  We present a new approach to constructing inflationary models in closed
universes. 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. The primordial curvature spectrum predicts the low-l fall-off
in the CMB power spectrum observed by WMAP. The spectrum also predicts a
fall-off in the matter spectrum at high k, relative to a power law. A further
prediction of our model is a large tensor mode component, with r~0.2.
