Table 1 presents a summary of several different
techniques for
measuring the matter density of the Universe. These techniques have
been developed over a wide range of scales, from galaxy (~ 100-200
kpc), through cluster (Mpc), on up to more global scales (redshifts of
a few). Excellent, recent reviews on determinations of
can be found in
Dekel, Burstein &
White (1997)
and Bahcall, Lubin
& Dorman 1995
and references therein. The first part of
the table lists _{m}
determinations that are independent of
_{}; the second part lists _{m}
determinations that are not independent of _{}; and
the third part of the table lists _{}
determinations. In
addition to listing the physical basis of the method, types of object
under study, and values of _{m} plus an estimated uncertainty,
Table 1 makes explicit some of the assumptions
that underlie each of
these techniques. Although in many cases, 95% confidence limits are
quoted, these estimates must ultimately be evaluated in the context of
the validity of their underlying assumptions. It is non-trivial to
assign a quantitative uncertainty in many cases, but in fact
systematic effects may be the dominant source of uncertainty. Several
of these assumptions and uncertainties are discussed further below.
They include, for example, diverse assumptions about mass tracing
light, mass-to-light ratios being constant, clusters being
representative of the Universe, clumping of X-ray gas, non-evolution
of type Ia supernovae, and the non-evolution of elliptical galaxies.
For methods that operate over very large scales (gravitational lensing
and type Ia supernovae), assumptions about _{} or
_{total} are currently
required to place limits on
_{m}.

Sample | Method | Scale | Assumptions | _{m}
| Error |

_{} Independent Methods
| |||||

Galaxies | dyn. M/L ratio | 100 kpc | galaxies representative | ~ 0.1 | |

M/L constant | |||||

Clusters | dyn. M/L ratio | < few Mpc | clusters representative | ~ 0.2 | |

M/L constant | |||||

Clusters | X-ray M/L ratio | < few Mpc | hydrostatic eqm | ~ 0.2 | |

Clusters | baryon fraction | clusters representative | 0.3-0.5 | ||

no clumping | |||||

Clusters | morphology | model dept. | > 0.3 | ||

Local Group | Least action principle | 1 Mpc | LG representative | ~ 0.15 | |

no external torques | |||||

model uniqueness | |||||

Galaxies | Virial Theorem | 1-300 Mpc | mass-indept. biasing | 0.2-0.4 | |

(pairwise velocities) | point masses | ||||

Galaxies | Peculiar velocities | 100 Mpc | biasing | > 0.3 | 95% |

Sample | Method | Scale | Assumptions | _{m}
| Error |

_{} Dependent Methods
| |||||

Type Ia SNae | Hubble diagram | z < 0.5 | _{} = 0 | 0.88 | 90% |

no evolution effects | |||||

_{tot} = 1 | > 0.49 | 95% | |||

Lensed QSO's | lensing statistics | global | _{} = 0 | > 0.15 | 90% |

dark matter distrib. | |||||

slow galaxy evolution | |||||

dust small effect | |||||

6 lenses | strong lensing | global | _{} =
0 | low | |

model dependent | |||||

CMB | multipole analysis | global | CDM | 0.3-1.5 | |

Sample | Method | Scale | Assumptions | _{} | Error |

Type Ia SNae | z < 0.5 | _{tot} = 1 | < 0.51 | 95% | |

Lensed QSO's | lensing statistics | global | _{tot} = 1 | < 0.66 | 95% |

6 lenses | strong lensing | global | _{tot} = 1
| < 0.9 | 95% |

H_{0} t_{0} | age discrepancy | 100 Mpc | H_{0} > 65 | > 0.5 | 66% |

t_{0} > 13 Gyr
| |||||

Since lower values of the matter density tend to be measured on
smaller spatial scales, it has given rise to the suspicion that the
true, global value of _{0} must be measured on scales beyond
even those of large clusters, i.e., scales of greater than
~ 100 Mpc (e.g.,
Dekel 1994).
In that way, one might
reconcile the low values of _{m} inferred locally with a
spatially flat Universe. However, recent studies
(Bahcall, Lubin &
Dorman 1995)
suggest that the *M / L* ratios of galaxies do not continue
to grow beyond a scale size of about ~ 200 kpc (corresponding to
the sizes of large halos of individual galaxies). In their Jeans
analysis of the dynamics of 16 rich clusters,
Carlberg et
al. (1997)
also see no further trend with scale. Hence, currently the
observational evidence does not indicate that measurements of
_{m} on cluster size
scales are biased to lower values than the true global value.

A brief description of several techniques for measuring the matter density is given below. These methods are discussed in the context of both their strengths and weaknesses, paying particular attention to the underlying assumptions. An excellent and more complete review on this topic is given by Dekel, Burstein & White (1997); also see Trimble (1987).