= Logarithmic cooling during cannibalism
{title2=$T_\phi\sim m/[3\log a]$}
The <nonrelativistic entropy at zero chemical potential> and $s_\phi a^3=\mathrm{constant}$ give $x+\frac12\log x=3\log a+\mathrm{constant}$. Using this leading nonrelativistic approximation, differentiation yields
$$
\frac{d\log T_\phi}{d\log a}\simeq-\frac3{x+1/2}.
$$
This is much smaller in magnitude than the relativistic-bath cooling rate $-1$ when $x\gg1$ and its entropy degrees of freedom are fixed. Thus <cannibal phase of a decoupled sector> cooling is logarithmic. Once number-changing reactions freeze out, a number-conserving nonrelativistic gas instead cools as $a^{-2}$ during adiabatic expansion.
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