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Chapter 19: Hot Universe (Nucleosynthesis)

19.2 Baryogenesis​

Why is there a universe made of matter rather than a symmetric, sterile sea of radiation? This section provides a deductive derivation of Baryogenesis via Leptogenesis in the QBD framework, demonstrating that the chirality of the graph's pre-geometric arrow of time naturally selects matter over antimatter.

19.2.1 Theorem: Sakharov Compliance​

Compliance with Sakharov Conditions through Chiral Braid Decay under Causal Timestamp Monotonicity
  • Baryon & Lepton Violation: The unified SU(5) dynamics of the graph (§9.2.1) support leptoquark rewrite rules (X/Y bosons) that allow transitions between quark and lepton ribbon topologies while conserving B−LB-L (§9.3.1).
  • CP Violation: Topological rewrite rules are chiral: Parity (P) inverts crossings, while Charge Conjugation (C) inverts writhe. Because the underlying causal graph is timestamp-monotone (tLt_L), the loop interference phase δ\delta differs for particles and antiparticles, causing decay rates to split: Γ(NR→LH)≠Γ(NˉR→LˉHˉ)\Gamma(N_R \to L H) \neq \Gamma(\bar{N}_R \to \bar{L} \bar{H}).
  • Out-of-Equilibrium Departure: The rapid expansion of the scale factor at the end of inflation ensures that the Hubble rate HH exceeds the decay rate (H>ΓdecayH > \Gamma_{decay}), freezing out the heavy neutrino states and preventing inverse washout reactions from restoring symmetry.

19.2.2 Lemma: CP-Asymmetry Parameter​

Derivation of CP Asymmetry Parameter from Topological Chirality of Braid crossings
  • Interference Phase: The microscopic CP asymmetry parameter ϵCP\epsilon_{CP} is derived from the interference of tree-level and loop-level self-energy braid diagrams.
  • Braid Twist Angle: The parameter scales with the Majorana mass scale MRM_R and the twist angle δ\delta of the 3-ribbon braid: ϵCP∝316πmνMRv2sin⁡(δ)\epsilon_{CP} \propto \frac{3}{16\pi} \frac{m_\nu M_R}{v^2} \sin(\delta)
  • Topological Invariant: The phase δ\delta is not a free fitting parameter; it is a topological invariant determined by the crossing angles of the ribbon embedding.

19.2.3 Proof: CP-Asymmetry Parameter​

Verification of Baryon Asymmetry Magnitude through Interference Calculation of Braid Decay Amplitudes
  • Quantitative Derivation: The proof calculates the asymmetry parameter using Seesaw parameters (mν≈0.05m_\nu \approx 0.05 eV, MR≈1016M_R \approx 10^{16} GeV).
  • Observation Match: Integrating the CP-violating decay rates over the cooling history yields the baryon-to-photon ratio: η=nB−nBˉnγ∼10−10\eta = \frac{n_B - n_{\bar{B}}}{n_\gamma} \sim 10^{-10} This matches the observed value ηobs≈6×10−10\eta_{obs} \approx 6 \times 10^{-10} within order-of-magnitude precision.

19.2.4 Theorem: Sphaleron Conversion​

Redistribution of Lepton Excess into Baryon Numbers via Emergent SU(2) Sphaleron Tunneling
  • Emergent SU(2) Topology: In the high-temperature plasma, the emergent SU(2)SU(2) electroweak sector (§8.5) supports non-trivial vacuum configurations (Sphalerons).
  • Symmetry Conversion: Sphaleron transitions correspond to topological updates that violate BB and LL conservation while strictly preserving the B−LB-L invariant.
  • Redistribution Flow: This electroweak tunneling converts the lepton asymmetry generated by heavy neutrino decay into a stable baryon excess, seeding the universe with quarks.

19.2.5 Proof: Sphaleron Conversion​

Verification of Sphaleron Conversion Efficiency through Numerical Evaluation of SU(2) Topological Charge Flux
  • Conversion Factor: The proof calculates the equilibrium distribution of charges in a hot plasma with Nf=3N_f = 3 generations and NH=1N_H = 1 Higgs doublet, deriving the conversion factor: Csph=8Nf+4NH22Nf+13NH=2879≈0.354C_{sph} = \frac{8N_f + 4N_H}{22N_f + 13N_H} = \frac{28}{79} \approx 0.354
  • Baryon Fraction: It proves that approximately 35%35\% of the initial lepton number is converted into baryon number, establishing the final matter abundance.