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 CMB ( cosmic microwave background ) radiation data .
The primordial curvature spectrum provides a possible explanation for the low- \ell fall-off in the CMB power spectrum observed by WMAP ( Wilkinson anisotropy probe ) .
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 \approx 0.2 .