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Combining the degenerate neutrino and antineutrino energy density with a critical density bound constrains the neutrino degeneracy parameter through the fourth root of the available energy budget. The numerical bound depends on $T_{\nu ,0}/T_{\gamma ,0}$ and the number of degenerate flavors. The standard ratio $(4/11{)}^{1/3}$ assumes neutrino decoupling before electron-positron annihilation in cosmology; it should be stated as a thermal-history assumption if the degeneracy is very large. The massless formula is inappropriate for a species that is nonrelativistic today.

A standard thermal Cosmic neutrino background with one populated neutrino and one antineutrino helicity per species has $n_{\nu }+n_{\bar{\nu }}=3n_{\gamma }/11$ after electron-positron annihilation in cosmology. If this species is nonrelativistic today and dominates a spatially flat matter universe, equate $m_{\nu }(n_{\nu }+n_{\bar{\nu }})$ to the critical density $3H_0^2{M}_P^2/(8\pi )$, using the unreduced Planck mass. The photon number density gives the displayed conditional mass. Extra populated states or several massive species change the abundance count; vacuum-dominated cosmology changes the density available for matter.

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.

Use natural units, assume negligible neutrino chemical potentials, and approximate neutrino decoupling as occurring before electron-positron annihilation in cosmology. At high temperature, weak interactions maintain a common temperature for neutrinos, antineutrinos, Electrons, Positrons and photons. Since ${\Gamma }_{\nu }/H\propto T^3$, the reaction rate eventually loses against expansion. Decoupled relativistic neutrinos then free stream; their momenta redshift as $a^{-1}$, so their distribution retains a temperature parameter satisfying $a{T}_{\nu }=\mathrm{constant}$.

The still-coupled electromagnetic bath receives the entropy of annihilating Electrons and Positrons. Its effective number of relativistic entropy degrees of freedom is initiallywhere the two photon polarizations count as bosonic states and the four electron-positron spin states have the fermion entropy weight $7/8$. After the pairs disappear, $g_{\ast s}^{\mathrm{EM}}=2$. Cosmological entropy conservation in this coupled bath gives $g_{\ast s}^{\mathrm{EM}}(a{T}_{\gamma }{)}^3=\mathrm{constant}$. The three already decoupled neutrino species are excluded from this bath's entropy count. At decoupling $T_{\nu }=T_{\gamma }$, henceSubsequent expansion preserves this ratio in the idealized history. Residual interactions during annihilation give small corrections to the instantaneous-decoupling approximation. If a relic species later becomes massive, $T_{\nu }$ here remains the redshifting momentum-distribution parameter; it need not be its nonrelativistic kinetic temperature.

A single ordinary neutrino species includes one populated neutrino helicity and one antineutrino helicity. For negligible chemical potential, integration of the Fermi-Dirac distribution givesTherefore the number density of a relativistic neutrino relative to photons, with the antiparticle included, isThe factor $3/4$ is the fermionic number-density weight, distinct from the energy and entropy weight $7/8$. This relation continues to hold after this species becomes nonrelativistic if its comoving particle number is conserved.

Under the proposed matter-dominated, spatially flat model, the Friedmann equation requires the present relic-neutrino energy density to be approximately the critical density. Using the unreduced Planck mass $M_P=G^{-1/2}$ givesThus the critical-density mass of a thermal relic neutrino isTaking the paper's rounded values, $T_{\gamma 0}=2\times 10^{-13}\mathrm{GeV}$, $H_0=h/(4.5\times 10^{41}{\mathrm{GeV}}^{-1})$ and $M_P\sim 10^{19}\mathrm{GeV}$, givesFor $0.5<h<1$, this is of order $30$--$100\mathrm{eV}$ with these approximate constants. More precise constants change the coefficient, not the requested mass scale. This mass is much larger than the present momentum scale $T_{\nu 0}\simeq 1.4\times 10^{-4}\mathrm{eV}$, validating the nonrelativistic energy density approximation. The result assumes that one thermal relic species dominates the matter density; it is conditional on that model rather than a measurement of a neutrino mass.

Before electron-positron annihilation in cosmology, photons contribute two helicity states; Electrons and positrons contribute two spin states each. Each of the three light neutrino flavours contributes one active neutrino helicity and one antineutrino helicity. At a common temperature the effective number of relativistic energy degrees of freedom is thereforeUsing $G_F=1.17\times 10^{-5}{\mathrm{GeV}}^{-2}$ and $M_{\mathrm{Pl}}\simeq 2.435\times 10^{18}\mathrm{GeV}$ in the scaling formula alone gives $T_F\sim 2\mathrm{MeV}$. Including the explicit expansion coefficient from (ii), but setting the unknown rate coefficient $A$ to one, gives $T_F\simeq 1.5\mathrm{MeV}$. Thus the warranted conclusion is a freeze-out temperature of order $1\mathrm{MeV}$, above the $0.511\mathrm{MeV}$ Electron mass. The omitted weak-rate coefficient prevents a more precise prediction from this estimate.

Neutrino decoupling occurs before most electron-positron pairs annihilate. Afterwards neutrinos redshift freely, $T_{\nu }a=\mathrm{constant}$, while the photons receive the pair entropy. For the still-coupled photon-pair gas, cosmological entropy conservation gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$. Its entropy degrees of freedom drop from $2+(7/8)4=11/2$ to $2$. Comparing this with the freely cooling neutrinos givesAfter annihilation, photon and neutrino temperatures differ. To write the total energy density as $({\pi }^2/30)g_{\ast }{T}_{\gamma }^4$, the neutrino contribution must include that temperature ratio:This is the paper's post-annihilation ${\mathcal{N}}_{\mathrm{eff}}$. Merely deleting the Electron term while keeping all species at the same temperature would be incorrect. For comparison the entropy count, normalized to $T_{\gamma }^3$, is $g_{\ast s}=2+(7/8)6(4/11)=43/11\simeq 3.91$. These results assume instantaneous neutrino decoupling and relativistic neutrinos; the two counts represent different thermal moments.

Before electron-positron annihilation in cosmology, the pair has two spin states, just as photons do, and the fermionic number integral gives $n_{\nu +\bar{\nu }}/n_{\gamma }=3/4$ at equal temperature. The factor follows by expanding $(e^x+1{)}^{-1}$ into an alternating exponential series: its $x^2$ integral is $3/4$ of the bosonic one.

After neutrino decoupling, cosmological entropy conservation in the photon-electron plasma changes its relativistic entropy count from $2+(7/8)4=11/2$ to $2$. The decoupled neutrinos do not share this heating. HenceThese counts include both the new neutrino and its antineutrino; another factor of two would double count them. This is the number density of a relativistic neutrino relative to photons, evaluated after photon heating.

They are nonrelativistic today, so the relic-neutrino energy density is dominated by rest mass:The number follows from the supplied critical density, without a Hubble-unit convention. It is conditional on the thermal history stated above; a sterile, colder or nonthermally produced state need not have this abundance.

Let $r_{\nu }=T_{\nu ,0}/T_0$, and retain the assumption that these neutrinos remain massless. Constancy of the neutrino degeneracy parameter gives $|{\mu }_{\nu ,0}|=|{\xi }_{\nu }|r_{\nu }{T}_0$. The condition on the critical density therefore yields the critical-density bound on massless neutrino degeneracyFor the standard instantaneous-neutrino decoupling approximation before electron-positron annihilation in cosmology, cosmological entropy conservation heats photons relative to the decoupled neutrinos. The electromagnetic entropy degrees of freedom change from $2+(7/8)4=11/2$ to $2$, givingThis numerical value requires that thermal-history assumption; the bound in terms of $r_{\nu }$ is the general answer if large degeneracy changes decoupling or subsequent entropy transfer. Setting $T_{\nu ,0}=T_0$ would instead give $26.2$. For $N$ equally degenerate flavors with the same temperature, the leading bound becomes $36.8N^{-1/4}$ under the standard temperature assumption. The neglected thermal corrections make a small change near this loose energy-density limit; this is not a detailed Big Bang nucleosynthesis bound.

In the local orthonormal tetrad, the components of the photon number current areThey are respectively the photon number density and the photon number flux. Thus $J^{\mu }$ is the photon number four-current, not an energy current or an electric charge current. Photons carry no electric charge. For the isotropic background,Integrating the covariant Boltzmann equation over the invariant mass-shell measure, with vanishing momentum-boundary flux, giveswhere $C[f]$ is the collision operator in the covariant streaming equation. ThereforeCollisionless photon propagation satisfies this condition, as do Thomson scattering and ordinary Compton scattering when all participating photons are counted: each collision has one incoming and one outgoing photon. Photon emission, photon absorption, bremsstrahlung, double Compton scattering and electron-positron annihilation in cosmology or its inverse generally give a nonzero number source. A cancellation of net creation and destruction can nevertheless make the integrated source vanish even when individual number-changing reactions occur. Cosmological redshift itself changes photon energy, not photon number; in an expanding homogeneous universe a conserved photon number current gives ${\bar{n}}_{\gamma }\propto a^{-3}$.

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.

For a relativistic species the supplied relation $P=ϵ/3$ makes its entropy density proportional to $T^3$. The fermion-to-boson thermal integral ratio gives the factor $7/8$ for each fermionic degree of freedom, so the electromagnetic plasma immediately before electron-positron annihilation in cosmology hasAfter annihilation only the two photon polarizations remain, so $g_{\ast s,\mathrm{after}}=2$.

The neutrinos have already undergone neutrino decoupling. They therefore receive none of the electron-positron entropy and cool as $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, cosmological entropy conservation gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$. If $T_d$ and $a_d$ denote the common temperature and scale factor just before annihilation, then at a later scale factor $a$,Dividing gives the Cosmic neutrino background temperature ratio

At neutrino decoupling, neutrinos, electrons, positrons, and photons share one temperature. The neutrinos subsequently free stream, so $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, the effective entropy degrees of freedom change during electron-positron annihilation in cosmology fromto $g_{\ast s,\mathrm{after}}=2$. Separate cosmological entropy conservation in that plasma gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$, while $T_{\nu }a$ remains constant. Consequently the Cosmic neutrino background temperature obeys

Before electron-positron annihilation in cosmology, photons and electron-positron pairs share a temperature. Their effective entropy degrees of freedom areAfter annihilation the electromagnetic plasma contains only the two photon polarizations, so $g_{\ast s}^{\mathrm{after}}=2$. The already decoupled neutrinos receive none of this entropy and simply cool as $T_{\nu }\propto a^{-1}$. Applying cosmological entropy conservation separately to the coupled electromagnetic plasma giveswhereas $T_{\nu }^3{a}^3=T_{\mathrm{before}}^3{a}_{mbefore}^3$. Division yields the Cosmic neutrino background temperature

In an orthonormal tetrad, $J^{\hat{0}}=\int f{d}^{3}p$ is the photon number density and $J^{\hat{i}}=\int f{e}^i{d}^{3}p$ is its number flux. Integrating the Boltzmann equation gives ${\nabla }_{\mu }{J}^{\mu }=\int C[f]d^{3}p/E$ when momentum-boundary terms vanish. Collisionless propagation and photon-number-conserving scattering give zero divergence. Photon emission, photon absorption, electron-positron annihilation in cosmology, bremsstrahlung and double Compton scattering generally do not. This is a number current, not an electric charge current: photons are neutral.

A relativistic species decoupling before ordinary neutrino decoupling cools as $T_X\propto a^{-1}$. Cosmological entropy conservation in the remaining bath gives $T_X/T_{\nu }=[10.75/g_{\ast s,D}{]}^{1/3}$, where $g_{\ast s,D}$ counts that bath just after the early species has left it. Subsequent electron-positron annihilation in cosmology gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. For the ideal high-temperature relativistic entropy content of the Standard Model, the resulting ratio is about $0.332$.