One of the problems of standard cosmology is the near flatness of the Universe. For the present total energy density in the Universe to be within one order of magnitude of the critical density, i.e. total ~ 1 today, at the Planck time (10-43 s) the value of total could not deviate from unity by more than 10-60. This is one of the problems that can be solved by assuming the existence of an inflationary period in the very early Universe. The simplest models of inflation predict that total should in practice be equal to unity today, i.e. the Universe to be spatially flat. Therefore, these models would receive strong support if it was shown that presently total = 1. Note that because there are inflationary models which predict the Universe not to be presently flat, then proof that total is different from unity today would not disprove the inflationary paradigm, but simply be evidence that if inflation indeed occurred than it did so in a more complicated fashion than it is generally assumed.
Recently, tantalising evidence has appeared that seem to indicate that the Universe is flat. There are several methods that directly or indirectly probe the geometry of the Universe. Those which presently provide the cleanest constraints on the geometry are the position of the first acoustic peak on the CMBR temperature anisotropy angular power spectrum and the magnitude-distance relation for Supernovae type Ia. Two other methods provide limits essentially on the total amount of non-relativistic matter in the Universe, m, the evolution with redshift of the abundance of rich galaxy clusters and deviations from Gaussianity measured either through the galaxy or the cluster velocity fields. Finally, the number of observed gravitational lensed high-redshift objects puts limits mainly on the possible contribution to the total matter density by a classical cosmological constant, . Given that most analysis assume only these two possible contributions to the total energy density in the Universe, non-relativistic matter, p 0, and a cosmological constant, p = - , I will not consider other eventual contributions with different equations of state, for example arising from an evolving scalar field.
Let me then summarise what we presently know about total, m and . This list does not pretend to be exhaustive, as some of the results cannot be expressed through a simple function of m and . Some of the limits were determined by (69) from results in the references given.
From the angular scale of the first acoustic peak in the CMBR anisotropy spectrum, under the assumption of Gaussian adiabatic initial perturbations: total > 0.85 (40); total = 1.15 ± 0.20 (50); total = 0.90 ± 0.15 (2); total = 1.11 ± 0.07 (43). From the magnitude-distance relation for Supernovae type Ia: 0.8m - 0.6 = - 0.2 ± 0.1 (68). From the cluster abundance evolution with redshift, under the assumption of Gaussian initial perturbations: m = 0.2+0.3-0.1 (1); m = 0.45 ± 0.20 (26); m > 0.3 (79); m = 0.45 ± 0.10 (33); m = 0.75 ± 0.20 (5). From the cosmic velocity field, m > 0.3 (21) at more than 95 per cent confidence from the amplitude of diverging flows of galaxies from voids (22) and from the skewness of the velocity field assuming the initial density distribution to be Gaussian (64). From the gravitational lensing of objects at high-redshift: = 0.70 ± 0.16 (14); m > 0.26 (27); m < 0.62 (16); -1.78 < - m < 0.27 (35).
The quoted results indicate that the situation is still too confusing for one to be able to say with any degree of certainty which is the value of either m or . However, it seems clear that the best explanation for the combined data is an Universe which is spatially flat (69, 70).