshift-angular size (Eq. [69]) and
redshift-magnitude (Eq. [7]) relations are physically
equivalent. ^{77}

The best present measurement of the redshift-magnitude relation
uses supernovae of Type Ia.
^{78}
The results are inconsistent with the Einstein-de
Sitter model, at enough standard deviations to make it clear
that unless there is something quite substantially and
unexpectedly wrong with the measurements the Einstein-de Sitter
model is ruled out. The data require
> 0 at two to
three standard deviations,
depending on the choice of data and method of analysis
(Leibundgut, 2001;
Gott et al., 2001).
The spatially-flat case with
_{M0} in the
range of Eq. (59) is a good fit for
constant . The
current data do not provide interesting
constraints on the models for evolving dark energy
density. ^{79}
Perlmutter et al.
(http://snap.lbl.gov/)
show that a
tighter constraint, from supernovae observations to redshift *z* ~ 2,
by the proposed SNAP satellite, is feasible and capable of giving
a significant detection of
and maybe its
evolution. ^{80}

Counts of galaxies -- or of other objects whose number density
as a function of redshift may be modeled -- probe the volume
element (*dV* / *dz*)*dz*
defined by a solid
angle
in the sky and a
redshift interval *dz*. The volume is fixed by the angular size
distance (Eq. [69]), which determines the area subtended
by the solid angle, in combination with the redshift-time relation
(Eq. [64]), which fixes the radial distance belonging
to the redshift interval.

Sandage (1961a) and Brown and Tinsley (1974) showed that with the technology then available galaxy counts are not a very sensitive probe of the cosmological parameters. Loh and Spillar (1986) opened the modern exploration of the galaxy count-redshift relation at redshifts near unity, where the predicted counts are quite different in models with and without a cosmological constant (as illustrated in Figure 13.8 in Peebles, 1993).

The interpretation of galaxy counts requires an understanding of
the evolution of galaxy luminosities and the gain and loss of
galaxies by merging. Here is an example of the former in a
spatially-flat cosmological model with
_{M0} =
0.25. The expansion time from high redshift is *t*_{3} =
2.4 Gyr at redshift
*z* = 3 and *t*_{0} = 15 Gyr now. Consider a galaxy
observed at *z* = 3.
Suppose the bulk of the stars in this galaxy formed at time
*t*_{f},
and the population then aged and faded without significant later
star formation. Then if
*t*_{f} << *t*_{3} the ratio of the observed
luminosity at *z* = 3 to its present luminosity would be
(Tinsley, 1972;
Worthey, 1994)

(75) |

If *t*_{f} were larger, but still less than
*t*_{3}, this ratio
would be larger. If *t*_{f} were greater than
*t*_{3} the galaxy
would not be seen, absent earlier generations of stars.
In a more realistic picture significant
star formation may be distributed over a considerable range of
redshifts, and the effect on the typical galaxy luminosity at a
given redshift accordingly more complicated. Since there are many
more galaxies with low luminosities than galaxies with
high luminosities, one has to know the luminosity evolution quite
well for a meaningful comparison of galaxy counts at high and low
redshifts. The present situation is illustrated by the rather
different indications from studies by
Phillipps et al. (2000)
and Totani et
al. (2001).

The understanding of galaxy evolution and the interpretation of galaxy counts will be improved by large samples of counts of galaxies as a function of color, apparent magnitude, and redshift. Newman and Davis (2000) point to a promising alternative: count galaxies as a function of the internal velocity dispersion that in spirals correlates with the dispersion in the dark matter halo. That could eliminate the need to understand the evolution of star populations. There is still the issue of evolution of the dark halos by merging and accretion, but that might be reliably modeled by numerical simulations within the CDM picture. Either way, with further work galaxy counts may provide an important test for dark energy and its evolution (Newman and Davis, 2000; Huterer and Turner, 2001; Podariu and Ratra, 2001).

**6. The gravitational lensing rate**

The probability of production of multiple images of a quasar or
a radio source by
gravitational lensing by a foreground galaxy, or of strongly
lensed images of a galaxy by a foreground cluster of galaxies,
adds the relativistic expression for the deflection of light to
the physics of the homogeneous cosmological model.
Fukugita, Futamase, and
Kasai (1990)
and Turner (1990)
point out the value of this test: at small
_{M0} the
predicted lensing rate is considerably larger in a flat model with
than in an
open model with
= 0 (as
illustrated in Fig 13.12 in
Peebles, 1993).

The measurement problem for the analysis of quasar lensing
is that quasars that are not lensed are not magnified by
lensing, making them harder to find and the correction for
completeness of detection harder to establish. Present estimates
(Falco, Kochanek, and
Muñoz, 1998;
Helbig et al., 1999)
do not seriously constrain
_{M0} in an
open model, and in a flat model
(_{K0} = 0)
suggest
_{M0} >
0.36 at
2. This is close to the
upper bound in Eq. (59).
Earlier indications that the lensing rate in a flat model with
constant requires
a larger value of
_{M0} than is
suggested by galaxy dynamics led
Ratra and Quillen
(1992)
and Waga and Frieman
(2000)
to investigate the inverse power-law
potential dark energy scalar field case. They showed this
can significantly lower the predicted lensing rate at
_{K0} = 0
and small
_{M0}. The
lensing rate still
is too uncertain to draw conclusions on this point, but advances
in the measurement certainly will be followed with interest.

The main problem in the interpretation of the rate of strong lensing of galaxies by foreground clusters as a cosmological test is the sensitivity of the lensing cross section to the mass distribution within the cluster (Wu and Hammer, 1993); for the present still somewhat uncertain state of the art see Cooray (1999) and references therein.

**7. Dynamics and the mean mass density**

Estimates of the mean mass density from the relation
between the mass distribution and the resulting peculiar
velocities, ^{81} and from
the gravitational deflection of light, probe gravity physics and constrain
_{M0}. The
former is not sensitive to
_{K0},
_{0},
or the dynamics of the dark energy, the latter only through the angular
size distances.

We begin with the redshift space of observed galaxy angular
positions and redshift distances *z*/*H*_{0} in the radial
direction. The redshift *z* has a contribution from the radial
peculiar velocity, which is a probe of the gravitational acceleration
produced by the inhomogeneous mass distribution.
The two-point correlation function,
_{v}, in
redshift space is defined by the probability that a randomly chosen
galaxy has a neighbor at distance
*r*_{||} along the line of sight in
redshift space and perpendicular distance
*r*_{},

(76) |

where *n* is the galaxy number density.
This is the usual definition of a reduced correlation function.
Peculiar velocities make the function anisotropic. On
small scales the random relative
peculiar velocities of the galaxies broaden
_{v}
along the line of sight. On large scales the streaming peculiar velocity of
convergence to gravitationally growing mass concentrations flattens
_{v}
along the line of sight. ^{82}

At 10 kpc
*hr*_{}
1 Mpc the measured
line-of-sight broadening is prominent, and indicates the
one-dimensional relative velocity dispersion is close to independent of
*r*_{} at
~ 300 km
s^{-1}. ^{83}
This is about what would be expected if the mass two-and three-point
correlation functions were well approximated by the galaxy
correlation functions, the mass clustering on these scales were
close to statistical equilibrium, and the density parameter were
in the range of Eq. (59).

We have a check from the motions of the galaxies in and around
the Local Group of galaxies, where the absolute errors in the
measurements of
galaxy distances are least. The two largest group members are the
Andromeda Nebula (M 31) and our Milky Way galaxy. If
they contain most
of the mass their relative motion is the classical two-body
problem in Newtonian mechanics (with minor corrections for
, mass accretion
at low redshifts, and the tidal torques
from neighboring galaxies). The two galaxies are
separated by 800 kpc and approaching at 110 km s^{-1}. In the
minimum mass solution the galaxies have completed just over
half an orbit in the cosmological expansion time
*t*_{0} ~ 10^{10} yr. By this argument
Kahn and Woltjer (1959)
find the sum of masses of the two galaxies has to be an order of
magnitude larger than what is seen in the luminous parts. An
extension to the analysis of the motions and distances of the
galaxies within 4 Mpc distance from us, and
taking account of the gravitational effects of the galaxies out
to 20 Mpc distance, gives masses quite similar to what Kahn and
Woltjer found, and consistent with
_{M0} in the
range of Eq. (59)
(Peebles et al., 2001).

We have another check from weak lensing: the shear distortion of
images of distant galaxies by the gravitational deflection by
the inhomogeneous mass distribution.
^{84}
If galaxies trace mass these measurements say the
matter density parameter measured on scales from about 1 Mpc to
10 Mpc is in the range of Eq. (59). It
will be interesting to see whether these measurements can check
the factor of two difference between the relativistic
gravitational deflection of light and the naive Newtonian
deflection angle.

The redshift space correlation function
_{v}
(Eq. [76]) is measured well enough at
*hr*_{} ~ 10
Mpc to demonstrate the flattening effect, again consistent with
_{M0} in the
range of Eq. (59), if galaxies trace mass. Similar numbers follow
from galaxies selected as far infrared IRAS sources
(Tadros et al., 1999)
and from optically selected galaxies
(Padilla et al., 2001;
Peacock et al., 2001).
The same physics, applied to estimates of the mean relative
peculiar velocity of galaxies at separations ~ 10 Mpc,
yet again indicates a similar density parameter
(Juszkiewicz et al.,
2000).

Other methods of analysis of the distributions of astronomical
objects and peculiar velocities smoothed over scales
10 Mpc
give a variety of results for the mass density, some above the
range in Eq. (59), ^{85}
others towards the bottom end of the range
(Branchini et al., 2001).
The measurement of
_{M0} from
large-scale streaming
velocities thus remains open. But we are impressed by an apparently
simple local situation, the peculiar motion of the Local Group
toward the Virgo cluster of galaxies.
This is the nearest known large mass concentration, at distance ~ 20 Mpc.
Burstein (2000)
finds that our virgocentric velocity is *v*_{v} = 220 km
s^{-1}, indicating
_{M0}
0.2
(Davis and Peebles,
1983a,
Fig. 1). This leads us to conclude that the weight of the evidence from
dynamics on scales ~ 10 Mpc favors low
_{M0}, in
the range of Eq. (59).

None of these measurements is precise. But many have been under
discussion for a long time and seem to us to be reliably
understood. Weak lensing is new, but the measurements are
checked by several independent groups. The result, in our
opinion, is a well checked and believable network of evidence
that over two decades of well-sampled length scales, 100 kpc to
10 Mpc, the apparent value of
_{M0} is
constant to a factor of three or so, in the range
0.15
_{M0}
0.4. The key
point for the purpose of this review is that this result is
contrary to what might have been expected from biasing, or from a
failure of the inverse square law (as will be discussed in test [13]).

**8. The baryon mass fraction in clusters of
galaxies**

Abell (1958)
made the first useful catalog of the rich clusters
considered here and in the next test. A typical value of the
Abell cluster mass within the Abell radius
*r*_{a} = 1.5 *h*^{-1} Mpc is
3 × 10^{14} *h*^{-1}
*M*_{}. The
cluster masses are reliably
measured (within Newtonian gravity) from consistent results from
the velocities of the galaxies, the pressure of the intracluster
plasma, and the gravitational deflection of light from background
galaxies.

White (1992) and
White et al. (1993)
point out that rich clusters likely are large enough to contain
a close to fair sample of baryons and dark matter, meaning the
ratio of baryonic to total mass in a cluster is a good measure of
_{B0} /
_{M0}. With
_{B0} from
the model for light elements
(Eqs. [62]), this gives a measure of the mean mass
density. The baryon mass fraction in clusters is still under
discussion. ^{86}
We adopt as the most direct and so maybe most reliable approach
the measurement of the baryonic gas mass fraction of clusters,
*f*_{gas}, through the Sunyaev-Zel'dovich microwave decrement
caused by Thomson-Compton scattering of cosmic microwave background
radiation by the intracluster plasma. The
Carlstrom et al. (2001)
value for *f*_{gas} gives
_{M0} ~
0.25, ^{87}in the range of
Eq. (59). This test does not directly constrain
_{K0},
_{0},
or the dynamics of the dark energy.

In the CDM model rich clusters of galaxies grow out of the rare
peak upward fluctuations in the primeval Gaussian mass
distribution. Within this model one can adjust the amplitude
of the mass fluctuations to match the abundance of clusters at
one epoch. In the Einstein-de Sitter model it is difficult to see
how this one free adjustment can account for the abundance of
rich clusters now and at redshifts near unity.
^{88}

Most authors now agree that the low density flat
CDM
model can give a reasonable fit to the cluster abundances
as a function of redshift. The constraint on
_{M0} from the
present cluster abundance still is under discussion, but
generally is found to be close to
_{M0} ~ 0.3
if galaxies trace mass. ^{89}
The constraint from the evolution of the cluster number density also
is under discussion. ^{90}
The predicted evolution is slower in a lower density universe, and at given
_{M0} the
evolution is slower in an open model with
= 0 than in a
spatially-flat model with
(for the reasons
discussed in Sec. III.D).
Bahcall and Fan (1998)
emphasize that we have good
evidence for the presence of some massive clusters at *z* ~ 1,
and that this is exceedingly difficult to understand in the
CDM model in the Einstein-de Sitter cosmology (when biasing is
adjusted to get a reasonable present number density).
Low density models with or without
can account for the
existence of some massive clusters at high redshift.
Distinguishing between the predictions of the spatially curved
and flat low density cases awaits better measurements.

**10. Biasing and the development of nonlinear mass
density fluctuations**

Elements of the physics of cluster formation in test (9) appear
in this test of the early stages in the nonlinear growth of
departures from homogeneity. An initially
Gaussian mass distribution becomes skew as low density
fluctuations start to bottom out and high density fluctuations
start to develop into prominent mass peaks. The early
signature of this nonlinear evolution is the disconnected
three-point mass autocorrelation function,
<(, *t*)
(, *t*)
(, *t*)>,
where (, *t*) =
/
is the
dimensionless
mass contrast. If galaxies are useful mass tracers the galaxy
three-point function is a good measure of this mass function.

The form for the mass three-point function, for Gaussian initial
conditions at high redshift, in lowest nonzero order in
perturbation theory, is worked out in
Fry (1984), and
Fry (1994)
makes the point that measurements of the galaxy three-point
function test how well galaxies trace mass.
^{91}
There are now two sets of measurements of the galaxy three-point
function on scales ~ 10 to 20 Mpc, where the
density fluctuations are not far from Gaussian. One uses
infrared-selected IRAS galaxies,
^{92}
the other optically-selected galaxies
(Verde et al., 2002).
The latter is consistent with the
perturbative computation of the mass three-point function for
Gaussian initial conditions. The former says infrared-selected
galaxies are adequate mass tracers apart from the densest
regions, which IRAS galaxies avoid. That has a simple
interpretation in astrophysics: galaxies in dense
regions tend to be swept clear of the gas and dust that make
galaxies luminous in the infrared.

This test gives evidence of consistency of three ideas:
galaxies are useful mass tracers on scales ~ 10 Mpc, the
initial conditions are close to Gaussian, and conventional
gravity physics gives an adequate description of this aspect of
the growth of structure. It is in principle sensitive to
_{0},
through the suppression of the growth of small departures from
homogeneity at low redshift, but the effect is small.

**11. The anisotropy of the cosmic microwave background
radiation**

The wonderfully successful CDM prediction of the power spectrum of the angular distribution of the temperature of the 3 K cosmic microwave background radiation has converted many of the remaining skeptics in the cosmology community to the belief that the CDM model likely does capture important elements of reality.

Efstathiou (2002)
provides a useful measure of the information in
the present measurements ^{93}:
the fit to the CDM model significantly
constrains three linear combinations of the free parameters.
We shall present three sets of considerations that roughly follow
Efstathiou's constraints. We begin with reviews of the standard
measure of the temperature anisotropy and of the conditions at
redshift *z* ~ 1000 that are thought to produce the observed
anisotropy.

The 3 K cosmic microwave background temperature
*T*(,
)
as a function of position in the sky usually is expressed as an
expansion in spherical harmonics,

(77) |

The square of *T*
averaged over the sky is

(78) |

where |*a*_{l}^{m}|^{2} is statistically
independent of *m*. This may be rewritten as

(79) |

Since
*l*^{-1} is
close to
*d* ln
*l*,
*T*_{l}^{2} is
the variance of the temperature per logarithmic interval of *l*.
A measure of the angular scale belonging to the multipole index
*l* is that the minimum distance between
zeros of the spherical harmonic *Y*_{l}^{m}, in
longitude or latitude, is
=
/ *l*, except close to
the poles, where *Y*_{l}^{m} approaches zero.
^{94}

Now let us consider the main elements of the physics that
determines the 3 K cosmic microwave background anisotropy.
^{95}
At redshift *z*_{dec} ~ 1000 the
temperature reaches the critical value at which the primeval
plasma combines to atomic hydrogen (and slightly earlier to
neutral helium). This removes the coupling between baryons and
radiation by Thomson scattering, leaving the radiation to
propagate nearly freely (apart from residual gravitational
perturbations). Ratios of mass densities near the epoch
*z*_{dec} when matter and radiation decouple are worth
noting. At redshift *z*_{eq} = 2.4 × 10^{4}
_{M0}
*h*^{2} the mass
density in matter -- including the baryonic and nonbaryonic
components -- is equal to the relativistic mass density in
radiation and neutrinos assumed to have low masses. At decoupling
the ratio of mass densities is

(80) |

at the central values of the parameters in Eqs. (6) and (59). The ratio of mass densities in baryons and in thermal cosmic microwave background radiation -- not counting neutrinos -- is

(81) |

That is, the baryons and radiation decouple just as the expansion rate has become dominated by nonrelativistic matter and the baryons are starting to lower the velocity of sound in the coupled baryon-radiation fluid (presenting us with still more cosmic coincidences).

The acoustic peaks in the spectrum of angular fluctuations of the 3 K cosmic microwave background radiation come from the Fourier modes of the coupled baryon-radiation fluid that have reached maximum or minimum amplitude at decoupling. Since all Fourier components start at zero amplitude at high redshift -- in the growing density perturbation mode -- this condition is

(82) |

where *c*_{s} is the velocity of sound in the baryon-radiation
fluid. Before decoupling the mass density in radiation is greater
than that of the baryons, so the velocity of sound is close
to *c* / 3^{1/2}. The proper wavelength at the first
acoustic peak thus is

(83) |

The parameter dependence comes from Eq. (66).
The observed angle subtended by
_{peak} is
set by the angular size distance *r* computed from
*z*_{eq} to the present (Eq. [71]). If
_{K0} = 0 or
_{0} =
0 the angular size distance is

(84) |

If = 0 this
expression is analytic at large *z*_{eq}.
The expression for
_{K0} = 0 is
a reasonable approximation to the
numerical solution. So the angular scale of the peak varies with
the matter density parameter as

(85) (86) |

The key point from these considerations is that the angle defined
by the first peak in the fluctuation power spectrum is sensitive
to _{M0} if
= 0 (Eq. [85]),
but not
if _{K0} = 0
(Eq. [86]). ^{96}
We have ignored the sensitivity of
*z*_{dec} and *t*_{dec} to
_{M0}, but
the effect is weak. More detailed computations, which
are needed for a precise comparison with the data, show that the
CDM model predicts that the first and largest peak of
*T*_{l}
appears at multipole index
*l*_{peak}
220 _{M0}^{-1/2} if
= 0, and at
*l*_{peak}
220 if
_{K0} = 0
and 0.1
_{M0}
1.
^{97}

The measured spectrum ^{98}
peaks at
*T*_{l} ~
80 *µ*K at
*l* ~ 200, thus requiring small space curvature in the CDM
model. This is the first of Efstathiou's constraints. Because of
the geometric degeneracy this measurement does not yet seriously
constrain
_{M0} if
_{K0} = 0.

The second constraint comes from the spectrum of temperature
fluctuations on large scales,
*l* 30,
where pressure gradient
forces never were very important. Under the scale-invariant initial
conditions discussed in Sec. III.C the
Einstein-de Sitter model predicts
*T*_{l} is
nearly independent of
*l* on large scales. A spatially-flat model with
_{M0} ~ 0.3,
predicts
*T*_{l}
decreases slowly with increasing *l*
at small *l*. ^{99}
The measured spectrum is close to flat at
*T*_{l} ~
30 *µ*K, but not well enough constrained for a
useful measure of the parameters
_{M0} and
_{0}.
^{100} Because of the
simplicity of the physics on large angular scales, this
provides the most direct and so perhaps most reliable normalization of the
CDM model power spectrum (that is, the parameter *A* in
Eqs. [40] and [41]).

The third constraint is the baryon mass density. It affects the
speed of sound *c*_{s} (Eq. [82]) in the
baryon-radiation fluid prior to decoupling, and the mean free
path for the radiation at
*z* ~ *z*_{dec}. These in turn
affect the predicted sequence of acoustic peaks (see, e.g.,
Hu and Sugiyama, 1996).
The detected peaks are consistent with
a value for the baryon density parameter
_{B0} in a
range that includes what is derived from the light elements abundances
(Eqs. [62]). ^{101}
This impressive check may be much improved by the measurements
of *T*_{l} in
progress.

The measurements of
*T*_{l}
are consistent with a near scale-invariant power spectrum (Eq. [41]
with *n* 1) with
negligible contribution from gravity wave or isocurvature fluctuations
(Sec. III.C.1).
The 3 K cosmic microwave background temperature fluctuations
show no departure from a Gaussian random process.
^{102}
This agrees with the picture in test (10) for the
nonlinear growth of structure out of Gaussian initial mass
density fluctuations.

The interpretation of the cosmic microwave background temperature
anisotropy measurements assumes and tests general relativity and
the CDM model. One can write down other models for
structure formation that put the peak of
*T*_{l} at
about the observed angular scale -- an example is
Hu and Peebles (2000)
-- but we have seen none so far that seem likely to fit
the present measurements of
*T*_{l}. Delayed recombination
of the primeval plasma in an low density
= 0 CDM model
can shift the peak of
*T*_{l} to
the observed scale. ^{103}
The physics is valid, but the scenario
is speculative and arguably quite improbable. On the other hand,
we cannot be sure a fix of the challenges to CDM reviewed in
Sec. IV.A.2 will not affect our
assessments of such issues, and hence of this cosmological test.

**12. The mass autocorrelation function and nonbaryonic
matter**

If the bulk of the nonrelativistic matter, with density parameter
_{M0} ~
0.25, were baryonic, then under adiabatic initial
conditions the most immediate problem would be the strong
dissipation of primeval mass density fluctuations on the scale of
galaxies by diffusion of radiation through the baryons at
redshifts near decoupling.
^{104}
Galaxies could form by fragmentation of the
first generation of protocluster "pancakes," as
Zel'dovich (1978)
proposed, but this picture is seriously challenged by the
evidence that the galaxies formed before clusters of
galaxies. ^{105}
In a baryonic dark matter model we could accommodate the
observations of
galaxies already present at *z* ~ 3 by tilting the primeval
mass fluctuation spectrum to favor large fluctuations on small
scales, but that would mess up the cosmic microwave background
anisotropy. The search for isocurvature initial
conditions that might fit both in a baryonic dark matter model
has borne no fruit so far
(Peebles, 1987).

The most important point of this test is
the great difficulty of reconciling the power spectra of matter
and radiation without the postulate of nonbaryonic dark matter.
The CDM model allows hierarchical growth of structure, from
galaxies up, which is what seems to be observed, because the
nonbaryonic dark matter interacts with
baryons and radiation only by gravity; the dark matter
distribution is not smoothed by the dissipation of density
fluctuations in the baryon-radiation fluid at redshifts
*z*
*z*_{eq}.

As discussed in Sec. III.D, in the CDM
model the small scale part
of the dark matter power spectrum bends from the primeval
scale-invariant form
*P*(*k*)
*k* to
*P*(*k*)
*k*^{-3}, and
the characteristic length at the break scales inversely with
_{M0}
(Eq. [42]). Evidence of such a break in
the galaxy power spectrum *P*_{g}(*k*) has been known
for more than a decade ^{106};
it is consistent with a value of
_{M0} in the
range of Eq. (59).

**13. The gravitational inverse square law**

The inverse square law for gravity determines the relation
between the mass distribution and the gravitationally-driven
peculiar velocities that enter estimates of the
matter density parameter
_{M0}. The
peculiar velocities also
figure in the evolution of the mass distribution, and hence the
relation between the present mass fluctuation spectrum and the
spectrum of cosmic microwave background temperature fluctuations
imprinted at redshift
*z* ~ 1000. We are starting to see demanding tests of
both aspects of the inverse square law.

We have a reasonably well checked set of measurements of the
apparent value of
_{M0} on
scales ranging from
100 kpc to 10 Mpc (as reviewed under test [7]). Most agree with a
constant value of the apparent
_{M0},
within a factor of three or
so. This is not the precision one would like, but the subject
has been under discussion for a long time, and, we believe, is
now pretty reliably understood, within the factor of three or so.
If galaxies were biased tracers of mass
one might have expected to have seen that
_{M0} increases
with increasing length scale, as the increasing scale includes
the outer parts of extended massive halos. Maybe that is
masked by a gravitational force law that decreases more rapidly
than the inverse square law at large distance. But the much more
straightforward reading is that the slow variation of
_{M0}
sampled over two orders of magnitude in length scale agrees with
the evidence from tests (7) to (10) that galaxies are useful mass
tracers, and that the inverse square law therefore is a useful
approximation on these scales.

The toy model in Eq. (57) illustrates how a failure
of the inverse square law would affect the evolution of
the shape of the mass fluctuation power spectrum *P*(*k*,
*t*) as a
function of the comoving wavenumber *k*, in linear perturbation
theory. This is tested by the measurements of the mass and
cosmic microwave background temperature fluctuation power spectra.
The galaxy power spectrum *P*_{g}(*k*) varies
with wavenumber at
*k* ~ 0.1*h* Mpc^{-1} about as expected
under the assumptions that the mass distribution grew by gravity
out of adiabatic scale-invariant initial conditions, the mass is
dominated by dark matter that does not suffer radiation drag at
high redshift, the galaxies are useful
tracers of the present mass distribution, the matter density
parameter is
_{M0} ~ 0.3,
and, of course, the evolution is
adequately described by conventional physics
(Hamilton and Tegmark,
2002,
and references therein). If the inverse
square law were significantly wrong at
*k* ~ 0.1*h* Mpc^{-1},
the near scale-invariant form would have to be an accidental
effect of some failure in this rather long list of assumptions.
This seems unlikely, but a check certainly is desirable. We
have one, from the cosmic microwave background anisotropy
measurements. They also are consistent with near scale-invariant
initial conditions applied at redshift *z* ~ 1000. This
preliminary check on the effect of the gravitational inverse
square law applied on cosmological length scales and back to
redshift *z* ~ 1000 will be improved by better understanding of the
effect on
*T*_{l} of
primeval tensor perturbations to
spacetime, and of the dynamical response of the dark energy
distribution to the large-scale mass distribution.

Another aspect of this check is the comparison of values of the
large-scale rms fluctuations in the present distributions of mass
and the cosmic microwave background radiation. The
latter is largely set at decoupling, after which the former grows
by a factor of about 10^{3} to the present epoch, in the standard
relativistic cosmological model. If space curvature is negligible the
growth factor agrees with the observations to
about 30%, assuming galaxies trace mass.
In a low density universe with
= 0 the standard
model requires that mass is more smoothly distributed than galaxies,
*N* / *N* ~
3*M* / *M*, or
that the
gravitational growth factor since decoupling is a factor of three
off the predicted factor ~ 1000; this factor of three is about
as large a deviation from unity as is viable. We are not proposing this
interpretation of the data, rather we are impressed by the modest size
of the allowed adjustment to the inverse square law.

^{70} This was recognized by
Zel'dovich (1964),
R. Feynman, in 1964, and S. Refsdal, in 1965.
Feynman's comments in a colloquium are noted
by Gunn (1967).
Peebles attended Refsdal's lecture at the International Conference on
General Relativity and Gravitation, London, July 1965;
Refsdal (1970)
mentions the lecture.
Back.

^{71} The history of the discovery and
measurement of this radiation,
and its relation to the light element abundances in test (2), is
presented in Peebles
(1971,
pp. 121-9 and 240-1),
Wilkinson and Peebles
(1990),
and Alpher and Herman
(2001).
The precision spectrum measurements are summarized in
Halpern, Gush, and Wishnow
(1991)
and Fixsen et
al. (1996).
Back.

^{72} To see this, recall the normal modes argument used to get
Eq. (7). The occupation number in a normal mode with
wavelength at
temperature *T* is the Planck form
=
[*e*^{ c / kT
} - 1]^{-1}.
Adiabaticity says is
constant. Since the mode wavelength varies as
*a*(*t*),
where *a* is the expansion factor in Eq. (4), and
is
close to constant, the mode temperature varies as
*T*
1 / *a*(*t*). Since the same
temperature scaling applies to each mode, an initially
thermal sea of radiation remains thermal in the absence of
interactions. We do not know the provenance of this argument; it
was familiar in Dicke's group when the 3 K cosmic microwave
background radiation was discovered.
Back.

^{73} There is a long history of
discussions of this probe of the
expansion rate at the redshifts of light element production.
The reduction of the helium abundance to *Y* ~ 0.2 if the
expansion time is increased by the factor 2^{1/2} is seen
in Figs. 1 and 2 in
Peebles (1966).
Dicke (1968)
introduced the constraint on evolution of the strength of the gravitational
interaction; see
Uzan (2002)
for a recent review.
The effect of the number of neutrino families on the expansion
rate and hence the helium abundance is noted by
Hoyle and Tayler (1964)
and Shvartsman (1969).
Steigman, Schramm, and
Gunn (1977)
discuss the importance of this effect as a test of cosmology and
of the particle physics measures of the number of neutrino
families.
Back.

^{74} The predicted maximum age of star
populations in galaxies at redshifts
*z* 1 does
still depend on
_{0}
and
_{K0}, and
there is the advantage that the predicted
maximum age is a lot shorter than today. This variant of the
expansion time test is discussed by
Nolan et al. (2001),
Lima and Alcaniz (2001),
and references therein.
Back.

^{75} See Carretta et al. (2000),
Krauss and Chaboyer
(2001),
Chaboyer and Krauss
(2002),
and references therein.
Back.

^{76} The earliest discussion we know of
the magnification effect is by
Hoyle (1959).
In the coordinate system
in Eq. (15), with the observer at the origin, light
rays from the object move to the observer along straight radial
lines. An image at high redshift is magnified because the light
detected by the observer is emitted when the proper distance to
the object measured at fixed world time is small. Because the proper
distance between the object and source is increasing faster than
the speed of light, emitted light directed at the observer
is initially moving away from the observer.
Back.

^{77} For a review of measurements of the
redshift-magnitude relation (and other cosmological tests) we
recommend
Sandage (1988).
A recent application to the most luminous galaxies in
clusters is in
Aragón-Salamanca,
Baugh, and Kauffmann (1998).
The redshift-angular size relation is measured by
Daly and Guerra (2001)
for radio galaxies,
Buchalter et al. (1998)
for quasars, and
Gurvits, Kellermann, and
Frey (1999)
for compact radio sources. Constraints on the cosmological parameters from
the Gurvits et al. data are discussed by
Vishwakarma (2001),
Lima and Alcaniz
(2002),
Chen and Ratra (2003),
and references therein, and constraints
based on the radio galaxy data are discussed by
Daly and Guerra (2001),
Podariu et al. (2003),
and references therein.
Back.

^{78} These supernovae are characterized by
the absence of hydrogen lines
in the spectra; they are thought to be the result of explosive nuclear
burning of white dwarf stars.
Pskovskii (1977)
and Phillips (1993)
pioneered the reduction of
the supernovae luminosities to a near universal standard candle.
For recent discussions of their use as a cosmological test see
Goobar and Perlmutter
(1995),
Reiss et al. (1998),
Perlmutter et
al. (1999a),
Gott et al. (2001), and
Leibundgut (2001).
We recommend
Leibundgut's (2001)
cautionary discussion of astrophysical
uncertainties: the unknown nature of the trigger for
the nuclear burning, the possibility that the Phillips
correction to a fiducial luminosity actually depends on redshift
or environment within a galaxy, and possible obscuration by
intergalactic dust. There are also issues of physics that may
affect this test (and others): the
strengths of the gravitational or electromagnetic interactions
may vary with time, and photon-axion conversion may reduce the
number of photons reaching us. All of this is under active study.
Back.

^{79}
Podariu and Ratra (2000)
and Waga and Frieman
(2000)
discuss the
redshift-magnitude relation in the inverse power-law scalar field
model, and
Waga and Frieman
(2000) and
Ng and Wiltshire (2001)
discuss this relation in the massive scalar field model.
Back.

^{80} Podariu, Nugent, and Ratra (2001),
Weller and Albrecht
(2002),
Wang and Lovelace
(2001),
Gerke and Efstathiou
(2002),
Eriksson and Amanullah
(2002),
and references therein, discuss constraints on cosmological
parameters from the proposed SNAP mission.
Back.

^{81} Early
estimates of the mean mass density, by
Hubble (1936, p. 189)
and Oort (1958),
combine the galaxy number
density from galaxy counts with estimates of galaxy masses from
the internal motions of gas and stars.
Hubble (1936, p. 180)
was quite aware that this misses mass between the
galaxies, and that the motions of galaxies within clusters
suggests there is a lot more intergalactic mass
(Zwicky, 1933;
Smith, 1936).
For a recent review of this subject see
Bahcall et. al. (2000).
Back.

^{82} This approach grew out of the
statistical method introduced by
Geller and Peebles
(1973);
it is derived in its present form
in Peebles (1980b)
and first applied to a serious redshift sample in
Davis and Peebles
(1983b).
These references give the theory for the second moment
^{2} of
_{v} in
the radial direction -- the mean
square relative peculiar velocity -- in the small-scale stable
clustering limit. The analysis of the anisotropy of
_{v} in
the linear perturbation theory of large-scale flows (Eq. [55]) is
presented in
Kaiser (1987).
Back.

^{83} This measurement requires close
attention to clusters that
contribute little to the mean mass density but a broad and
difficult to measure tail to the distribution of relative velocities.
Details may be traced back through
Padilla et al. (2001),
Peacock et al. (2001),
and Landy (2002).
Back.

^{84} Recent studies include
Wilson, Kaiser, and
Luppino (2001),
Van Waerbeke et
al. (2002),
Refregier, Rhodes, and
Groth (2002),
Bacon et al. (2002),
and Hoekstra, Yee, and
Gladders (2002).
See
Munshi and Wang (2002)
and references therein for discussions
of how weak lensing might probe dark energy.
Back.

^{85} The methods and results may be traced
through
Fisher, Scharf, and Lahav
(1994),
Sigad et al. (1998), and
Branchini et al. (2000).
Back.

^{86} See Hradecky et al. (2000),
Roussel, Sadat, and
Blanchard (2000),
Allen, Schmidt, and Fabian
(2002),
and references therein.
Back.

^{87} This assumes
_{B0}
*h*^{2} = 0.014 from
Eqs. (62). For the full range of values in Eqs. (6) and (62),
0.1
_{M0}
0.4 at two
standard deviations.
Back.

^{88} Early discussions of this problem include
Evrard (1989),
Peebles, Daly, and
Juszkiewicz (1989),
and Oukbir and Blanchard
(1992).
Back.

^{89} For recent discussions see
Pierpaoli, Scott, and
White (2001),
Seljak (2001),
Viana, Nichol, and Liddle
(2002),
Ikebe et al. (2002),
Bahcall et al. (2002),
and references therein.
Wang and Steinhardt
(1998)
consider this test in the context of the
XCDM parametrization; to our knowledge it
has not been studied in the scalar field dark energy case.
Back.

^{90} Examples include
Blanchard et al. (2000),
and Borgani et
al. (2001).
Back.

^{91} Other notable contributions to the
development of this point include
Bernardeau and Schaeffer
(1992),
Fry and Gaztañaga
(1993),
and Hivon et al. (1995).
Back.

^{92} Two sub-samples of IRAS galaxies are
analyzed by
Scoccimarro et
al. (2001)
and by Feldman et
al. (2001).
Back.

^{93} Recent measurements are presented in
Lee et al. (2001),
Netterfield et
al. (2002),
Halverson et
al. (2002),
Miller et al. (2002a),
Coble et al. (2001),
Scott et al. (2002),
and Mason et al. (2002).
Back.

^{94} A more careful analysis
distinguishes averages across the sky from ensemble averages. By
historical accident the conventional normalization replaces 2*l* + 1
with 2(*l* + 1) in Eq. (79).
Kosowsky (2002)
reviews the physics of the polarization of the radiation.
Back.

^{95} The physics is worked out in
Peebles and Yu (1970)
and Peebles (1982).
Important analytic considerations are in
Sunyaev and Zel'dovich
(1970).
The relation of the cosmic microwave background anisotropy to the
cosmological parameters is explored in many papers; examples of
the development of ideas include
Bond (1988),
Bond et al. (1994),
Hu and Sugiyama (1996),
Ratra et al. (1997,
1999),
Zaldarriaga, Spergel, and
Seljak (1997),
and references therein.
Back.

^{96} This "geometrical degeneracy" is
discussed by
Efstathiou and Bond
(1999).
Marriage (2002)
presents a closer analysis of the effect.
Sugiyama and Gouda
(1992),
Kamionkowski, Spergel, and
Sugiyama (1994b),
and Kamionkowski et
al. (1994a)
are early discussions of the cosmic microwave background anisotropy in
an open model.
Back.

^{97}
Brax et al. (2000) and
Baccigalupi et al. (2000)
compute the angular spectrum of the cosmic microwave background
anisotropy in the dark energy scalar field model.
Doran et al. (2001)
discuss the angular scale of the peaks in this case, and
Corasaniti and Copeland
(2002),
Baccigalupi et
al. (2002),
and references therein, compare
model predictions and observations -- it is too early to draw profound
conclusions about model viability, and new data are eagerly anticipated.
Wasserman (2002)
notes that the cosmic microwave background anisotropy data
could help discriminate between different dark energy scalar field models
whose predictions do not differ significantly at low redshift.
Back.

^{98} For analyses see
Knox and Page (2000),
Podariu et al. (2001),
Wang, Tegmark, and
Zaldarriaga (2002),
Durrer, Novosyadlyj, and
Apunevych (2002),
Miller et al. (2002b),
and references therein.
Back.

^{99} The physics was first demonstrated by
Sachs and Wolfe
(1967) and
applied in the modern context by
Peebles (1982).
The intermediate
Sachs-Wolfe effect that applies if the universe is
not Einstein-de Sitter is shown in Eq. (93.26) in
Peebles (1980).
This part of the Sachs-Wolfe effect receives a contribution from
the low redshift matter distribution, so cross-correlating the
observed large-scale cosmic microwave background anisotropy with the
low redshift matter distribution could provide another test of the
world model
(Boughn and Crittenden,
2001,
and references therein).
Back.

^{100} See, e.g.,
Górski et
al. (1998).
This ignores the "low" value
of the cosmological quadrupole (*l* = 2) moment, whose value depends
on the model used to remove foreground Galactic emission (see, e.g.,
Kogut et al., 1996).
Contamination due to non-cosmic microwave background
emission is an issue for some of the anisotropy data sets (see, e.g.,
de Oliveira-Costa et al.,
1998;
Hamilton and Ganga,
2001;
Mukherjee et al., 2002,
and references therein). Other issues that need care
in such analyses include accounting for the uncertainty in the calibration
of the experiment (see, e.g.,
Ganga et al., 1997;
Bridle et al., 2002),
and accounting for the shape of the antenna pattern (see, e.g.,
Wu et al., 2001a;
Souradeep and Ratra,
2001;
Fosalba, Dore, and
Bouchet, 2002).
Back.

^{101} The
_{B0}
*h*^{2} values estimated from the cosmic microwave background
anisotropy measured by
Netterfield et
al. (2002),
Pryke et al. (2002),
and Stompor et
al. (2001),
are more consistent with the higher, deuterium based,
Burles et al. (2001)
range in Eqs. (62).
Back.

^{102}
Colley, Gott, and Park
(1996)
present an early discussion of the
situation on large angular scales; more recent discussions are in
Mukherjee, Hobson, and
Lasenby (2000),
Phillips and Kogut
(2001), and
Komatsu et al. (2002).
Degree and sub-degree angular scale
anisotropy data are studied in
Park et al. (2001),
Wu et al. (2001b),
Shandarin et al. (2002),
and Polenta et
al. (2002).
Back.

^{103} The model in
Peebles, Seager, and Hu
(2000)
assumes stellar ionizing radiation at *z* ~ 1000 produces
recombination Lyman
photons. These resonance photons
promote photoionization from the *n* = 2 level of atomic hydrogen.
That allows delayed recombination with a rapid transition to
neutral atomic hydrogen, as required to get the shape of
*T*_{l}
about right.
Back.

^{104} Early analyses of this effect are in
Peebles (1965),
and Silk (1967,
1968).
Back.

^{105} For example, our Milky Way galaxy
is in the Local Group, which
seems to be just forming, because the time for a group member
to cross the Local Group is comparable
to the Hubble time. The Local Group is on the outskirts of the
concentration of galaxies around the Virgo cluster. We and
neighboring galaxies are moving away from the cluster, but
at about 80 percent of the mean Hubble flow, as if the local
mass concentration were slowing the local expansion. That is,
our galaxy, which is old, is starting to cluster with
other galaxies, in a "bottom up" hierarchical growth of
structure, as opposed to the "top down" evolution of the pancake picture.
Back.

^{106} The first good evidence is discussed in
Efstathiou et
al. (1990);
for recent examples see
Sutherland et
al. (1999),
Percival et al. (2001),
and Dodelson et
al. (2002).
Back.