Radiation fluid is usually represented by the equation of state $p = \frac{1}{3} \, \rho$, where $p$ is the pressure and $\rho$ is the energy density. The local conservation of energy states that \begin{equation}\tag{1} \rho_\textrm{rad} = \rho_0 \, \frac{a_0^4}{a^4}, \end{equation} where $\rho_0$ and $a_0$ are constants and $a$ is the cosmological scale factor with units of length. This is a well known subject in standard cosmology. What puzzles me is the constant $\rho_0 \, a_0^4$. What is its rigorous interpretation ? For cold dust-like matter, it's easy : $\rho_\textrm{mat} \propto a^{- 3}$ and $M = \rho_0 \, a_0^3$ is interpreted as the conserved total proper mass of matter inside the volume $a^3$.
In case of radiation, energy $E_\textrm{rad} = \rho \, a^3$ is not conserved, since there is pressure and the photons wavelength is redshifted while the universe expands. But then what is the constant $\rho \, a^4$ ? I strongly suspect it's related to the total radiation entropy in the volume $a^3$, or maybe the number of ultra-relativistic particles, but I can't find any reliable source on this.
I know that in standard RWFL (Robertson-Walker-Friedmann-Lemaître) models, entropy is conserved because of the local conservation of energy-momentum, which is equivalent of saying \begin{equation}\tag{2} \mathrm dE = T \, \mathrm dS - p \, \mathrm dV = -\, p \, \mathrm dV. \end{equation} I also know that Stefan-Boltzmann law states that $\rho_\textrm{rad} \propto T^4$, and classical radiation entropy is $S \propto T^3 V$ within a volume $V$. Thus $\rho \, a^4 \propto T^4 \, V^{4/3} \propto S^{4/3}$, which is indeed conserved since entropy doesn't change. But I'm not totally satisfied by this reasoning and never saw this in my books on General Relativity and cosmology.
Anyone has a convincing argument that this indeed should be true ?
EDIT : The following is one aspect of the problem that puzzles me. The thermodynamics argument above gives \begin{equation}\tag{3} \rho_\textrm{rad} \, a^4 \propto S^{4/3}. \end{equation} The exponent $\frac{4}{3}$ coincides with the adiabatic index $\gamma = \frac{4}{3}$ of the ultra-relativistic gas. What is it doing here ? If I introduce the entropy density $\sigma = S/V$, then we could apparently write this : \begin{equation}\tag{4} \rho_\textrm{rad} \propto \sigma^{\gamma}, \end{equation} which recalls the polytropic gaz equation of state $p \propto \rho_\textrm{mass}^{\gamma}$. Why is that ? Can this be made more general (for $\gamma \ne \frac{4}{3}$) ? I never saw the relations (3) and (4) in my thermodynamics books, as far as I can tell. I need a more rigorous proof of these, and references if possible.
If $\rho \, a^4$ is related to the number of particles instead of entropy, then the polytropic gas state would make more sense, since $p \propto \rho_\textrm{rad} \propto n^\gamma$, where $n = N/V$ is the particles density.
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