A Spherical Boundary Cosmology: Junction Conditions, Effective Dynamics, and Observable Signatures

We construct a mathematically rigorous cosmological model in which the observable Universe is modeled as a finite, spherically symmetric fluid region bounded by a timelike hypersurface \( \Sigma \), embedded in an exterior Schwarzschild--de Sitter vacuum. The boundary dynamics are derived from the Darmois--Israel junction conditions, yielding an effective equation of motion \( \ddot{R} = \alpha/R^{2} + \mathcal{O}(R^{-3}) \), where \( \alpha \) is an effective parameter encoding the surface stress-energy of the boundary layer and the exterior cosmological constant. We derive the shell conservation law for a general equation of state \( p = w\sigma \), decompose \( \alpha \) into its physical constituents (\( \alpha_{\rm surface} \), \( \alpha_{\rm mass} \), \( \alpha_\Lambda \)), and present the exact implicit solution. A complete linear stability analysis is performed, proving that radial perturbations are stable if and only if \( \alpha > 0 \). We derive the luminosity-distance relation and demonstrate qualitative consistency with Type Ia supernova observations. The model is presented as a mathematically consistent toy model for exploring boundary effects in general relativistic cosmology, with the exterior cosmological constant \( \Lambda \) remaining an essential ingredient. Future work will focus on full relativistic perturbation theory with junction boundary conditions and rigorous observational constraints via Markov Chain Monte Carlo methods.