Eigenvalue Spectrum of the Three-Dimensional Wave Operator in Cosmological Perturbation Theory: Rigorous Derivation and Observational Verification

This study presents a detailed mathematical derivation of the spectrum of vibrational modes for baryon density perturbations in the primordial universe, expressed in comoving coordinates. Starting from the linearized fluid equations in an expanding Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, the authors obtain the governing wave equation and solve it exactly in both Cartesian and spherical coordinate systems. The results show that the natural frequencies of these oscillations follow the same pattern as acoustic resonances in three-dimensional cavities — as if the early universe behaved like a “cosmic instrument” vibrating in specific modes. This parallel is not merely an analogy: mathematically, both systems are governed by the same type of differential operator and share identical spectral properties. The large-scale cosmic web emerges naturally from the nodal structure of these eigenmodes, with voids corresponding to zero regions and filaments to regions of maximum gradient. 👉 Explore the eigenvalue spectrum of the 3D cosmic wave — a rigorous formulation of the universe’s structure. Link : https://revistazen10.github.io/PAPER//Simulation CATEGORICAL CORRESPONDENCES BETWEEN MATHEMATICAL STRUCTURES AND COSMOLOGICAL SPACETIMES: Complete Formalization with Variational Dynamics and Operational Empirical Protocols https://doi.org/10.5281/zenodo.18027745