gBefore
plain-language theorem explainer
Effective entropy degrees of freedom of the coupled photon–electron plasma just before e± annihilation equal 11/2. Cosmologists cite this as the pre-annihilation g_*s input to neutrino temperature dilution and to the present-day entropy-per-photon ratio. The definition is the rational sum of two photon polarizations plus the fermionic weight times four e± spin states.
Claim. The effective entropy degrees of freedom of the photon–electron plasma before electron–positron annihilation are $g_{\mathrm{before}} = g_\gamma + \frac{7}{8}\, g_{e^\pm} = 2 + \frac{7}{8}\cdot 4 = \frac{11}{2}$.
background
The EntropyPerPhoton module derives $s/n_\gamma = \pi^4 g_{*s}/(45\zeta(3))$ in a narrow window around 7.04 from three ingredients: a $\zeta(3)$ bound, a $\pi^4$ bound, and the present-day entropy dof $g_{*s}=43/11$. That last figure rests on how entropy is shared between photons and neutrinos after e± annihilation.
Photon internal dof are the two polarizations ($g_\gamma=2$). Electron–positron internal dof are two spin states times particle and antiparticle ($g_{e^\pm}=4$). Fermions enter entropy with weight $7/8$, the ratio of the Fermi–Dirac to Bose–Einstein integrals $\int x^3/(e^x+1),/,\int x^3/(e^x-1)=\eta(4)/\zeta(4)=1-2^{-3}$, now theorem-backed rather than a model postulate.
Before annihilation the plasma is photons plus e±; neutrinos have already decoupled and stream freely. The combination $g_\gamma+(7/8)g_{e^\pm}$ is therefore the coupled-sector entropy count that entropy conservation acts on.
proof idea
Pure definitional arithmetic on rationals: unfold the three named constants $g_\gamma=2$, fermion weight $7/8$, and $g_{e^\pm}=4$, then form $2+(7/8)\cdot 4$. No tactic proof; the value $11/2$ is the expanded rational.
why it matters
This constant is the denominator of the neutrino dilution factor. Downstream, dilutionCubed is defined as $g_{\mathrm{after}}/g_{\mathrm{before}}$ and proved equal to $4/11$; that identity is the arithmetic core of dilution_from_entropy_conservation, which states that free-streaming neutrinos plus conserved comoving entropy in the photon–e± sector force $(T_\nu/T_\gamma)^3=4/11$.
The NeutrinoDilution layer also equates the functional radiation-entropy density of a $(g_B,g_F)=(2,4)$ plasma to $(2\pi^2/45)\cdot g_{\mathrm{before}},T^3$, so the $11/2$ here is exactly the coefficient that appears in the thermodynamic integrals. Together with post-annihilation photons-only dof and the diluted neutrinos, it yields present-day $g_{*s}=43/11$, closing the third ingredient of the entropy-per-photon derivation used by the baryogenesis dynamical lane.
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