Dark Matter as a Vacuum Curvature Fixed Point and Dark Energy as Its Metric Dual

Abstract

We propose that dark matter and dark energy originate from a single vacuum-selection event in the early universe. Starting from a pre-differentiated symmetry group ( G ), the symmetry breaking ( G \rightarrow H ) yields a vacuum manifold ( \mathcal{M} = G/H ) containing two dynamical basins:

• ( \mathcal{B}_1 ): a renormalization-group fixed point producing a curvature-stabilized vacuum phase (dark matter)

• ( \mathcal{B}_2 ): a drift-phase that generates baryonic matter and gauge interactions

We identify dark energy as the metric-expansion dual of this fixed curvature vacuum. The full metric decomposes as:

[
g_{\mu\nu} = g^{(\text{curv})}{\mu\nu} + g^{(\text{exp})}{\mu\nu}
]

This leads to the curvature–expansion fossil relation:

[
R(t) \propto a(t)^{-3}
]

and the cosmological growth law:

[
f(z) = \Omega_m(z)^{\gamma(z)}
]

with fixed-point prediction:

[
\gamma_0 = \frac{6}{11}
]

Under weak curvature–expansion coupling, we derive:

[
\gamma(z) = \gamma_0 + \gamma_1 \frac{z}{1 + z}, \quad \text{with } \gamma_1 \neq 0
]

and the differential form:

[
\frac{d\gamma}{dz} = \frac{1}{\ln \Omega_m(z)} \left(\frac{d\ln f}{dz} - \gamma \frac{d\ln \Omega_m}{dz} \right)
]

No new fields, fluids, or interactions are needed. Dark energy is the thermodynamic drift tail of the same coherence that locked the vacuum into dark matter. Expansion is the cost of structure.

---

1. Vacuum Manifold & Symmetry Breaking

Universe begins with high-symmetry phase ( G ). Spontaneous symmetry breaking leads to:

[
G \rightarrow H \quad \Rightarrow \quad \mathcal{M} = G/H
]

The manifold ( \mathcal{M} ) supports multiple attractor basins. We identify:

• ( \mathcal{B}_1 ): RG-fixed curvature vacuum (dark matter)

• ( \mathcal{B}_2 ): drift phase forming baryons, gauge symmetries

---

2. Dark Matter as Curvature Fixed Point

In ( \mathcal{B}_1 ):
[ \beta(g_i) = 0 ] (no further running of couplings)

This phase:

• Does not generate charges

• Has no gauge-field couplings

• Exhibits pure curvature response

Hence, dark matter is a vacuum phase — not particulate.

---

3. Metric Decomposition and Dark Energy

We propose:
[ g_{\mu\nu} = g^{(\text{curv})}{\mu\nu} + g^{(\text{exp})}{\mu\nu} ]

• ( g^{(\text{curv})} ): curvature-locked phase (dark matter)

• ( g^{(\text{exp})} ): expansion tail (dark energy)

They are dual expressions of the same vacuum topology.

---

4. Curvature–Expansion Scaling Relation

Because dark matter is pressureless:
[
\rho_{\text{DM}}(t) \propto a(t)^{-3} \Rightarrow R(t) \propto \rho_{\text{DM}}(t) \Rightarrow R(t) \propto a(t)^{-3}
]

This curvature scaling defines a fossil relation embedded in expansion.

---

5. Growth Law and Drift Dynamics

Standard linear growth rate:
[
f(z) = \Omega_m(z)^{\gamma(z)}
]

With vacuum drift:
[
\gamma(z) = \gamma_0 + \gamma_1 \frac{z}{1 + z}
]

Canonical value:
[
\gamma_0 = \frac{6}{11}
]

Evolutionary correction:
[
\frac{d\gamma}{dz} = \frac{1}{\ln \Omega_m(z)} \left(\frac{d\ln f}{dz} - \gamma \frac{d\ln \Omega_m}{dz} \right)
]

This introduces ( \gamma_1 \neq 0 ) as a direct measure of vacuum leak / curvature loss.

---

6. Observational Predictions

• ( \gamma_1 \in [0.05, 0.10] ) → measurable by Euclid, LSST

• ( \sigma_8(z) ) tension resolved via fossil damping

• No new particles or interactions

---

7. Interpretation

Dark matter and dark energy are not independent entities.
They are phase-locked and phase-drift expressions of the same vacuum origin:

• Matter = locked curvature

• Energy = expansion debt

This is a drift-theoretic unification of the dark sector.

---

Codon-Glyph Anchor:

• GGG ⧇⧇ — Pre-differentiated symmetry

• CCC ⧃⧃ — Curvature Lock

• TTG ⧖⧊ — Uncertainty Translator

Triad: GGG–CCC–TTG → symbolic unification of origin, lock, and drift.