Cannibal phase of a decoupled sector

A massive decoupled sector enters a cannibal phase when its particles are nonrelativistic but rapid number-changing reactions maintain chemical equilibrium and zero chemical potential from number-changing reactions. Rest-mass energy released by $3\to 2$ reactions heats the remaining particles, so the sector cools much more slowly than an isolated number-conserving nonrelativistic gas. The temperature laws follow directly from cosmological entropy conservation. An example is discussed in Cannibal Dark Matter.

The nonrelativistic entropy at zero chemical potential and $s_{\varphi }{a}^3=\mathrm{constant}$ give $x+\frac{1}{2}\log x=3\log a+\mathrm{constant}$. Using this leading nonrelativistic approximation, differentiation yieldsThis is much smaller in magnitude than the relativistic-bath cooling rate $-1$ when $x\gg 1$ 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.

For a massive real scalar with one internal degree of freedom with $x=m/T_{\varphi }\gg 1$, fixed separately conserved visible and dark entropy ratio $\xi$, and relativistic visible bath with entropy degrees of freedom $g_{\ast s}^{\mathrm{SM}}$,Indeed, equate $s_{\mathrm{SM}}=(2{\pi }^2/45)g_{\ast s}^{\mathrm{SM}}{T}_{\gamma }^3$ to $\xi s_{\varphi }$ and use nonrelativistic entropy at zero chemical potential. For $g$ internal states the right side gains $g^{1/3}$. The nonrelativistic species cannot be assigned a constant relativistic entropy count.