First Order In ation in

I give a general formulation of the constraints on models of in ation ended by a rst order phase transition arising from the requirement that they do not produce too many large (observable) true vacuum voids { the `big bubble problem'. It is shown that this constraint can be satis ed by a simple model in Einstein gravity { a variant of `hybrid' or `false vacuum' in ation. The idea that in ation could be ended by a rst order phase transition is often dismissed as too problematic following Guth's original trouble in nding a graceful exit from the false vacuum [1]. The scalar eld, , whose self-interaction potential acts like a cosmological constant driving the expansion, is trapped in a local potential minimum so cannot evolve classically. A xed tunnelling rate to the true vacuum in a de Sitter universe leads to a time invariant state. One expects the phase transition to complete either at once or not at all. First order in ation was revived in the guise of extended in ation [2] which invoked an extended gravitational lagrangian. In Brans-Dicke gravity [3], where Newton's constant is replaced by a eld , the expansion rate decreases with time as the Planck mass, mPl, grows during in ation allowing the phase transition to eventually complete. However this too was shown to produce too many large true vacuum voids in any Brans-Dicke model that was su ciently close to general relativity to obey other observational limits [4, 5] { the `big bubble problem'. Even when the Brans-Dicke parameter ! is allowed to vary in time [6] it is di cult to achieve an acceptable model as the same variation of the Hubble rate required to produce a non-scale-invariant bubble spectrum also produces a non-scale-invariant conventional perturbation spectrum incompatible with models of large-scale structure formation [7]. My aim here is to show that these problems need not necessarily be present if it is the changing shape of the potential that varies during in ation rather than (or in addition to) the Hubble rate. Soon after La and Steinhardt's work it was pointed out by both Linde [8] and Adams & Freese [9] that a graceful exit from rst order in ation can also be achieved within general relativity. All that is required is the presence of a second scalar eld which can interact with and slow-rolls while the eld is trapped in the false vacuum. It is not obvious whether these models based on two interacting elds in general relativity should fair any better than the extended gravity models in avoiding the `big bubble problem'. Indeed, as far as I am aware, this issue has never been seriously addressed before. Here I will quantify the big bubble constraints on the model proposed originally by Linde [8] and give a more general formulation of the big bubble constraint in rst order in ation. For a wide range of parameter space, models of rst order in ation in general relativity easily satisfy these constraints while producing near-scale-invariant perturbation spectra. These results are based on work presented in [10]. Talk presented at \The Birth of the Universe" workshop, Rome, May 1994 Indeed this clearly must be the case at some level as La and Steinhardt's Brans-Dicke model is itself conformally related to general relativity plus another scalar eld.