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.

Anthony Lasenby and Chris Doran, **Closed Universes, de Sitter Space and Inflation, **Phys.Rev. D **71**, 063502 (2005)