IndisputableMonolith.Cosmology.EntropyConservationFRW
Derives the FRW continuity equation from the two Friedmann equations, then obtains comoving entropy conservation and the radiation aT law under equilibrium identities. Cosmologists citing neutrino dilution (Tν/Tγ)³ = 4/11 and g*s = 43/11 use the discharged hypotheses here. The argument differentiates Friedmann I, eliminates a″ via Friedmann II, and feeds continuity into Euler and Gibbs–Duhem.
claimFrom the Friedmann equations $a'^2 = (8\pi G/3)\rho a^2$ (along the evolution) and $a'' a = -(4\pi G/3)(\rho+3p)a^2$ (at the given time), the continuity equation $a\rho' = -3a'(\rho+p)$ follows by differentiation and elimination of $a''$, with no division. Under the Euler relation $T s = \rho + p$ and Gibbs–Duhem, comoving entropy $s a^3$ is conserved, radiation obeys $aT$ constant, and the neutrino dilution factors $(T_\nu/T_\gamma)^3 = 4/11$ and $g_{*s} = 43/11$ are recovered.
background
In flat FRW cosmology the scale factor $a(t)$ obeys two Friedmann equations relating energy density $\rho$ and pressure $p$ to $a'$ and $a''$. Continuity is often postulated from stress-energy conservation; here it is derived by differentiating the first Friedmann equation and cancelling $a''$ against the second. The common factor $(8\pi G/3)a^2$ drops with $G \neq 0$ and $a(t) \neq 0$, and no division is used.
Comoving entropy uses the Euler identity $T s = \rho + p$. Upstream, NeutrinoDilution treated comoving entropy conservation and the $1/a$ redshift law as named model hypotheses; its module doc states both are now discharged here from FRW continuity plus equilibrium identities.
The local setting is classical radiation-era thermodynamics on an FRW background. Radiation Euler and Gibbs–Duhem close the system so that $s a^3$ and $aT$ become constants of the evolution, unlocking the standard dilution arithmetic.
proof idea
The spine differentiates the first Friedmann equation along the evolution, substitutes $a''$ from the second, and cancels the common factor $(8\pi G/3)a^2$ to obtain continuity. Entropy conservation follows by feeding that continuity equation into the Euler relation $T s = \rho + p$ together with Gibbs–Duhem, so comoving entropy $s a^3$ is constant. Radiation specializations yield the conserved $aT$ law. Dilution and $g_{*s}$ results then package the standard neutrino-decoupling arithmetic, discharging the two named model hypotheses left open in NeutrinoDilution.
why it matters in Recognition Science
GrandPotential imports this module and builds Euler and Gibbs–Duhem from the grand potential. Its doc notes that comoving entropy conservation was already derived here from FRW continuity given those two equilibrium identities. The chain therefore closes NeutrinoDilution: $(T_\nu/T_\gamma)^3 = 4/11$ and $g_{*s} = 43/11$ become theorems rather than assumptions.
Within Recognition Science cosmology this is the thermodynamic backbone linking Friedmann dynamics to entropy bookkeeping for early-universe dilution and effective relativistic degrees of freedom. It sits between pure geometric FRW evolution and the grand-potential identities that justify the equilibrium relations used throughout.
scope and limits
- Does not derive the Friedmann equations from Recognition Science axioms.
- Does not treat anisotropic stress, spatial curvature, or non-equilibrium entropy production.
- Does not prove Euler or Gibbs–Duhem; those enter as named identities discharged downstream.
- Does not address late-time matter or dark-energy eras beyond the radiation identities used.
- Does not fix numerical values of G or initial temperatures; only structural conservation laws.
used by (1)
depends on (1)
declarations in this module (10)
-
theorem
continuity_from_friedmann -
theorem
comoving_entropy_conserved -
theorem
radiation_aT_conserved -
theorem
radiation_euler -
theorem
radiation_gibbs_duhem -
theorem
entropy_conserved_from_friedmann -
theorem
comoving_entropy_constant -
theorem
radiation_aT_constant -
theorem
dilution_from_frw -
theorem
gStarS_from_frw