= Solution
During the <cannibal phase of a decoupled sector>, <cosmological entropy conservation> and the leading result of part (c) give
$$
a^3x^{-1/2}e^{-x}=\mathrm{constant},\qquad
x+\frac12\log x=3\log a+\mathrm{constant}.
$$
Differentiating and using $T_\phi=m_\phi/x$ yields <logarithmic cooling during cannibalism>:
$$
\boxed{\frac{d\log T_\phi}{d\log a}\simeq-\frac3{x+1/2},\qquad T_\phi\sim\frac{m_\phi}{3\log(a/a_*)}.}
$$
Here $a_*$ 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 $x\gg1$ the scalar particles cool much more slowly than the photons and become relatively hotter.
The physical reason is that $3\phi\to2\phi$ 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 $T_\phi\propto a^{-2}$ under adiabatic expansion, which is faster cooling than the photons. The paper supplies no rates, so that transition time cannot be determined.
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