Conditions for successful extended inflation.

We investigate, in a model-independent way, the conditions required to obtain a satisfactory model of extended in ation in which in ation is brought to an end by a rst-order phase transition. The constraints are that the correct present strength of the gravitational coupling is obtained, that the present theory of gravity is satisfactorily close to general relativity, that the perturbation spectra from in ation are compatible with large scale structure observations and that the bubble spectrum produced at the phase transition doesn't con ict with the observed level of microwave background anisotropies. We demonstrate that these constraints can be summarized in terms of the behaviour in the conformally related Einstein frame, and can be compactly illustrated graphically. We con rm the failure of existing models including the original extended in ation model, and construct models, albeit rather contrived ones, which satisfy all existing constraints.