Cosmological entropy conservation

Let $s$ be proper entropy density in a homogeneous FLRW metric and $u^{\mu }$ the comoving fluid four-velocity. Reversible adiabatic expansion has a conserved entropy current,Since a fixed comoving volume has physical volume proportional to $a^3$, the covariant divergence givesThus the conserved quantity is comoving entropy; the proper entropy density falls as $a^{-3}$. This dilution does not require a decrease of the total entropy in the selected material volume.

For equilibrium relativistic matter at negligible chemical potentials, $s$ is proportional to $g_{\ast s}{T}^3$, where $g_{\ast s}$ is the effective number of relativistic entropy degrees of freedom. Conservation gives $aT{g}_{\ast s}^{1/3}=\text{constant}$, so $T\propto a^{-1}$ only while $g_{\ast s}$ is constant. Annihilation into a still-coupled bath can change its temperature relative to a decoupled species while conserving the bath's comoving entropy.^[ref]

Irreversible entropy production replaces the zero divergence by a nonnegative source. Separately decoupled sectors need their own entropy balances; imposing one common temperature after decoupling would add an assumption.

For $u=A{T}^4$, $P=u/3$ and physical cell volume $R^3$, the first law integrates to $S=4u{R}^3/(3T)$ with zero chemical potentials and a fixed relativistic species set. Entropy conservation through a species threshold instead requires tracking $g_{\ast s}{T}^3{R}^3$.

A relativistic relic that decouples between pion and muon annihilation keeps $a{T}_H$ constant. The coupled bath initially contains photons, electron-positron pairs, muon-antimuon pairs and three neutrino-antineutrino species, with entropy degrees of freedom $57/4$. After muon annihilation this is $43/4$, giving the displayed temperature ratio. After neutrino decoupling, electron-positron annihilation heats photons by $(11/4{)}^{1/3}$, so the final relic-to-photon ratio becomes $(172/627{)}^{1/3}$. This counting assumes the species are relativistic when counted and treats annihilations and decoupling as sharp equilibrium transitions.

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.

If a nonrelativistic relic decays entirely into promptly thermalized photons, energy conservation at fixed scale factor gives ${\rho }_{\gamma }^{+}={\rho }_{\gamma }^{-}+{\rho }_{\chi }^{-}$. Since thermal photons have ${\rho }_{\gamma }\propto T_{\gamma }^4$, their temperature rises by the displayed factor while an already decoupled Cosmic neutrino background is not heated. Before decay the relic-to-photon density ratio grows as $a$.

The effective number of relativistic entropy degrees of freedom is defined byA relativistic bosonic degree of freedom contributes one, while a relativistic fermionic degree of freedom contributes $7/8$ when all species share a temperature.

In the ideal-gas high-temperature limit, the Standard Model has 28 boson states: 16 gluon states, eight electroweak gauge boson states and four Higgs field states. Its 90 fermion states are 72 quark states, 12 charged lepton states and six neutrino states. Counting a relativistic fermion state with weight $7/8$ gives the displayed effective number of relativistic entropy degrees of freedom. The effective number of relativistic energy degrees of freedom has the same value when all species share a temperature. Mass thresholds and interactions change the ideal-gas count.

A particle species thermally decouples when its interaction rate falls below the Hubble parameter. Its collisionless relativistic distribution subsequently redshifts with temperature proportional to $a^{-1}$, while annihilations among still-coupled species can heat the remaining plasma.

If a species and the remaining bath share a temperature at decoupling, their initial entropy ratio is the ratio of their entropy-weighted degrees of freedom. Subsequent conservation of each sector's comoving entropy keeps that fraction constant, even when the bath heats relative to the relic. For a two-state neutrino species leaving an ideal Standard Model bath, the fraction is $(7/4)/(106.75+7/4)\simeq 1.61\%$.

If relic particle number and reference-gas entropy are separately conserved in a comoving volume, their ratio $Y=n/s$ is constant. A decay or entropy injection can change it. The reference entropy is that of the specified thermal sector, not necessarily the entropy of the decoupled relic itself.

A species freezing out while relativistic retains its thermal comoving number. If it is nonrelativistic today, ${\rho }_x\simeq m_x{n}_x$. Comparing with photons gives ${\Omega }_x/{\Omega }_{\gamma }=m_x(n_x/n_{\gamma })/⟨E_{\gamma }⟩$. An upper density fraction therefore bounds its mass inversely with its relic number density. The relation does not apply unchanged to a species whose abundance is depleted by nonrelativistic annihilation.

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.

Kinetic decoupling occurs when momentum-exchange collisions cease to maintain the temperature and velocity dispersion of a particle species in equilibrium with a surrounding bath. Afterwards, a freely moving nonrelativistic particle has momentum proportional to $a^{-1}$ and effective kinetic temperature proportional to $a^{-2}$, where $a$ is the scale factor.

If a relativistic species decouples when the coupled plasma has $g_{\ast s,\mathrm{dec}}$ entropy degrees of freedom, then after the plasma reaches $g_{\ast s}$ degrees of freedom,The ratio follows from separate redshifting of the decoupled species and cosmological entropy conservation in the coupled plasma.

An extra relativistic neutrino that decouples before ordinary neutrinos misses two stages of plasma heating. Until ordinary decoupling, entropy conservation gives a temperature ratio $(43/(4g_d){)}^{1/3}$. Electron-positron annihilation then heats photons but neither neutrino population, multiplying the ratio by $(4/11{)}^{1/3}$. Here $g_d$ counts the coupled bath at the extra species' decoupling, with no subsequent exotic entropy injection.

The Cosmic neutrino background is the relic population of neutrinos left after thermal decoupling in cosmology. As the universe expands, their momenta redshift; the decoupled distribution retains a temperature parameter proportional to the inverse scale factor, even after massive neutrinos become nonrelativistic.

The standard ideal temperature ratio follows from cosmological entropy conservation. Assume instantaneous neutrino decoupling before electron-positron annihilation and negligible chemical potentials. The coupled photon–electron–Positron sector initially has entropy degrees of freedom $2+(7/8)4=11/2$, dropping to two after annihilation. Conserving its comoving entropy heats the photons relative to the freely redshifting neutrinos, givingFor example, a photon temperature of $2.725\mathrm{K}$ corresponds to a relic-neutrino temperature parameter near $1.95\mathrm{K}$ in this idealization. Residual interactions during non-instantaneous neutrino decoupling slightly modify the distribution, so the simple ratio is an approximation.^[ref]

With negligible chemical potential, the Fermi-Dirac distribution has $3/4$ of the boson number density per degree of freedom at equal temperature. A species with $g_{\nu }$ populated states therefore has $n_{\nu }/n_{\gamma }=(3g_{\nu }/8)(T_{\nu }/T_{\gamma }{)}^3$, since photons have two states. This remains the number-density ratio after the neutrino becomes massive if its comoving particle number is conserved.

For one relic-neutrino species with two internal states, physical momentum $p$, comoving temperature $T_{\nu }$, and mass $m_{\nu }$,It approaches $7{\pi }^2(T_{\nu }/a{)}^4/120$ while relativistic and $n_{\nu }{m}_{\nu }$ after becoming nonrelativistic.

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.

For three standard thermal relic mass eigenstates, each neutrino-antineutrino pair has number density $3n_{\gamma }/11$ after pair annihilation. Their rest-mass density is this common per-species abundance times the mass sum. Requiring it not to exceed the near-critical total density gives the displayed conservative bound. Dividing instead by the total abundance over three species bounds the mean mass. With $T_{\gamma 0}=2.725$ K and $H_0=72$ km s${}^{-1}$ Mpc${}^{-1}$ the instantaneous-decoupling estimate is about $49$ eV/$c^2$ for the sum, conditional on standard abundances and nonnegative other energy components.

Expanding the relic-neutrino energy density at $T_{\nu }/a\ll m_{\nu }$ givesThe thermal speed decreases as $a^{-1}$ after the neutrinos become nonrelativistic.

The cosmic neutrino background last-scattering surface is the locus from which relic neutrinos that reach an observer underwent their final appreciable scattering. Although neutrino decoupling occurs before photon decoupling, a massive neutrino travels slower than light after becoming nonrelativistic, so its last-scattering surface need not be farther away than the cosmic microwave background last-scattering surface.

Neutrino decoupling occurs when weak interactions become slower than cosmic expansion. The neutrino distribution then free streams and cools as $T_{\nu }\propto a^{-1}$.

Instantaneous neutrino decoupling followed by electron-positron annihilation conserves electromagnetic entropy. The electromagnetic entropy count falls from $11/2$ to $2$, giving $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$ after annihilation.

Electromagnetic comoving entropy conservation gives $a^3{g}_{\ast s}(T_{\gamma })T_{\gamma }^3$ constant. Differentiation proves the displayed relation. During equilibrium electron-positron annihilation, $g_{\ast s}$ increases with temperature, so the absolute photon temperature still falls as space expands. The comoving temperature $a{T}_{\gamma }$ rises, and the asymptotic relativistic-to-pair-free entropy count heats photons relative to decoupled neutrinos by $(11/4{)}^{1/3}$. This relative heating must not be mistaken for an absolute temperature rise along the smooth expanding history.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.

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 $0.332$.

After neutrino decoupling, electron-positron annihilation transfers entropy to photons but not to neutrinos. Comparing the electromagnetic entropy degrees of freedom before and after annihilation gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$.

For $T\ll m_e$ and negligible chemical potentials, the combined equilibrium density of electrons and positrons divided by the photon number density isThe exponential suppression eventually makes pair equilibrium incompatible with the residual electrons required by charge neutrality.

The neutrino degeneracy parameter is the dimensionless chemical potentialIn thermal equilibrium the corresponding antineutrino has chemical potential $-{\mu }_{\nu }$.

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.

For one massless neutrino flavor with one helicity per member and $|{\mu }_{\nu }|\gg T_{\nu }$, the Fermi-Dirac distribution fills momentum states up to the positive chemical potential. If ${\mu }_{\nu }>0$, the neutrino density is ${\mu }_{\nu }^4/(8{\pi }^2)$ and the antineutrino contribution is approximately $3T_{\nu }^4{e}^{-{\mu }_{\nu }/T_{\nu }}/{\pi }^2$. For negative chemical potential the roles reverse. Thus the leading total energy density is even in ${\mu }_{\nu }$ and has no factor of two from the two species. Extra populated internal degrees of freedom do multiply it. This distinction prevents using a positive-degeneracy approximation for an exponentially dilute negative-chemical-potential distribution.

A degenerate relic neutrino has ${\xi }_{\nu }\gg 1$. In the ultrarelativistic limit its Fermi-Dirac distribution approaches a filled momentum sphere, giving for one internal statewhile the antineutrino density is exponentially suppressed.

A nonzero neutrino chemical potential produces unequal neutrino and antineutrino densities and hence a cosmological lepton asymmetry. It changes the radiation density, and an electron-neutrino asymmetry also changes neutron--proton weak equilibrium during Big Bang nucleosynthesis.

The effective number of neutrino species parametrizes radiation beyond photons through

If a real relativistic scalar decouples when the other coupled particles have entropy degrees of freedom $g_{\ast s}(T_d)$, thenEarlier decoupling and the subsequent annihilation of more coupled species make the scalar colder and suppress its contribution.