One of the fundamental questions of astronomy is "How did this all get here?" (Kleinmann 1988 private communication). This is deeply tied to observations bearing on the formation, evolution, and distribution of galaxies. First we develop some basic consideration on the "standard" big-bang picture, then deal with the relevant observations of the extragalactic universe.

The **Hubble expansion** (conventionally interpreted as an expansion of
spacetime with galaxies carried along for the ride) and cosmological
principle together imply one of two kinds of universe:

There are several immediately relevant data to help us choose a best bet scheme:

The most useful description of cosmological models involves general
relativity. Its geometric basis allows a natural treatment of light
propagation along geodisics, which is how we get most of our information.
Static solutions to Einstein's equations are unstable unless there is some
repulsive force giving rise to nonzero cosmological constant
Λ. A
description of this kind has two locally observable parameters, in terms of
the behavior of a scale factor *R*. This is defined so that the evolution
of the distance between two comoving points (that is with no peculiar
velocities superimposed on those from the cosmological expansion)
evolves as *r*_{12} = *r*_{0} *R(t)*.
These parameters are the Hubble
ratio (not quite constant since it will in general change with cosmic time)

and the density or deceleration parameter

For both of these, a subscript 0 indicates their evaluation at the present
epoch.
*q* is related to the open or closed nature of the universe. The critical
closure density is
ρ_{0}
= 3 H² / 8
ρG
as may be easily derived from Newtonian physics. Consider a volume element
of gas in a uniform medium of density
ρ at
distance *r* from some
arbitrary central point. Using a theorem of Newton, the gravitational
attraction of a uniform medium outside *r* has no net effect, so the
gravitational force due to the material within *r* generates a potential
- GM(< r)ρ
dV/*r* = -4G
ρ *r*²
ρ²
*dV*/3
for the matter in the volume element *dV*. Its kinetic energy for a
Hubble-like flow is *mv*²/2 or
ρ
*dV H² r² /2*. At critical density,
the net energy is zero or alternately the material in *dV* is marginally
bound: 4 *G*
π *r*²
ρ>² *dV* /3 =
ρ
*dV* H² *r*² /2
This is satisfied for the value of
ρ_{0}
given above.
The density in units of
ρ_{0}
is often denoted by
Ω. For
Λ=0,
*q* = Ω/2
so that *q*_{0} = 1/2 is the critical value.

Enormous effort has gone into the determination of H_{0} and *q*_{0} via
"classical" tests using galaxies as standard candles. Measurement of H_{0}
has been discussed in
the distance-scale lecture. The
problem with measuring the deceleration
parameter is that the universe is close to the critical density, so that
departures from this critical value become apparent only over such large
path lengths that galaxy evolution becomes the dominant factor. What little
we know about galaxy evolution says that galaxies won't be standard
candles. These tests are well documented in the review by Sandage 1988
(ARA&A 26, 561) in an eloquent testimonial to the ultimate failure of most
of them. These rely on the propagation of radiation in an expanding metric
and the consequent breakdown of the inverse-square law for intensities and
the inverse angular diameter-distance relation (more later on what
distance means in this context). Further discussion of these tests may be
found in Weinberg, *Gravitation and Cosmology*, Peebles, *Physical
Cosmology*, and ch. 3 of Weedman, *Quasar Astronomy*.

**Hubble diagram or magnitude-redshift test.** Assume we have a set
of galaxies with unchanging and known intrinsic brightness. It has been
popular to take radio galaxies or third-brightest cluster members for this;
radio galaxies in the K-band are especially well-behaved. At small
redshifts, the Hubble expansion makes the magnitude-log *z* relation
linear, with some scatter due to galaxy properties and our ignorance of
just how to select a perfectly uniform set of galaxies. At high *z*, the
relation may curve for various *q*_{0}. To see how, consider various measures
of distance to an object of redshift *z*. The *proper distance* is that
traversed by a photon in its own frame from there and then to here and now,
which is to say *D _{P}* = 1/

For the standard (Friedman) model,

This is one of a set of closed forms worked out by Mattig (1958 Astron.
Nachr. 284, 109) in a demonstration of analytical virtuosity starting from
the Robertson-Walker metric and curvature scale in a Friedmann model.
[I tried to reproduce this derivation as an inquisitive grad student,
and gave up after two days when it wasn't getting any closer. George
Blumenthal said it took him three days.]
Then we have *D _{L} = (1+z) D_{P}*
and

In principle, the curvature in the extended Hubble diagram for standard
candles can give *q*_{0} as shown by Weinberg p. 448 quoting Sandage.
Big news has been the finding that high-redshift type Ia supernovae
show evidence of upward curvature in the analogous magnitude-redshift
relation, implying a nonzero cosmological constant. Type Ia objects are]
useful because they have a narrow absolute-magnitude dispersion to
begin with, and much of this dispersion is corralated with
independently measurable fading timescale, plus the fact that
they should all have the same kind of progenitor which formed its own
heavy elements so that initial-metallicity effects should be small.
The evidence is shown in
Fig. 4 of Riess et al. (1998 AJ 116, 1009, reproduced by permission of the
AAS):

The differences among various values of *q*_{0} become important
only for
*z* > 1. Applying this test to real data on galaxies
is further complicated by various effects, some of whioch don't enter
for the supernova test.
The *K-correction* accounts for the fact that one is no longer
observing the same part of the spectrum at various redshifts, and for the
decrease in photon arrival rate (by a factor *1+z*) even if one follows the
same emitted-frame wavelength with *z*. Galaxy evolution, both stellar and
dynamical, turns out to be so strong that it's much easier to measure than
*q*_{0}. For cluster galaxies, richness effects enter - at high redshift it's
easier to pick out rich clusters whose *n*th brightest members are then
brighter than expected. It is not clear that this is the path to *q*_{0}
because of the dominant role of galaxy luminosity and spectral evolution.

The **angular diameter - redshift test** looks for a breakdown
of the inversion relation between distance and angular diameter of some set
of standard measuring rods (say galaxy isophotes or radio-galaxy lobe
separations). Surface brightnesses must be corrected for dimming by
*(1+z)*^{4} due to expansion of space plus photon energy and arrival-time
decreases. The form of this has been used as a test (the Tolman test)
that redshifts really correspond to an expansion, and not to once-popular
"tired-light"
phenomena (Sandage and Perelmuter 1990 ApJ 350, 481; 361, 1; 1991 ApJ 370,
455, Wirth 1997 PASP 109, 344). Here again, one might have to deal with
evolutionary effects in such
objects as radio sources - have they always been the same size? At least
with galaxy structures one has some control over how much dynamical
evolution might have gone on.

It is interesting that, for any given positive value of
*q*_{0}, there exists a
minimum in the
θ(*z*)
relation for any linear size; things at higher redshift look
bigger again because they were quite nearby when the light was emitted.
For *q*_{0}=1/2, the angular diameter goes through a
minimum at *z*=5/4. Using the analytical form listed, for example,
in Lang, *Astrophys. Formulae*, eqn. 5-314, we have:

(If you want to experiment with this, here's some
simple IDL code which generated the plot).
This effect may be in part responsible for the large apparent sizes of
very-high-redshift radio galaxies (see Djorgovski in *Nearly Normal
Galaxies*, for example). In interpreting sizes of high-redshift
objects, though, surface-brightness dimming can be a dominant effect.

**Galaxy number counts** may be used to trace the history of the
volume per galaxy, so that the *N(z)* relation implies *R(t)* since
the scale factor *R* scales as
*(1+z)*^{-1}. If we could count a conserved
population over a wide redshift range, we could learn *q*_{0}
directly from its definition. An early attempt used six-color mapping to
estimate photometric redshifts (Loh and Spillar 1986 ApJLett 307, L1).
Modelling with a standard luminosity function, they get
Ω= 0.9
^{+0.7}_{-0.5}
from 1000 "field" galaxies. This is independent of color evolution, but is
dependent on luminosity evolution and merging; both these processes
could change the number of galaxies in a given luminosity range with
redshift. As discussed by Sandage 1988, most applications of this test are
more sensitive to galaxy evolution than to cosmology (because the universe
is old enough for us to be here talking about it).

There are additional constraints from local (or indeed laboratory)
observations. Most prominent is the **cosmic microwave background**. This
is the radiation field at the epoch of (re)combination, when the universe
first became transparent enough for radiation to propagat freely over
cosmologically interesting distances. At the high densities then, the
radiation field was fully themalized (blackbody). Since then the observed
temperature has dropped due to redshifting (a redshifted blackbody is
another lower-temperature blackbody). See Peebles, chapter 7, for some
complications. Recombination was somewhere around *z* ~ 10^{4} set by
the temperature at which hydrogen ionizes and the observed CMB temperature
(which we know from COBE to be 2.785 K, Mather et al 1990 APJLett 354, L37
and Smoot et al 1991 ApJLett 371, L1).
The CMB is highly isotropic (except for a dipole term thought to reflect our
motion with respect to the large-scale velocity defined by the CMB emission
surface) and embarrassingly smooth. Its spectrum is as perfect a blackbody
as can be measured: from Fig. 4 of Fixsen et al. 1996 (ApJ 473, 576,
by permission of the AAS),

We would expect some lumps
corresponding to protogalaxies and -clusters, since it is hard enough to
understand how
galaxies form by *z* ~ 5 from inhomogeneities at recombination.
These were finally detected in a convincing way by the COBE group,
with subsequent confirmation by ground-based observations from
Tenerife and Antarctica as well as balloon-borne instruments.
The fluctuations, as measured at a resolution of a few degrees,
are at the level
Δ T / ~ 3
× 10^{-6},
which is (to factors "of order unity") the density contrast
Δ ρ / ρ
of fluctuations. Linear
development of perturbations won't cut it to clump matter fast enough;
there
is something major here that we don't know about making galaxies. I
am holding out for nonbaryonic matter already clumped at recombination
and providing seeds for forming baryonic structures, but then I
could be convinced otherwise if anything resembling evidence
shows up. The CMB
was one of the major downfalls of a steady-state picture; something has
clearly changed since the time when space was uniformly filled with
4000-K plasma.

Absorption of the CMB by hot gas in cluster is expected (the
Sunyaev-Zeldovich effect) and has been observed after years of upper
limits - this is interesting as
direct confirmation that we are not seeing some unknown local effect, since
the CMB comes from behind clusters at substantial redshift.
Note that the CMB temperature should scale as *1+z*; measurements
of low-energy fine-structure levels (specifically C II^{*}) in QSO
absorption-line systems indeed show this effect. As shown by
Ge, Bechtold, and Black 1997 ApJ 474, 67, (courtesy of the AAS),

In retrospect, the first hint of the CMB was the excitation temperature of interstellar CN seen in absorption at 3874 Å against galactic stars, with observations tracing back to Adams 1941 (ApJ 93, 11; see Thaddeus 1972 ARA&A 10, 305 for a fuller history). As discussed by Roth, Meyer, and Hawkins (1993 ApJL 413, L67), these lines are excited by absorbing radiation at 1.3 and 2.6mm. These observations have some philosophical interest in showing the uniformity of this radiation on galactic scales.

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