Act 4 The Reaction Zone

Behind the detonation front, radiation converts to matter on a logistic sigmoid. The entire radiation era is the CJ reaction zone.

Molten Glass Cooling

Fuel to Product Conversion

The blast wave tears through the superfluid at 85% of light speed. Behind it, everything is chaos — a roiling mess of quasiparticles at relativistic speeds.

Think of molten glass cooling. At high temperature, the glass is liquid — energy flows freely. As it cools, molecules slow and lock into place. Pure energy congeals into matter.

The key asymmetry is what makes it irreversible: a lobster trap. Getting into the ground state is frictionless (Landau criterion). Getting out is viscous. Energy falls in easily but cannot climb back out.

Four descriptions, one epoch

The conversion finishes at $z \approx 3400$. Four independent frameworks all identify this moment:

  1. CJ detonation: reaction zone ends
  2. Israel-Stewart viscosity: Deborah number crosses 1
  3. Logistic conversion: inflection point ($f = 0.5$)
  4. Standard cosmology: matter-radiation equality

Four frameworks. Zero tuning. One epoch.

The Logistic Conversion

The matter fraction $f = \rho_m / \rho_{\text{total}}$ evolves as:

$$\frac{df}{d\ln a} = f(1-f) \times \frac{34}{27}$$

Kinematic term (rate = 1): Matter ($\rho_m \propto a^{-3}$) dilutes slower than radiation ($\rho_r \propto a^{-4}$).

Viscous term (rate = 7/27): The geometric bulk viscosity actively converts radiation to matter. The factor $7/27 = (d^2-2)/d^3$.

Total: $1 + 7/27 = 34/27$.

The Deborah Number

$$De = \frac{\alpha_B}{4} = \frac{27/7}{4} = 0.964$$

$De \approx 1$ means the relaxation timescale matches the expansion timescale — the definition of a reaction zone. $De$ crosses 1 at $z_{\text{eq}} \approx 3400$.

Derivation: The Logistic Equation

Matter fraction evolution

Define $f = \rho_m / (\rho_m + \rho_r)$. Dilution rates: $d\rho_m/d\ln a = -3\rho_m$, $d\rho_r/d\ln a = -4\rho_r$.

Differentiate $f$
$$\frac{df}{d\ln a} = \frac{\rho_r(-3\rho_m) - \rho_m(-4\rho_r)}{(\rho_m+\rho_r)^2} = \frac{\rho_m\rho_r}{(\rho_m+\rho_r)^2} = f(1-f)$$
Pure differential dilution gives rate = 1.
Add viscous conversion
$$\frac{df}{d\ln a} = f(1-f) \times \left(1 + \frac{d^2-2}{d^3}\right) = f(1-f) \times \frac{34}{27}$$
Positive feedback: more matter → more viscosity → faster conversion. Saturation from fuel depletion gives $(1-f)$.
Canon Result
$$De = 0.964, \quad \text{rate} = \frac{34}{27}$$

Both from $\alpha_B = 27/7$. The radiation era IS the CJ reaction zone.

Expert Notes

Lobster trap thermodynamics

The irreversibility is thermodynamic. Landau criterion guarantees frictionless flow toward the ground state. Above the ground state, excitations experience bulk viscous friction. Falling into the condensate is lossless; climbing out is dissipative.

Logistic form emerges, not imposed

$f(1-f)$ emerges from self-consistent viscosity feedback. Structurally identical to population dynamics: growth rate $\propto$ population $\times$ remaining carrying capacity.

Interactive: Logistic Conversion

What Comes Next

During the conversion, nucleosynthesis occurs — the three-minute reactor. UFC's non-uniform detonation solves the lithium problem.