Initial conditions are imposed using harmonic functions derived from analytic eigenmodes of the Laplacian operator on curved manifolds:
\Box \Phi(x, t) = 0 \quad \text{with} \quad \Phi(x, 0) = \sum_n A_n \psi_n(x)
where:
-
is the scalar field representing spacetime density modulation,
are spatial eigenfunctions,
are mode amplitudes determined by spectral energy distribution,
Time evolution follows linear or nonlinear wave equations depending on the test case.
The boundary mode structure simulates initial resonant imprinting at the Planck epoch or before inflationary smoothing, if any.
6.2 Computational Tools and Techniques
To simulate the evolution and structure emergence from these resonant conditions, we utilize a hybrid computational framework:
