Boundary-Driven Cosmic Expansion: A Purely Geometric Theory of Inflation

Abstract

Cosmic inflation remains the leading paradigm for explaining the earliest stage of cosmological evolution, yet its underlying physical mechanism continues to motivate alternative theoretical descriptions. In this work, we introduce a purely geometric framework in which cosmic expansion emerges from the collective evolution of an infinite ensemble of independently expanding universes represented as three-dimensional spheres embedded in a common Euclidean space. Unlike conventional inflationary models, the proposed formulation introduces neither scalar fields, spacetime curvature, vacuum energy, nor modified gravitational dynamics. Instead, expansion is governed exclusively by the availability of unconstrained boundary. The central concept of the theory is the
free boundary fraction, a dimensionless geometric functional that measures the proportion of each universe’s boundary remaining free from overlap with neighbouring universes. Beginning from a small set of geometric axioms, we derive a closed boundary-driven dynamical system, establish the existence, boundedness, monotonicity, and limiting properties of the free boundary functional, and show that the resulting constitutive law naturally yields an initial phase of maximum boundary-driven expansion followed by a progressive, self-regulated reduction as collective boundary interactions accumulate. The resulting dynamics provide a mathematically self-contained geometric mechanism that qualitatively reproduces several characteristic features commonly associated with the inflationary epoch while remaining fundamentally distinct from conventional exponential inflation. Rather than attributing primordial expansion to additional physical fields or phenomenological potentials, the proposed framework demonstrates that collective boundary geometry alone is sufficient to generate heterogeneous expansion histories and intrinsic self-regulation. This work establishes a new geometric perspective on early-universe evolution and provides a mathematical foundation for future theoretical development, numerical simulation, and observational investigation.