Hyperextended inflation: Dynamics and constraints

In order to reconcile the notion of extended inflation with limits from measurements of the microwave background anisotropies, several authors have considered extending the gravitational action beyond the Brans-Dicke one used in the original extended inflation model. The most popular such extension has become known as hyperextended inflation, and has appeared in two different forms. In each case one allows the Brans-Dicke parameter $\omega$ to depend on the value of the gravitational scalar $\Phi$. We numerically investigate these models and enhance our previous investigation of the extended inflation void spectrum in order to constrain these models. We find each model to be more stringently constrained than previously thought. In a model where one simply writes $\omega \left(\Phi \right)={\omega }_0+{\omega }_m{\Phi }^m$, we find that the power $m$ must be at least 5 to allow the microwave background bounds to be satisfied. An alternative model where a scalar $\phi$ is coupled to gravity via a term $f\left(\phi \right)\mathcal{R}$ requires terms of even order up to sixth order in $f$ for a successful implementation without fine-tuning the initial conditions. Each model is even more strongly constrained if one requires density fluctuations from hyperextended inflation to seed large-scale-structure formation, and in neither case can one readily produce a spectrum of bubble sizes of astrophysical interest.