Generalized Cosmic Energy Distribution: From Homogeneous FLRW Spacetimes to Non-Homogeneous Frameworks

This work presents a unified and first-principles derivation of the cosmic energy distribution across all physical scales of the universe from homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology to fully general non-homogeneous spacetimes. Starting from the Einstein–Hilbert action, the paper systematically derives the Einstein field equations and constructs the complete cosmic energy budget directly from microscopic physics. The homogeneous FLRW framework is developed rigorously, with energy components derived from fundamental principles including particle number conservation, photon kinetic theory, scalar field dynamics, dark matter freeze-out processes, and Big Bang nucleosynthesis. This yields a fully derived homogeneous energy distribution equation consistent with modern cosmological observations. The framework is then extended beyond the assumptions of homogeneity and isotropy. Using the ADM (3+1) decomposition, imperfect fluid dynamics, multi-component interactions, perturbative cosmology, and non-linear collapse models, the paper develops a general non-homogeneous formulation capable of describing structure formation, gravitational collapse, black hole formation, and strong-field regimes. Semi-classical corrections, including vacuum polarization, Hawking radiation, and cosmological particle production, are incorporated to bridge classical general relativity and quantum field effects in curved spacetime. The resulting formalism provides a comprehensive theoretical framework for understanding the universe’s energy distribution from the smooth early universe to highly localized astrophysical structures. This work serves both as a foundational synthesis of cosmological energy components derived from first principles and as a unified bridge between background cosmology, gravitational collapse, numerical relativity, and semi-classical gravity.