**2.1. The Physics of the Friedmann-Lemaître Model**

The second to the last column in the table summarizes elements of the physics of the relativistic Friedmann-Lemaître cosmological model we want to test.

The starting assumption follows Einstein in taking the observable universe to be close to homogeneous and isotropic. This agrees with the isotropy of the radiation backgrounds and of counts of sources observed at wavelengths ranging from radio to gamma rays. Isotropy allows a universe that is inhomogeneous but spherically symmetric about a point very close to us. I think it is not overly optimistic to consider this picture unlikely, but in any case it is subject to the cosmological tests, through its effect on the redshift-magnitude relation, for example.

It will be recalled that, if a homogeneous spacetime is described by a single metric tensor, the line element can be written in the Robertson-Walker form,

(1) |

This general expression contains one constant, that measures the
curvature of sections of
constant world time *t*, and one function of *t*, the expansion
parameter *a*(*t*). Under conventional local physics the de Broglie
wavelength of a freely moving particle varies as
*a*(*t*). In
effect, the expansion of the universe
stretches the wavelength. The stretching of the wavelengths of
observed freely propagating electromagnetic radiation is measured
by the redshift, *z*, in terms of the ratio of the observed
wavelength of a spectral feature to the wavelength measured at
rest at the source,

(2) |

At small redshift the difference
*t* of world times at
emission and detection of the light from a galaxy is relatively
small, the physical distance between emitter and observer is
close to *r* = *c*
*t*, and the rate of
increase of the proper distance is

(3) |

evaluated at the present epoch, *t*_{o}. This is Hubble's law for
the general recession of the nebulae.

Under conventional local physics Liouville's theorem applies. It
says an object at redshift *z* with radiation surface brightness
*i*_{e}, as measured by an observer at rest at the object,
has observed surface brightness (integrated over all wavelengths)

(4) |

Two powers of the expansion factor can be ascribed to aberration, one to the effect of the redshift on the energy of each photon, and one to the effect of time dilation on the rate of reception of photons. In a static ``tired light'' cosmology one might expect only the decreasing photon energy, which would imply

(5) |

Measurements of surface brightnesses of galaxies as functions of redshift thus can in principle distinguish these expanding and static models (Hubble & Tolman 1935).

The same surface brightness relations apply to the 3 K thermal
background radiation
(the CBR). Under equation (4) (and generalized to
the surface brightness per frequency interval), a thermal
blackbody spectrum remains thermal, the temperature varying as
*T*
*a*(*t*)^{-1}, when the universe is optically thin. The
universe now is optically thin at the Hubble length at CBR
wavelengths. Thus we can imagine the CBR was thermalized at
high redshift, when the universe was hot, dense, and optically
thick. Since no one
has seen how to account for the thermal CBR spectrum under
equation (5), the CBR is strong evidence for the
expansion of the universe. But since our imaginations are
limited, the check by the application of the Hubble-Tolman
test to galaxy surface brightnesses is well motivated
(Sandage 1992).

The constant *R*^{-2} and function *a*(*t*) in
equation (1) are measurable in principle, by the
redshift-dependence of counts of objects, their angular
sizes, and their ages relative to the present. The metric theory
on which these measurements are based is testable from
consistency: there are more observable functions than theoretical ones.

The more practical goal of the cosmological tests is to
over-constrain the parameters in the relativistic equation for
*a*(*t*),

(6) |

The total energy density,
_{t}, is the
time-time
part of the stress-energy tensor. The second line assumes
_{t} is a sum
of low pressure matter, with energy density that varies as
_{m}
*a*(*t*)^{-3}
(1 + *z*)^{3}, and a
nearly constant component that acts like Einstein's cosmological
constant . These terms, and
the curvature term, are
parametrized by their present contributions to the square of the
expansion rate, where Hubble's constant is defined by
equation (3).

**2.2. Applications of the Tests**

When I assembled Table 1 the
magnificent program of
application of the redshift-magnitude relation to type Ia
supernovae was just getting underway, with the somewhat mixed
preliminary results entered in line 2c
(Perlmutter et
al. 1997).
Now the supernovae measurements clearly point to low
_{m}
(Reiss 2000
and references therein), consistent with most other results entered in
the table.

The density parameter
_{m} inferred from the
application
of equation (6), as in the redshift-magnitude
relation, can be compared to what is derived
from the dynamics of peculiar motions of gas and stars, and from
the observed growth of mass concentrations with decreasing
redshift. The latter is assumed to reflect the theoretical
prediction that the expanding universe is gravitationally
unstable. Lines 1a and 1b show estimates of the density
parameter _{m} based on
dynamical interpretations of
measurements of peculiar velocities (relative to the uniform
expansion of Hubble's law) on relatively small and large scales.
Some of the latter indicated
_{m} ~ 1. A constraint from
the evolution of clustering is entered in line 3c. The overall
picture was pretty clear then, and now seems well established:
within the Friedmann-Lemaître model
the mass that clusters with the galaxies almost certainly is well
below the Einstein-de Sitter case
_{m} = 1. I
discuss the issue of how much mass might be in the voids between
the concentrations of observed galaxies and gas clouds in
Section 4.1.

All these ideas were under discussion, in terms we could
recognize, in the 1930s. The entry in line 2e is based on
the prediction of gravitational lensing. That was well known in
the 1930s, but the recognition that it provides a cosmological
test is more recent. The entry refers to the multiple imaging of
quasars by foreground galaxies
(Fukugita &
Turner 1991).
The straightforward reading of the evidence from this strong lensing still
favors small , but with broad
error bars, and it does not yet seriously constrain
_{m}
(Helbig 2000).
Weak lensing
- the distortion of galaxy images by clustered foreground mass
- has been detected; the
inferred surface mass densities indicate low
_{m}
(Mellier et al. 2001),
again consistent with most of the other constraints.

The other considerations in lines 1 through 4 are tighter, but the situation is not greatly different from what is indicated in the table and reviewed in Lasenby, Jones & Wilkinson (2000).