1. INTRODUCTION

We were asked to debate the value of , the fundamental energy density parameter of cosmology, and in particular its mass component, _m. Is the universe flat and marginally bound with _m = 1 in accordance with the simplest cosmological model? Is _m clearly smaller than unity as seems to be indicated by several observations? Unfortunately, we cannot provide a clear answer at this point because there is conflicting evidence. Entertaining the audience with our biased views on the subject might not be very constructive. Instead, it may be more interesting to lay out the various methods used to measure _m, mention new developments and current estimates, and focus on the promising prospects versus the associated difficulties. In the critical discussion that follows we try to shed light on the nature of the uncertainties that may be responsible for the current span of estimates for _m.

We divide the methods into the following four classes:

• Global measures. Based on properties of space-time that constrain combinations of _m and the other cosmological parameters (, H₀, t₀).

• Virialized Systems. Methods based on nonlinear dynamics within galaxies and clusters on comoving scales 1 - 10 h^-1Mpc.

• Large-scale structure. Measurements based on mildly-nonlinear gravitational dynamics of fluctuations on scales 10 - 100 h^-1Mpc of superclusters and voids, in particular cosmic flows.

• Growth rate of fluctuations. Comparisons of present day structure with fluctuations at the last scattering of the cosmic microwave background (CMB) or with high redshift objects of the young universe.

The methods and current estimates are discussed below and summarized in Figure 1 and Table 1. The estimates based on virialized objects typically yield low values of _m ~ 0.2 - 0.3. The global measures, large-scale structure and cosmic flows typically indicate higher values of _m ~ 0.4 - 1.