Resonant Genesis and Post LCDM Geometry: Toward Harmonic Framework of Cosmic Structure Formation

Appendix A: Full Mathematical Derivations

This section expands the core equations introduced in Sections 3 and 4, presenting complete derivations and intermediary steps for clarity and rigor.

A.1. Derivation of Resonant Field Modes

We solve the generalized Klein-Gordon equation in curved spacetime:

\Box \Phi + V(\Phi) = 0

In a metric of the form:

ds^2 = -dt^2 + a^2(t)\left[dr^2 + r^2 d\Omega^2\right]

Assuming separable solutions:

\Phi(t, \vec{x}) = \sum_{n, l, m} A_{n l m}(t) \psi_{n l m}(\vec{x})

We apply spherical harmonics and radial Bessel modes to decompose the spatial part. Boundary conditions at horizon scale are imposed:

\psi_{n l m}(r) = j_l\left(k_{nl} r\right), \quad \text{with} \quad k_{nl} = \frac{\pi n}{R_H}