**2.1. The Cosmological Principle**

It is meaningful to talk of cosmic or universal radiation fields, rather than the radiation received from this or that system in a hierarchy of structures, because the evidence is that the universe we can see back to high redshift is close to homogeneous and isotropic. I discuss here some of the evidence and the measures of large-scale departures from homogeneity.

A useful foil to the standard homogeneous world picture models the large-scale
distribution of matter as a scale-invariant fractal with dimension
*D*. If *D* < 2 the mean
column density of material along a line of sight converges, meaning
there is no Olbers
paradox of unbounded sky brightness even if the universe were static and
stars lived
forever. This case is easy to rule out, however, because the angular
distribution of the
column density of material integrated to distance *R* has the same
fractal dimension
*D*, independent of *R* (as one must expect, for in a
scale-invariant fractal there is no
characteristic length to set the scale for *R*). That is, the
angular fluctuations in counts
of objects across the sky are statistically independent of the depth to
which the objects are counted, which is quite contrary to the
observations. (This prediction and its
observational tests are discussed in sections 3 and 7 of *Principles
of Physical Cosmology*,
Peebles 1993,
hereafter PC).

If 2 < *D* < 3 the line-of-sight integral for the background
radiation surface brightness
has to be cut off at some limiting distance *R*_{lim}, by
the finite lifetime of the sources
or by the cosmological redshift. We know an appreciable fraction of the X-ray
background (the XRB) is produced by sources with redshifts on the order
of unity, that is,
that in this case *R*_{lim} is on the order of the Hubble
distance *cH*_{0}^{-1} = 3000*h*^{-1}
Mpc (where *H*_{0} = 100*h* km s^{-1}
Mpc^{-1}). It is an easy exercise to check that in a
scale-invariant fractal model with 2 < *D* < 3, and with sources
turned off at distances
greater than *R*_{lim}, the quadrupole anisotropy of the
integrated background is

(1) |

again independent of *R*_{lim}. The large-scale anisotropy
of the XRB is not more than a
few tenths of one percent, meaning that the fractal dimension of the
large-scale space
distribution of X-ray sources differs from *D* = 3 by only a few
parts in 1000.

The bound on *D* is removed if we are close to the center of a
universe spherically
symmetric about one position, but since our galaxy and its near
neighbors appear to
be quite ordinary this does not seem to be a very sensible model.

The large-scale isotropy of the 3 K thermal cosmic background radiation
(the CBR)
gives an even better limit on large-scale mass fluctuations, if we are
willing to assume
general relativity theory is a useful approximation to physics on the
scale of the Hubble
length. Here a useful foil assumes mass cluster with galaxies on scales
~ 10*h*^{-1} Mpc,
and that the mass autocorrelation function
(*r*) =
< ( +
)
() > /
< >^{2} - 1
vanishes at
larger separations. Then the large-scale rms density fluctuations are
characterized by the integral

(2) |

The second line is based on the galaxy two-point correlation
function. In this model
the rms fluctuation in the mass in a randomly placed sphere of radius
*r* large compared
to the support of is

(3) |

and the rms value of the velocity averaged over the material within the sphere is (PC Sections 13 and 21)

(4) |

The cosmological density parameter is the ratio of the mean mass density to the critical Einstein-de Sitter value. The indicated scaling with is a good approximation if the cosmological constant vanishes or if is present and space curvature is negligible. The numbers in equations (3) and (4) are comparable to the direct estimates from galaxy counts and galaxy peculiar motions.

In this model for the mass distribution, and with an Einstein-de Sitter cosmological model, the CBR quadrupole anisotropy produced by the gravitational potential fluctuations at the Hubble distance (the Sachs-Wolfe effect) is

(5) |

an order of magnitude above the COBE quadrupole anisotropy Mather and Vittorio discuss in these Proceedings.

The lesson is that our model gives a not unreasonable picture for the mass fluctuations on scales 100 Mpc, but it significantly overestimates fluctuations on larger scales. An equivalent way to put it is that the power spectrum of mass fluctuations has to decrease with increasing scale at wavelengths 100 Mpc. This was predicted by the inflation theory incorporated in the adiabatic Cold Dark Matter cosmogony (Frenk 1991 and references therein), an impressive success. One should bear in mind, however, that the wanted decrease in the mass fluctuation spectrum at large separations appears in other scenarios, such as cosmic strings and textures (Turok 1991).

Equation (5) does assume general relativity theory, and it would be very
interesting
to have a check on this constraint from a direct measurement of the
fluctuations in
the distribution of sources on the scale of the Hubble length. The
constraint says the
fluctuations are less than one part in 10^{4}, however, which
will not be easy to get at.

A related issue is the interpretation of the CBR dipole anisotropy. The COBE measurements described by Mather show that the CBR anisotropy is very close to a pure dipole in thermodynamic temperature. This is consistent with the usual assumption that the dipole is caused by our peculiar motion relative to the general Hubble expansion of the universe, that is, that the rest frame defined by the CBR agrees with the rest frame defined by the mean motion of the matter within the Hubble length. Detection of the departure from a pure dipole due to the second-order Doppler shift would be an important check, but I gather not yet within reach.

A detection of the mass fluctuations whose gravitational pull drive our
motion relative
to the CBR would be a key test of the velocity interpretation. One can
estimate the
peculiar gravitational acceleration caused by the galaxies taken to be
tracers of the mass
within distance *R*. If the direction and magnitude of the
acceleration are unchanged
as *R* is increased, and agree with our motion relative to the CBR
for a sensible value
of the mass per tracer galaxy, one has reason to believe the mass
fluctuations whose
gravity produced our peculiar motion are detected with *R*. In a
recent application,
Strauss et al. (1992)
find that the direction of the gravitational acceleration of IRAS
galaxies taken to be mass tracers converges to within 20° of the
CBR dipole direction at
*H*_{0}*R* ~ 3000 km s^{-1}, and stays within
that angular distance as one adds IRAS galaxies
out to *H*_{0}*R* ~ 20,000 km s^{-1}. This
suggests the source of the motion of our Local Group
of galaxies is within 3000 km s^{-1} distance. If that were so,
however, one would expect
to find an appreciably smaller mean peculiar velocity in the average
over the material
within a sphere of radius larger than 3000 km s^{-1}, contrary
to the present evidence.
The mean peculiar velocity relative to the CBR for the galaxies within
6000 km s^{-1} is
about 400 km s^{-1}, directed ~ 40° away from the motion of
the Local Group relative to the CBR
(Faber and Burstein
1988;
Mould et al. 1993).
The Postman-Lauer
(1993)
mean peculiar flow for Abell clusters at *R* < 15,000 km
s^{-1} has a similar magnitude
but swings to ~ 90° from the motion of the Local Group relative to
the CBR. In short,
our picture for the large-scale peculiar velocity field and the mass
density fluctuations that may have caused it remains confused.

It would be very helpful to have a measurement of the XRB dipole. If the CBR dipole were the result of our peculiar motion caused by the gravitational pull of mass fluctuations well within the Hubble length then we would expect to see a parallel dipole anisotropy in the XRB. If on the other hand the CBR dipole were due to a very smooth gradient in the primeval entropy per unit mass the X-ray dipole from sources within the Hubble length would not be expected to stand out above the other low order multipole moments of the XRB, or to be aligned with the CBR dipole. I gather the experts are not yet willing to make a definite pronouncement on this important test.

**2.2 The Mean Optical Luminosity Density**

My conclusion is that we have a good case for Einstein's cosmological
principle,
that the universe is close to homogeneous and isotropic in the
large-scale average. ^{(1)}
This means we can consider the mean mass density of the universe, and the mean
density of radiation as a function of wavelength. Of particular interest
is the quantity
de Vaucouleurs (1949)
considered, the density of starlight, as a check on our inventory
of luminous objects in the universe
(Peebles and
Partridge 1967).
This is motivated
by the notorious mass problem in cosmology, that the starlight in the
known galaxies,
assigned a mass-to-light ratio characteristic of familiar star
populations, yields a mass
insufficient to account for the motions of galaxies within groups and
clusters, and two
orders of magnitude below the Einstein-de Sitter density usually
considered to be the
most reasonable possibility. The commonly discussed interpretation is
that the mass of
the universe is dominated by nonbaryonic matter, perhaps massive
neutrinos or some
other weakly interacting particle, maybe something even more exotic. But
it behooves
us to consider the possibility that our sums have missed luminous
objects too compact
to be distinguished from stars, or too low in surface density to be seen
against the light
of the night sky. We have a test, from the comparison of the mean
surface brightness
contributed by the known galaxies and the measured extragalactic sky
brightness (after
correction for airglow, the zodiacal light, and the diffuse light of the
Milky Way).

Tyson's (1990) value for the mean surface brightness of the sky at 4500 Å from the sum of observed galaxies is

(6) |

This is de Vaucouleurs' (1949) computation; it is impressive that the result has not changed by more than a factor of about two. The equivalent mean luminosity density is

(7) |

Mattila's (1990) survey indicates the measured extragalactic sky brightness satisfies

(8) |

The difference between this and the surface brightness contributed by ordinary galaxies (eq. [6]) leaves room for new classes of objects with luminosity density an order of magnitude greater than that of the known galaxies. With conventional mass-to-light ratios for star populations this leaves the bound on the mass density of luminous material an order of magnitude below the Einstein de-Sitter value.

The relation between equations (6) and (7) is not sensitive to the cosmological model, because the cosmological redshift suppresses the contribution from ordinary galaxies at the Hubble distance. But we see that there is considerable room for galaxy evolution that could have produced ultraviolet emission which has been redshifted into the visible band. Advances in the art of measuring the extragalactic optical and near infrared background, as discussed here by Hauser, thus will be followed with great interest.

^{1} A lesson from the inflation cosmology
is that the universe at great distances could
be very different from what we see. The more careful statement thus is
that mass
density variations across the Hubble length have to be less than about
one part in 10^{4}.
The issue seldom debated is whether Einstein was right for the right
reason, and the universe really is close to uniform everywhere.
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