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Take a homogeneous comoving cell of physical volume $V_c=a^3{V}_0$ and total entropy $S=s{V}_c$. At vanishing chemical potential, the reversible first law of thermodynamics readsFor an extensive equilibrium system, the thermodynamic Euler relation is $E=TS-P{V}_c+\mu N$. At $\mu =0$ this givesThe supplied pressure identity verifies consistency directly: $dP=(\rho +P)dT/T$ impliesThe thermodynamic Euler relation fixes the additive normalization that a differential identity alone would leave undetermined.

For adiabatic cosmological expansion, the cosmological perfect-fluid continuity equation gives $\dot{\rho }+3H(\rho +P)=0$, while ${\dot{V}}_c=3H{V}_c$. HenceThis proves cosmological entropy conservation in the stated perfect-fluid approximation. Kinetic equilibrium alone does not guarantee adiabatic evolution: entropy injection, dissipative stresses or out-of-equilibrium reactions would require additional source terms. Those effects are implicitly excluded in this calculation.

When the number-changing reaction is in chemical equilibrium, its chemical balance condition is $2{\mu }_{\varphi }=3{\mu }_{\varphi }$. Therefore zero chemical potential from number-changing reactions gives $\boxed{{\mu }_{\varphi }=0}$. Elastic scattering maintains kinetic equilibrium but imposes no independent condition on ${\mu }_{\varphi }$. The conclusion requires the number-changing rate to remain large compared with the Hubble parameter; merely listing an allowed reaction does not guarantee this after freeze-out.

Before thermal decoupling in cosmology, the sectors have a common temperature $T$. For relativistic species at that temperature their entropy densities aresoThese are the effective number of relativistic degrees of freedom for entropy. The paper's $g_{\ast }$ agrees with $g_{\ast s}$ for relativistic components at a common temperature. For one real scalar field, $g_{\ast s}^{\varphi }=1$. At temperatures where all Standard Model species are relativistic, both counts are constant, and so is $\xi$ before decoupling; thresholds can change this equilibrium ratio if they occur while the sectors remain coupled.

After decoupling, adiabatic isolation gives separately conserved visible and dark entropy: $s_{\mathrm{SM}}{a}^3$ and $s_{\varphi }{a}^3$ are individually constant. Thus $\boxed{\xi (T<T_d)=\xi (T_d)}$ even when either sector's entropy degrees of freedom change. Changes in particle content then alter their temperature ratio rather than the ratio of their comoving entropies. This assumes no late inter-sector transfer or entropy injection.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.