Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 3 / d / Solution

During the cannibal phase of a decoupled sector, cosmological entropy conservation and the leading result of part (c) giveDifferentiating and using $T_{\varphi }=m_{\varphi }/x$ yields logarithmic cooling during cannibalism:Here $a_{\ast }$ absorbs the entropy normalization, and the logarithmic expression is a large-$x$ asymptotic relation. With fixed visible entropy degrees of freedom the photon bath has $T_{\gamma }\propto a^{-1}$. Thus for $x\gg 1$ the scalar particles cool much more slowly than the photons and become relatively hotter.

The physical reason is that $3\varphi \to 2\varphi$ converts rest-mass energy into kinetic energy of the remaining particles. Elastic scattering distributes this energy, offsetting much of the cooling caused by expansion. The total comoving entropy remains constant; the number density does not merely dilute as $a^{-3}$. This explanation applies only while number-changing chemical equilibrium persists. If those reactions freeze out, an isolated number-conserving nonrelativistic gas instead has $T_{\varphi }\propto a^{-2}$ under adiabatic expansion, which is faster cooling than the photons. The paper supplies no rates, so that transition time cannot be determined.