2. _m - The Matter Density

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.

Table 1. SUMMARY OF _m and DETERMINATIONS

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%
H0 t0	age discrepancy	100 Mpc	H0 > 65	> 0.5	66%
			t0 > 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).