3. THE DENSITY PARAMETER

3.1.1. Luminosity Density x < M/L >

The mass density can be obtained by multiplying the luminosity density with galaxy's average mass to light ratio < M/L >. The local luminosity density, evaluated by integrating the luminosity function, is reasonably well converged to L_B = (2.0 ± 0.4) x 10⁸ h L Mpc^-3 from many observations. The M/L_B of galaxies generally increases with the scale. When the mass is integrated to 100 kpc, a typical M/L_B is about (100-200)h in solar units, and it may still increase outward (e.g., Faber & Gallagher 1979; Little & Tremaine 1987; Kochanek 1996; Bahcall et al. 1995; Zaritsky et al. 1997). The virial radius in a spherical collapse model is r = 0.13 Mpc ^-0.15 [M / 10¹² M]^1/2_{< 100 kpc}. If the dark matter distribution is isothermal within the virial radius, the value of M/L_B inside the virial radius is (150-400)h for L* galaxies. This is about the value of M/L_B for groups and clusters, (150-500)h. Multiplying the two values we get = 0.20 x 2^±1. See also Fukugita, Hogan & Peebles (1998) for variants of this argument.

Carlberg et al. (1996, 1997a) tried to make the argument more quantitative using their cluster sample and a built-in field galaxy sample. They estimated M / L_r (210 ± 60)h for field galaxies from the cluster value (289 ± 50)h. Their luminosity density of field galaxies is L_r = (1.7 ± 0.2) x 10⁸ h L Mpc^-3, and therefore ₀ = 0.19 ± 0.06. Note that M / L_B 1.4 x M / L_r in solar units for the respective pass bands.

The important assumption for these calculations is the absence of copious matter outside the clusters. This is a question difficult to answer, but the observation of weak lensing around the clusters indicate that the distributions of dark mass and galaxies are similar at least in the vicinity of clusters (Tyson & Fischer 1995; Squires et al. 1996).

Some attempts have also been made to estimate the mass on a supercluster scale. Small et al. (1998) inferred M / L_B 560h for the Corona Borearis supercluster, by applying the virial theorem (inspired by an N body simulation). On the other hand, Kaiser et al. (1998) estimated M / L_B 250 from a mesurement of the gravitational shear of weak lensing caused by a supercluster MS0302+17 ⁽³⁾; the result is not well convergent, but it seems unlikely that is larger than 0.5.

³ They suggest 0.04 on the basis that only early-type galaxy population traces the mass distribution and the luminosity density is multiplied by the fraction of early-type galaxies (20%). It seems possible that late type galaxies reside in low density regions, causing only a small shear, which is buried in noise, and escaped from the measurement. Back.