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Assume standard radiation-dominated expansion, negligible chemical potential, one neutrino helicity and one antineutrino helicity, and a stable species $X$ with no later entropy injection. Use natural units and unreduced Planck mass $m_{\mathrm{pl}}\simeq 1.22\times 10^{19}\mathrm{GeV}$. The relativistic energy density is ${\rho }_R={\pi }^2{g}_{\ast }{T}^4/30$. The Friedmann equation then givesFor thermal decoupling in cosmology, equate the interaction rate $G_A^2{T}_D^5$ to $H(T_D)$:This order-of-magnitude estimate assumes the supplied rate law remains applicable there.

A decoupled relativistic species cools as $T_X\propto a^{-1}$, while annihilations heat the coupled bath. Treat the supplied high-temperature count $100$ as the remaining bath's entropy count, approximately equal to its energy count. Cosmological entropy conservation down to ordinary neutrino decoupling givesSubsequent electron-positron annihilation in cosmology heats photons but neither neutrino population. Therefore the two-stage entropy dilution of an early-decoupled neutrino givesThe ordinary Cosmic neutrino background has $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. If the rounded count $100$ instead includes the new two helicities, the remaining bath count is $98.25$; this changes the answer only slightly, to about $0.341$. Keeping the extra species' entropy separate is essential in either convention.

For the two relativistic fermion states, $g_{\ast s,X}=(7/8)2=7/4$. Photon and three standard neutrino species have present entropy degrees of freedom, referred to $T_{\gamma }$,Writing $\theta =T_X/T_{\gamma }$, the fractional entropy of a relativistically decoupled species isThus approximately 1.7% of the thermal particle entropy is in the new neutrinos and antineutrinos. If $100$ counts all states before the new decoupling, the fraction is $1.75\%$. These are the same rounded estimate. The question concerns the radiation/relic thermal entropy budget; it does not supply a model for entropy in later black holes or other gravitational structures.

If $X$ has a small mass but decoupled relativistically, its comoving particle number is still conserved. The relativistic-neutrino number-density ratio givesOnce nonrelativistic, ${\rho }_X\simeq m_X{n}_X$. The density bound on a relativistically decoupled massive relic is thereforeUsing the standard present photon abundance and critical-density normalization, this is approximately ${\Omega }_X{h}^2\simeq m_X/(9\times 10^2\mathrm{eV})$, where $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})$ is the present Hubble normalization, not a magnetic field. A stable relic must not exceed the permitted matter density; for example, if ${\Omega }_M{h}^2=0.15$, even assigning all matter to it gives only an order-$10^2\mathrm{eV}$ mass ceiling.

A stronger restriction can follow from neutrino free streaming and the observed formation of large-scale structure of the universe: a light thermal population acts as hot rather than cold dark matter, erasing structure below its streaming scale. If at most a fraction $f_{\max }$ of the matter may be in this population, replace the overclosure ceiling by $m_X\lesssim 9\times 10^2{f}_{\max }{\Omega }_M{h}^2\mathrm{eV}$. The allowed fraction is observational input and is not supplied, so the paper does not determine a unique tighter mass limit. A species still relativistic today is radiation rather than matter; its contribution to the effective number of neutrino species is $\Delta N_{\mathrm{eff}}=(T_X/T_{\nu }{)}^4\simeq 0.051$, and must instead satisfy the radiation-density constraints. These conclusions assume stable thermally populated states; decay or nonstandard entropy histories change the bounds.

The new neutrino species undergoes thermal decoupling in cosmology when its interaction rate becomes comparable to the Hubble parameter. Equating the given rates givesWith $g_{\ast }\simeq 100$, $G_A=10^{-12}{\mathrm{GeV}}^{-2}$ and $m_{\mathrm{Pl}}\simeq 10^{19}\mathrm{GeV}$,This is an order-of-magnitude electroweak-scale estimate under the given approximation. The mass in this formula is the unreduced Planck mass, not the reduced Planck mass.

Assume the new neutrino is stable and retains the abundance established by relativistic thermal decoupling in cosmology. While it remains relativistic, its contribution to the effective number of neutrino species isThus its effect on the radiation density and expansion during Big Bang nucleosynthesis is reduced relative to a fully heated extra neutrino species.

After it becomes nonrelativistic, its energy density is approximately $m_X{n}_X$. A two-state relativistic fermion relic has $n_X/n_{\gamma }=(3/4)(T_X/T_{\gamma }{)}^3\simeq 0.0275$. The density bound on a relativistically decoupled massive relic then givesHere ${\Omega }_{X,\max }$ is the allowed present matter-density fraction and $⟨E_{\gamma }⟩={\pi }^4{T}_{\gamma ,0}/(30\zeta (3))$. Choosing the total available matter fraction yields the basic overclosure bound. Its mass cannot be raised freely at a fixed relic number density.

There is also a perturbation constraint. Neutrino free streaming suppresses clustering below the distance traveled before the species becomes nonrelativistic. The ratio $T_X/m_X$ controls this distance, and the relic matter fraction controls the strength of the suppression. A significant hot or warm relic component must therefore satisfy the observed structure-formation constraints in addition to the abundance bound. Its allowed dark-matter contribution is limited jointly by these two effects. Numerical observational bounds depend on the chosen matter budget and perturbation data; the temperature and abundance dilution derived here are the factors needed to apply them.

Let $T_{-}$ be the residual gas temperature immediately before decay. This is the $T_d$ that enters the target abundance formula, not the higher reheating temperature after decay. Before the decay, separately conserved relic number and gas entropy give the conserved cosmological relic abundance $Y=n_{\varphi }/s_{-}$. The cold nonrelativistic relic has negligible kinetic energy, henceAssume its density dominates over that of the residual gas and that curvature and dark energy are negligible in the Friedmann equation. Using the reduced Planck mass $M_{\mathrm{Pl}}^2=(8\pi G{)}^{-1}$ gives $3M_{\mathrm{Pl}}^2{H}_d^2\simeq {\rho }_{\varphi }(t_d^{-})$. Under the allowed sudden decay of a dominant nonrelativistic relic convention $H_d\simeq \Gamma =t_d^{-1}$, the temperature immediately before decay isThe coefficient inherits the approximation $H_d\simeq 1/t_d$: the result is a sudden-decay estimate, not an exact age relation for a matter-only universe. If $t_d$ were instead the exact cosmic age in an uninterrupted spatially flat matter era, $H_d=2/(3t_d)$ would replace the numerator $3M_{\mathrm{Pl}}^2$ by $4M_{\mathrm{Pl}}^2/3$.

The assumptions also require thermal decoupling in cosmology and negligible prior decay so that $Y$ remains conserved, a fixed relativistic count $g_{\star }$, and rapid thermalization after the energy release. In the same approximation, the reheating temperature $T_{+}$ follows from $c_2{g}_{\star }{T}_{+}^4\simeq 3M_{\mathrm{Pl}}^2/t_d^2$:Its absence of $Y$ at fixed decay rate makes clear why the two temperatures must be distinguished.

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.

Before electron-positron annihilation, photons, electrons, and positrons share one temperature. Their effective entropy degrees of freedom areThe neutrinos have already undergone thermal decoupling in cosmology, so $T_{\nu }a$ remains constant and they receive none of the electron-positron entropy. In the still-coupled electromagnetic plasma, cosmological entropy conservation givesThe decoupled neutrino temperature at the same later time is $T_{\nu }=T_{\mathrm{before}}{a}_{\mathrm{before}}/a$. Dividing the two relations gives the Cosmic neutrino background temperatureThis instantaneous-decoupling calculation neglects the small reheating correction from non-instantaneous neutrino decoupling.

After thermal decoupling in cosmology, the collisionless relativistic species has $T_{X}a=\text{constant}$. The still-coupled plasma separately conserves entropy, so $g_{\ast s}{T}_{\gamma }^3{a}^3$ remains constant. At decoupling $T_X=T_{\gamma }$, while at late times the coupled plasma contains only the two photon polarizations. Therefore the temperature of a decoupled relativistic relic isHere $g_{\ast }$ is understood as the effective entropy degrees of freedom in the plasma that remains coupled after $X$ decouples.

After thermal decoupling in cosmology, two homogeneous sectors that evolve adiabatically without entropy injection satisfy $s_{\mathrm{SM}}{a}^3=\mathrm{constant}$ and $s_{\varphi }{a}^3=\mathrm{constant}$, hence their entropy density ratio $\xi$ is fixed. At a common relativistic temperature before decoupling, $\xi =g_{\ast s}^{\mathrm{SM}}/g_{\ast s}^{\varphi }$. Internal annihilations can change each sector's temperature and effective entropy degrees of freedom without changing its total comoving entropy.