Wave function of the Universe as a sum over eventually inflating universes

We consider a proposal to define the wave function of Universe as sum over spacetimes that eventually inflate. In minisuperspace model, we explicitly show simple family initial conditions, parametrized by positive real number $a_0$, can be imposed realise this prescription. The resulting is found proportional Hartle-Hawking and its dependence on $a_0$ only through an overall phase factor. Motivated observation, ask whether it possible analytically extend extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining form function. use condition for convergence path integral recent theorem due Kontsevich Segal, further Witten, find $\bar{\mathcal{D}}$. Interestingly, special point boundary recovers exact no-boundary Following that, our prescription leads quantum states perturbations, which give rise scale-invariant power spectra. If demand, extra ingredient prescription, matching at "no-boundary point" $\bar{\mathcal{D}}$, converge unique state perturbations.