L27: Defect Cosmology — UGP Physics

Phase Transitions in the Early Universe

When liquid water freezes, tiny ice crystals nucleate independently across the lake. Each crystal grows outward, but adjacent crystals were not in contact when they started — they independently chose their crystal orientation. Where they meet, a grain boundary (a defect) forms.

The same happens in the early universe. As the universe expanded and cooled, it underwent phase transitions — moments when the symmetric high-temperature behavior gave way to a broken-symmetry configuration. If multiple equally-valid ground states exist (vacuum degeneracy), different causally disconnected regions independently choose different vacua. Their boundaries are topological defects.

The GTE field Φ_MDL has exactly seven degenerate ground states, indexed by Z₇ = {0,1,2,3,4,5,6}. This means it will produce domain walls.

k=0
physical

k=1

k=2

k=3

k=4

k=5

k=6

The seven Z₇ vacua of Φ_MDL. Today the field sits everywhere in k=0. Particles are topological kinks winding between sectors.

The Kibble Mechanism and the Domain-Wall Problem

At temperatures above the Z₇ ordering-crossover T_G ≈ 0.70 GeV, thermal fluctuations are large enough to wash out the distinctions between the seven vacua. As the universe cools through T_G, each causally disconnected region independently "freezes" into one of the seven vacua with probability ≈ 1/7 per vacuum. Adjacent regions that chose different vacua are now separated by a domain wall. This is the Kibble mechanism.

The catastrophe: A domain-wall network has energy density ρ_wall ∝ σ/t, where σ is the wall tension. This decays more slowly than matter (∝ t⁻²) or radiation (∝ t^−8/3). Eventually walls dominate the energy budget and collapse the universe. For Φ_MDL, the wall tension σ = 0.29010 GeV³ gives catastrophic domination at z ≈ 832 — exactly the recombination epoch. The ZKO anisotropy bound would be violated by a factor of 3400. Every theory with discrete vacuum degeneracy must solve this problem.

Quantity	Value	Implication
Ordering crossover T_G	≈ 0.70 GeV	Between QCD scale (0.2 GeV) and EW scale (100 GeV)
Wall tension σ	0.29010 GeV³	From BPS kink integral (canonical normalization)
Domination redshift z_dom	≈ 832	Recombination era — catastrophic CMB violation
ZKO bound violation	×3400	The walls must be gone before this epoch

The GTE Resolution: Automatic Annihilation

The resolution is built into the GTE Lagrangian — not added to solve the problem. The Φ_MDL field couples canonically to a second field χ (the color sector):

        Vcoupling = ε · φ² · (Dμχ)²
      

with coupling constant ε = 7/9 (machine-certified, CatAL). This coupling is not Z₇-shift invariant: different vacua give different values of φₖ², and thus different kinetic normalizations for the χ field. The free energies of the seven vacua are no longer equal — the physical vacuum k=0 is preferred. This energy bias is the key.

The bias magnitude: |ΔV(0→1)|(T_G) ≈ 3.8 × 10⁻² GeV⁴. With wall tension σ = 0.29 GeV³, the critical collapse radius R_c = σ/ΔV ≈ 7.7 GeV⁻¹. Any wall larger than this radius collapses. The entire foam is gone in t_wall ≈ 5 × 10⁻²⁴ s — approximately 10⁻²³ seconds total.

T ≈ 0.70 GeV — Universe age ≈ 10⁻²³ s

Z₇ ordering crossover. Kibble foam forms: seven vacua populated with equal probability. Domain walls appear everywhere.

T ≈ 0.70 GeV — ~10⁻²⁴ s later

Coupling bias ΔV drives walls to collapse inward at near-lightspeed. Adjacent domains convert to k=0. Foam annihilates.

T ≈ 0.20 GeV — QCD confinement

Domain walls are long gone. No trace in the baryon asymmetry. Normal QCD physics proceeds.

T ≈ 1 MeV — Big Bang nucleosynthesis

The domain walls were annihilated 700× before nucleosynthesis even began. Zero cosmological damage.

T ≈ 0.26 eV — Recombination (z ≈ 1100)

The CMB is released. No Z₇ domain-wall anisotropy signal — consistent with Planck 2018 at full precision.

The structural point: The coupling V_coupling with ε = 7/9 is not introduced to solve the domain-wall problem. It is the canonical coupling forced by the MDL uniqueness of the Φ_MDL Lagrangian — the same coupling that generates the particle spectrum and color sector. Solving the domain-wall problem is automatic.

Observational Signatures and Predictions

Although the macroscopic domain-wall foam annihilates instantly, microscopic kink configurations survive — they are the particles of the theory. Upcoming surveys will probe the remnant signatures of the Z₇ phase transition:

CMB-S4 and LiteBIRD: The GTE primordial spectrum predicts r = 0 (no tensor modes) and n_s = 0.96488 with no running. Any detected r > 0.001 falsifies the framework.
Euclid and DESI large-scale structure: The dark energy equation of state should be w = −1 exactly (Λ-CDM). Time-varying w would indicate a different mechanism from the PSC adjudication floor.
Gravitational wave background: Annihilation of a cosmological domain-wall network at T_G ≈ 0.70 GeV produces a gravitational wave stochastic background peaking near 10⁻⁸ Hz — within the NANOGrav / LISA frequency band. This is a falsifiable prediction.

Key Points

The Φ_MDL field has seven Z₇ degenerate vacua. When the universe cooled through T_G ≈ 0.70 GeV, the Kibble mechanism produced a domain-wall foam — a universal cosmological catastrophe for theories with discrete vacuum degeneracy.
The GTE Lagrangian contains a canonical coupling ε·φ²·(D_μχ)² with ε = 7/9 (Lean-certified). This coupling breaks the Z₇ degeneracy, biasing the system toward k=0 with |ΔV| ≈ 3.8 × 10⁻² GeV⁴.
The entire Kibble foam collapses in ≈ 5 × 10⁻²⁴ s — about 700 times before Big Bang nucleosynthesis. The domain-wall problem is solved automatically by the same coupling that generates the particle spectrum. No new ingredients are added.
The zero-relic prediction: today the universe is in the unique k=0 vacuum everywhere. Lean cert: vacuum_unique_temporal_fixed_point_ring (CatAL).
Falsifiable signatures: r=0 from LiteBIRD, w=−1 from Euclid, and a gravitational wave background peak near 10⁻⁸ Hz from NANOGrav/LISA.

Defect Cosmology

Phase Transitions in the Early Universe

The Kibble Mechanism and the Domain-Wall Problem

The GTE Resolution: Automatic Annihilation

Observational Signatures and Predictions

Key Points

See Also