Cosmic Web Formation in the DRUMS Framework

1. Superfluid Cosmological Medium

The DRUMS framework models the universe as a coherent superfluid field:

\[ \Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)} \]

Velocity field:

\[ \mathbf{v} = \frac{\hbar}{m} \nabla \theta \]

2. Governing Hydrodynamics

The system evolves under:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \]
\[ m \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v} \right) = -\nabla (P + Q) \]
\[ Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} \]

3. Linear Instability and Growth

Introduce perturbations:

\[ \rho = \rho_0 + \delta \rho \]

Linearization yields:

\[ \frac{\partial^2 \delta \rho}{\partial t^2} = c_s^2 \nabla^2 \delta \rho \]

Including self-interaction leads to growth for long wavelengths:

\[ \omega^2 = c_s^2 k^2 - G_{eff} \rho_0 \]

Instability occurs when:

\[ k < k_J = \sqrt{\frac{G_{eff} \rho_0}{c_s^2}} \]

4. Anisotropic Collapse from Phase Gradients

The velocity field derives from a scalar phase, producing irrotational flow:

\[ \nabla \times \mathbf{v} = 0 \]

This enforces collapse along principal gradient directions. Density evolves anisotropically:

\[ \frac{d^2 x_i}{dt^2} = -\partial_i (P + Q) \]

Eigenvalue decomposition of the deformation tensor yields directional collapse rates.

5. Sheet Formation (Pancakes)

Collapse first occurs along the largest eigenvalue direction:

\[ \lambda_1 > \lambda_2 > \lambda_3 \]

This produces 2D sheet-like structures where density increases sharply:

\[ \rho \propto \frac{1}{(1 - \lambda_1 t)} \]

6. Filament Formation

Subsequent collapse along the second eigenvalue forms filaments:

\[ \rho \propto \frac{1}{(1 - \lambda_1 t)(1 - \lambda_2 t)} \]

Flow becomes confined to 1D structures.

7. Node (Halo) Formation

Final collapse along the third direction creates nodes:

\[ \rho \propto \frac{1}{\prod_{i=1}^3 (1 - \lambda_i t)} \]

These correspond to galaxy clusters or halos.

8. Role of Quantum Pressure

The quantum term prevents singular collapse and sets structure thickness:

\[ \nabla Q \sim \frac{\hbar^2}{m} \nabla \left( \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} \right) \]

This stabilizes sheets and filaments at finite width.

9. Emergent Network Topology

The combination of:

naturally produces a connected network of sheets, filaments, and nodes.

10. Scaling and Correlation

The characteristic scale is set by the instability length:

\[ \lambda_J = \frac{2\pi}{k_J} \]

This determines spacing between filaments and nodes.

11. Final Interpretation

Within the DRUMS framework, the cosmic web emerges inevitably from:

The observed large-scale structure is thus a direct manifestation of coherent superfluid dynamics rather than requiring additional dark matter constructs.