Researchers tested whether a local bump in the inflaton field can seed cosmic inflation when the rest of space is not inflating. Using numerical relativity in spherically symmetric, asymptotically flat spacetimes, they evolved initial field bumps and the full gravitational constraints to see which ones grow into inflation and which do not. Their main result echoes earlier work by Goldwirth and Piran: a fluctuation must span several times the characteristic inflationary length scale, (G ρ_infl)^{-1/2}, for inflation to succeed.

To set up the problem the authors pick a simple inflation model with a quadratic potential V(φ)=½ m^2 φ^2 and a Gaussian initial field profile φ(r)=φ0 exp(−r^2/σ^2). They choose vanishing initial field momentum and impose asymptotically flat boundary conditions so the spacetime becomes ordinary (non‑inflating) far away. The initial gravitational data must satisfy Einstein’s constraint equations, so they solve for the conformal factor of the spatial metric and the extrinsic curvature numerically. To explore different ways that the same energy can source spatial curvature, they introduce a parameter ε that splits the potential energy between intrinsic curvature (shape of the spatial slice) and extrinsic curvature (how the slice sits in spacetime).

Because the constraint equations are nonlinear, the authors find families of allowed initial data only up to a critical overdensity. Below that critical point there are two branches of solutions, labeled “weak‑field” and “strong‑field,” that join at the maximum allowed overdensity. Working in the asymptotically flat setting lets them follow the evolution through strong gravity events such as black‑hole formation and the pinch‑off of small inflating regions, sometimes described as baby universes.