Act 9 Galaxies

The Ringing Drum

Strike a drumhead and watch what happens. The membrane vibrates, but not uniformly — it forms a pattern of still lines (nodes) and vibrating regions (antinodes). Sprinkle sand on the drum and the grains collect at the nodes, tracing out the vibration pattern. These are called Chladni patterns.

A galaxy is the same idea in three dimensions. The superfluid is the drumhead. The seed black hole at the center is the boundary condition — the edge of the drum. Gravity provides the tension. And the "sand" is ordinary matter: gas, dust, stars.

The superfluid settles into a standing wave pattern around the black hole. It does not orbit like planets around a star — it rings like a bell. The density profile that emerges from this ringing is not arbitrary: it is dictated by the equation of state ($\gamma = 2$) and the boundary condition (the black hole).

Why rotation curves are flat

When you measure how fast stars orbit at different distances from a galaxy's center, you expect them to slow down farther out — just as outer planets orbit the Sun more slowly. But they do not. Stars at the edge of a galaxy orbit just as fast as stars halfway in. The rotation curve is flat.

Standard cosmology explains this with an invisible halo of dark matter particles. UFC explains it with the acoustic structure of the superfluid: the density profile that naturally arises from a $\gamma = 2$ BEC produces exactly flat rotation curves. No new particle. No free parameter. Just fluid mechanics.

Acoustic Cavities with Black Hole Boundary Conditions

A galaxy in UFC is an acoustic cavity — a region of superfluid bounded by the central black hole on the inside and the cosmic web on the outside. The superfluid settles into the lowest-energy vibrational mode of this cavity.

Bessel modes

In a spherically symmetric cavity, the natural modes are Bessel functions. The density profile of the superfluid ground state in such a cavity takes the form of a solitonic core transitioning to an isothermal envelope:

• Solitonic core: Near the center, the BEC wavefunction forms a coherent soliton — a stable, self-reinforcing density peak. This is the quantum ground state of the BEC in the gravitational potential.
• Isothermal envelope: Beyond the core radius, the density follows an isothermal profile where the velocity dispersion $\sigma$ is constant. This is the thermalized region where the superfluid has reached virial equilibrium.

The transition between these two regimes is smooth and occurs at a radius $r_c$ determined by the BEC healing length — itself a function of the particle mass and the local density.

Flat rotation curves from the equation of state

The isothermal envelope has a density profile such that the circular velocity $v_{\text{circ}}$ is constant with radius. This is not assumed — it is derived from the condition of hydrostatic equilibrium in a $\gamma = 2$ BEC:

The gravitational potential $\Phi(r)$ and the density $\rho(r)$ are related by hydrostatic equilibrium. For an isothermal system with velocity dispersion $\sigma$, this gives $\rho \propto e^{-\Phi/\sigma^2}$, which yields a flat rotation curve with $v_{\text{circ}} = \sqrt{2}\,\sigma$.

Derivation: Rotation Curves from BEC Hydrostatics

Expert Notes

Dimensional dependence of rotation curves

The flatness of rotation curves is not a universal feature — it is specific to $d = 3$ spatial dimensions. For a self-gravitating isothermal fluid in $d$ dimensions, the circular velocity scales as:

This gives:

• $d = 2$: $v \propto r^{1/2}$ — rising rotation curves
• $d = 3$: $v \propto r^0$ — flat rotation curves
• $d = 4$: $v \propto r^{-1/2}$ — falling rotation curves

Flat rotation curves are a signature of three spatial dimensions. This is a deep result: the very existence of galaxies with flat rotation curves tells you $d = 3$.

Tully-Fisher relation from dimensionality

The baryonic Tully-Fisher relation states that the baryonic mass of a galaxy scales as $M_b \propto v_f^\alpha$ with $\alpha \approx 4$. In UFC:

At $d = 3$: $\alpha = 4$, exactly as observed. This is not a fit — it is a consequence of the virial theorem in $d$ dimensions applied to a $\gamma = 2$ BEC. The Tully-Fisher relation is another dimensional fingerprint.

Core-halo mass relation

The solitonic core mass and the halo mass are related by:

This is a prediction of the Gross-Pitaevskii soliton in a self-gravitating BEC and matches the empirical core-halo relation inferred from dwarf galaxy observations. In $\Lambda$CDM with particle dark matter, this relation requires fine-tuned baryonic feedback prescriptions.

No cusp-core problem

$\Lambda$CDM $N$-body simulations generically predict cuspy density profiles (NFW: $\rho \propto r^{-1}$ at small $r$). Observations of dwarf galaxies consistently show cores ($\rho \sim \text{const}$ at small $r$). This "cusp-core problem" has persisted for over two decades. In UFC, cores are automatic: the solitonic ground state of the BEC always produces a flat central density. No baryonic feedback required.