Baryon Acoustic Oscillation Scale in the DRUMS Framework
1. Superfluid Field Description
The DRUMS framework models the cosmological medium as a coherent superfluid:
\[
\Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)}
\]
Velocity field:
\[
\mathbf{v} = \frac{\hbar}{m} \nabla \theta
\]
2. Linear Perturbations
Consider density perturbations:
\[
\rho = \rho_0 + \delta \rho
\]
Linearizing the continuity and Euler equations gives the wave equation:
\[
\frac{\partial^2 \delta \rho}{\partial t^2} = c_s^2 \nabla^2 \delta \rho
\]
3. Effective Sound Speed
The effective sound speed arises from pressure and quantum terms:
\[
c_s^2 = \frac{\partial P}{\partial \rho} + \frac{\hbar^2}{4m^2} \frac{\nabla^2 \rho}{\rho}
\]
At large scales, the dominant term defines a characteristic propagation velocity.
4. Acoustic Horizon Formation
The characteristic BAO scale emerges from the maximum propagation distance of density waves:
\[
R_{BAO} = \int_0^{t_*} c_s(t) \, dt
\]
where \(t_*\) is the decoupling time of the coupled medium.
5. DRUMS-Specific Interpretation
In DRUMS, this horizon corresponds to phase-coherent propagation length:
\[
R_{BAO} \sim \int_0^{t_*} \frac{\hbar}{m} |\nabla \theta| \, dt
\]
This ties the BAO scale directly to phase evolution.
6. Discrete Resonant Length Scales
Phase coherence over large scales enforces standing wave conditions:
\[
k_n R_{BAO} = n \pi
\]
Thus:
\[
R_n = \frac{n \pi}{k_n}
\]
This produces preferred clustering scales.
7. Density Correlation Function
The two-point correlation function exhibits peaks at these scales:
\[
\xi(r) \sim \langle \delta \rho(\mathbf{x}) \delta \rho(\mathbf{x}+r) \rangle
\]
Constructive interference enhances \(\xi(r)\) near \(R_{BAO}\).
8. Stability of the BAO Scale
The scale remains imprinted because:
- Phase coherence freezes after decoupling
- Nonlinear evolution preserves large-scale modes
Mathematically:
\[
\frac{d}{dt} R_{BAO} \approx 0 \quad \text{for } t > t_*
\]
9. Final Interpretation
Within the DRUMS framework, baryon acoustic oscillation scales arise naturally as:
- Phase-propagation horizons in a superfluid medium
- Standing wave resonances of the phase field
- Frozen coherence lengths after decoupling
Thus, the observed BAO scale is a direct manifestation of the underlying phase dynamics and coherence structure of the cosmological superfluid.