Lecture 25

We continued analysis begun last time connecting the expansion (rising $R$) of the universe and the cooling (falling $T$) of its particle populations, in a radiation-dominated universe. We recalled that the primary constituent, the thermal bath of relativistic particles, remains in equilibrium as the universe expands, at a temperature $T \propto 1/(g_*^{1/3}\ R)$. This is the temperature of all coupled particles in the universe, relativistic and (because they equilibrate with the dominant relativistic component) nonrelativistic as well.

We then examined the behavior of particles that decouple from the thermal plasma.

We considered first the case where a particle species decouples while relativistic -- as occurs both for neutrinos and for photons. Separate conservation of the species' entropy requires that $T \propto R^{-1}$, while the particle momenta redshift with the expansion of the universe. We showed that this implies that the decoupled particle remains in an equilibrium momentum distribution, with redshifting temperature $T \propto R^{-1}$. For most of the time, while $g_*$ remains constant, the decoupled particle has a temperature which is redshifting exactly parallel to that of the radiation-dominated thermal bath in the universe.

We discussed the one exception to this parallel track: when a particle in thermal contact with the bath becomes nonrelativistic, transferring its entropy to the bath. We calculated how the temperature of the bath is increased by this transfer, over what it would have been had it simply continued redshifting. Since the decoupled species do simply continue redshifting (they cannot interact to participate in this transfer), this raises the temperature of the universe above that of the decoupled species. We calculated the difference between the neutrino and photon temperatures due to this effect. Here the neutrinos decouple from the electron/positron/photon plasma before the electrons and positrons become nonrelativistic, reheating the photons above the neutrino temperature.

We considered next the case where a particle species decouples while nonrelativistic. Here we had to work harder to retain the leading nontrivial consequences of $dS = 0$, and eventually obtained that $T \propto R^{-2}$. There was a memory glitch here, so I will recast next time. To preview, though, the second law of thermodynamics applied to a comoving volume says that $TdS = dE + pdV = d(\rho V) + pdV$. This was actually used in deriving our expression for the entropy, $S = (\rho + p)V/T$. However that derivation, which we skipped, is slightly nontrivial, it involves satisfying an integrability condition equating mixed partial derivatives of the entropy, which relates small changes $dp$ to $dT$. So, in setting $dS = 0$, it's easiest to go back as a starting point to the second law of thermodynamics, which states that $dS = 0$ when $d(\rho V) + pdV =0$. Imposing this relates $dR$ and $dT$, with solution $T \propto R^{-2}$. Using the particle's redshift ( $\vert\vec{p}\vert \propto R^{-1}$), we showed again that the particle species remains in an equilibrium distribution, here with redshifting temperature $T \propto R^{-2}$. This placed a constraint on the evolution of $\mu$, which is enforced by the species' particle number conservation.

Note that in the final case, where a particle species decouples at $T \approx m$, our three facts -- number and entropy conservation and redshifting particle momenta -- do not lead to a continued equilibrium distribution for the decoupled species.

We will consolidate these general principles in the standard big bang cosmology's ``thermal history of the universe,'' beginning at $t \approx 10^{-2}\ {\rm s}$ in a radiation-dominated universe, next time.

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