  Conformal embedding of closed-universe models in a de Sitter background
suggests a quantisation condition on the available conformal time. This
condition implies that the universe is closed at no greater than the 10% level.
When a massive scalar field is introduced to drive an inflationary phase this
figure is reduced to closure at nearer the 1% level. In order to enforce the
constraint on the available conformal time we need to consider conditions in
the universe before the onset of inflation. A formal series around the initial
singularity is constructed, which rests on a pair of dimensionless,
scale-invariant parameters. For physically-acceptable models we find that both
parameters are of order unity, so no fine tuning is required, except in the
mass of the scalar field. For typical values of the input parameters we predict
the observed values of the cosmological parameters, including the magnitude of
the cosmological constant. The model produces a very good fit to the most
recent CMBR data, predicting a low-l fall-off in the CMB power spectrum
consistent with that observed by WMAP.

Astrophysics;General Relativity and Quantum Cosmology