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:
0.8_{m} -
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).