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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 a relativistic equilibrium species with temperature $T_i$, the entropy density is $s_i=({\rho }_i+P_i)/T_i=4{\rho }_i/(3T_i)$. In units $k_B=\hslash =c=1$, bosons have ${\rho }_i={\pi }^2{g}_i{T}_i^4/30$, while fermions have $7/8$ times this expression. Relative to a reference bath temperature $T$, define the effective number of relativistic degrees of freedom for entropy byThe sums here concern relativistic species, or their effective entropy contributions. Since a physical comoving patch has volume proportional to $a^3$, cosmological entropy conservation givesfor an isolated adiabatic bath. After thermal decoupling in cosmology, one must apply this law separately to sectors that no longer exchange entropy.

Before electron-positron annihilation in cosmology, the electromagnetic bath contains two photon polarizations and four electron/positron spin states. After neutrino decoupling, its entropy degrees of freedom are initiallyand after annihilation only the two photon polarizations remain. Electromagnetic entropy conservation therefore givesThe decoupled Cosmic neutrino background receives none of this entropy; its distribution temperature obeys $a{T}_{\nu }=\text{constant}$, and $T_{\nu }=T_{\gamma }$ before annihilation. ThusThis is the temperature of a decoupled relativistic relic in the instantaneous-decoupling approximation. The ratio remains the same afterwards in the absence of additional photon heating. At late times $T_{\nu }$ denotes the redshifted relic distribution temperature even when neutrino masses become relevant; it need not be a kinetic-equilibrium temperature.