**1.4. The Value of
_{0}**

The luminous matter itself accounts for only
_{lum} ~
0.005*h*^{2}
(Faber and Gallagher,
1979).
This was discovered long ago to be insufficient to
account for either the flat rotation curves of disk galaxies (the dark
massive halo problem,
Rubin (1988)),
or for the velocity dispersions
of groups and clusters of galaxies (the virial mass discrepancy problem,
Zwicky (1933)).
It later became apparent from cosmic
nucleosynthesis arguments that the baryonic density of the universe
was substantially higher than the density inferred from the luminous
material. There is "dark (nonluminous) baryonic material" in some form
or other, perhaps warm gas, or even very low luminosity stars. The
amount of baryonic dark matter inferred from nucleosynthesis appears
to be just about enough to explain the cluster virial mass discrepancy
problem in most clusters of galaxies. However, this would not be
sufficient to make
_{0} =
1. Whether we care about
_{0}
1 is a central
issue of cosmology, so I shall discuss briefly the various ways we get
at _{0} and
why we should strive to get
_{0} = 1.

But first a word of caution which we will continually return to
throughout these lectures. Determinations of
_{0} are
frustrated by the fact that
_{0}
describes the quantity of gravitating matter in the
universe, whereas we only see the luminous material which is but a
fraction of the total mass density. If the luminosity density were
everywhere proportional to the mass density, this would not prove a
problem since it would only be necessary to discover what the sealing
factor is. However, it is evident that the mass and light are
distributed differently on different scales and some other hypothesis
is needed.

The simplest hypothesis of this kind is that the *fluctuations* in
mass density about the mean are proportional to the *fluctuations* in
light density. The constant of proportionality is referred to as the
*biasing parameter* and it is denoted by the symbol *b*. We shall
encounter this frequently in what follows. Note that the constancy of
the biasing parameter is merely a simplifying hypothesis, the actual
situation could be far more complicated.

**1.4.1. The Deceleration Parameter
q_{0}**

The central task of classical cosmology was to determine the cosmic
expansion rate, *H*_{0} and the *deceleration
parameter* *q*_{0}:

(19) |

*H*_{0} was seen as the slope of the velocity-distance
relationship and *q*_{0}
as the deviation from the linear Hubble law, its curvature, due to the
gravitational deceleration of the cosmic expansion.

Note that by virtue of equation (5) and the definition of
_{0}
(equations (11) and (12))

(20) |

with =
/
3*H*_{0}^{2}. This relationship between
_{0}
and *q*_{0} holds only as long
as equations (5, 6, 7) or (14) are valid; that is, provided there is
no cosmic pressure. The Einstein de Sitter universe has
*q*_{0} = 1/2 (since
= 0 and
_{0} = 1).

We can calculate the relationships between the redshift of a galaxy
and various observed properties such as brightness, look-back time and
surface brightness. For example, the *look-back time*, measuring the
time elapsed since photon was emitted at time *t*_{E}, to
redshift *z* is

(21) |

for small *z*.

The apparent brightness *l* is related to the intrinsic luminosity
*L* by

(22) |

and for not too distant galaxies (*z* < 1), this simplifies to

(23) |

This expression is also exact in the limits *q*_{0} = 0 and
*q*_{0} = 1. The
first terms are simply the standard *r*^{-2} inverse square
law, the correction due to the *q*_{0} term is due to the
deceleration of the expansion. In astronomical units this is

(24) |

where *m* is the apparent magnitude of a galaxy of absolute magnitude
*M* seen at a redshift *z*. (Technically, these are the
luminosities or
magnitudes integrated over the whole spectrum of the emitted light. If
the measurements are done in a restricted spectral band, then other
terms come into this relationship, these are the so-called
K-correction terms). This expresses the Hubble Law directly in terms
of a magnitude-redshift relationship.

Note that any intrinsic evolution of the quantity *L* (or the absolute
magnitude *M*) will introduce non-geometric effects into the
relationship and so confuse the determination of
*q*_{0}. We can
approximate this by assuming that the luminosity evolves as

(25) |

when expressed as a function of look-back time *t* -
*t*_{0}. Relating look-back time to redshift then yields

(26) |

showing an extra linear dependence on *z*. Thus if this relationship is
used to measure *q*_{0}, the (unknown) evolutionary
correction biases *q*_{0}
downward by
*H*_{0}^{-1}.

In the small *z* limit, we can also calculate the number of galaxies
*N(m)* we would see in a galaxy survey down to apparent magnitude
*m*, again under the assumption that there are no evolutionary
effects. This is the classical *number-magnitude relationship*.

In the pre-1965 days of cosmology the central issue in cosmology was
the values of *H*_{0} and *q*_{0}. Cosmology
was simply "a search for two
numbers". Today, that view has changed. Cosmology is properly a branch
of physics and the values of these two parameters are regarded simply
as important parameters that observations will eventually determine to
characterize our Universe.

Unfortunately, the program of measuring the curvature of the Hubble
Law directly has not provided any strong constraints on
*q*_{0}. This is
largely because the curvature of the relationship is influenced by non
geometric effects (galaxy luminosities evolve with time in an unknown
way) and because there is considerable scatter in the
magnitude-redshift diagram. Indeed, the tendency today is to use the
Hubble diagram and the number-magnitude relationship together to
determine the evolutionary history of galaxies! (See
Guideroni and
Rocca-Volmerange, 1990;
Rocca-Volmerange and
Guideroni, 1990).
As we shall see below, there is a possibility that this will yield limits on
_{0} and
as a by-product
since faint galaxy counts are sensitive to
_{0}.

**1.4.2. The classical approach again**

In order to discriminate cosmological models, the magnitude redshift
relationship needs a large sample of redshifts out to at least *z*
= 0.5 and preferably as far beyond *z* = 1 as possible.
Loh and Spillar (1986)
have used a galaxy survey to determine approximate redshifts for ~
1000 galaxies out to a redshift of *z* ~ 1. On the basis of this
survey,
they look at the redshift-volume relationship and conclude that
_{0} =
0.9 ± 0.3 if
= 0.
Caditz and Petrosian (1989)
argue that the luminosity function history assumed by Loh and Spillar is not
consistent with their data. Taking this into account, Caditz and
Petrosian derive
_{0}
0.2 with considerable
uncertainty due to such things as incompleteness of the sample.
Yoshii and Takahara (1989)
make a detailed model for the luminosity evolution based on merger
driven evolution and discuss the problems associated with such methods
of getting at
_{0}.

The number magnitude relationship provides an alternative probe of
cosmological models and galaxy evolution and has generated a great
deal of interest since we can now survey galaxies down to extremely
faint magnitudes in many wavebands. In recent years we have seen faint
galaxy counts by
Tyson (1988)
and by Jones et al. (1991)
and Metcalf et al. (1991).
The latter surveys penetrate to *B*-magnitudes
*B* < 25. The interpretation of such counts and the galaxy
evolution models that are used have been discussed by
Koo (1990) and by
Guideroni and
Rocca-Volmerange (1990).
It seems that the present data in the R and B
bands can be largely understood in terms of current models of galactic
evolution. However,
Cowie (1991)
has recently presented some infrared
counts of galaxies which confuse the situation somewhat by appearing
to demand a non-zero cosmological constant! This is also the
conclusion of the analysis of counts by
Fugikita et al. (1990).

Cosmic nucleosynthesis sets strong bounds on the amount of baryonic material in the Universe (Boesgard and Steigman, 1985; Pagel 1991a, b). Standard Big Bang nucleosynthesis implies that

(27) |

where _{B}
is the contribution of baryons to the total mass density.
(See chapter 4 of the
Kolb and Turner (1990)
book for an excellent
discussion of this). There is a need already here to have ten times as
much mass in the baryonic dark matter as is accounted by the luminous
mass in galaxies.

The nucleosynthesis question is fully discussed elsewhere in this volume. There is a couple point that should be emphasised here. The low baryonic density implied by nucleosynthesis causes a problem in the hydrogen-helium cooling of the pregalactic gas: the density may simply be too low. This is a point about star formation, but it does have a bearing on what we observe on the largest scales since we can only observe what is luminous.

**1.4.4.
_{0} from
Hubble flow deviations**

The large scale peculiar motions of galaxies are clearly related to
the density inhomogeneities, since it is those inhomogeneities that
give rise to the peculiar motions.
is involved in that
relationship
and so in principle it could be estimated by comparing density
excursions with peculiar velocities. The problem arises because we
cannot directly observe the fluctuations in mass density, but only the
fluctuations in luminosity density. Another parameter relating mass
density fluctuations to luminosity density fluctuations comes into the
game. This parameter, *b*, is called the *bias parameter* and
it might
depend on the location, the morphological type of the galaxies
involved, or any number of other things. In this spirit of ignorance
the simplest assumption we can make is that *b* is a universal
constant. Then we can in principle determine the combination
_{0}
*b*^{-5/3}. We
shall have more to say about this below (see sections 2.1.3 and 4.2.1).

On the assumption that light traces mass (*b* = 1), most dynamical
determinations of
_{0}
converge on
0.1 < _{0}
< 0.3
(Peebles, 1987;
Shanks et al., 1989).
Stavely-Smith and Davies
(1989)
report _{0} =
0.08 ± 0.05. (The latter authors remark that some `biasing' is
demanded by their data, bringing
_{0} up to
at least 0.25.)

More recently, deep redshift surveys of galaxy samples drawn from
the IRAS catalog have provided another route to
_{0}. The
limits from
these surveys involve another parameter, *b*, the biasing parameter,
whose value is largely unknown (and may not even be a constant):

with large error bars. More will be said about this approach later on (section 2.3.2).