Entropy Conservation and Comoving Entropy

The thermal history of the early universe rests on two conservation laws. One is energy conservation, encoded in the Friedmann and fluid equations. The other is subtler and, for bookkeeping, more powerful: the conservation of entropy inside a comoving volume — a coordinate cube that stretches with the expansion. On any timescale short compared with the Hubble time $H^{-1}$ , particle collisions hold the plasma in local thermal equilibrium, so the expansion is quasi-static, reversible, and generates no net entropy. The total entropy in a comoving box is therefore frozen even as the box grows and its contents cool. Quantitatively the entropy density of a relativistic plasma is $s=(\rho+p)/T$ , which collapses to $s=\frac{2\pi^2}{45}g_{*s}T^3$ for a gas of relativistic species, where $g_{*s}$ counts effective degrees of freedom weighted by the cube of each species' temperature. Conservation of $S=s\,a^3$ then reads $g_{*s}T^3a^3=\text{const}$ . This one relation upgrades the naive $T\propto1/a$ into a precise statement: the temperature falls as $1/a$ only while $g_{*s}$ holds fixed, and falls more slowly whenever a species annihilates and pours its entropy into the particles it is still colliding with. The signature consequence is the offset between photons and neutrinos. Neutrinos decouple near $T\sim1\ \mathrm{MeV}$ ; moments later, when $T$ drops below the electron mass, electron-positron pairs annihilate and dump their entropy into photons but not into the already free-streaming neutrinos. Applying entropy conservation separately to the two sectors fixes $T_\nu=(4/11)^{1/3}T_\gamma\approx0.7138\,T_\gamma$ — a number that flows straight into $N_\mathrm{eff}$ , big-bang nucleosynthesis, and the predicted cosmic neutrino background. Entropy conservation is the accounting device that ties decoupling microphysics to the relic densities we measure today.

Key takeaways

In an expanding universe the entropy per comoving volume $S=s\,a^3$ is conserved to superb accuracy, because equilibrium interactions run reversibly on the expansion timescale.
Conservation gives $g_{*s}(T)\,T^3a^3=\text{const}$ : as long as $g_{*s}$ is fixed the photon temperature redshifts as $T\propto1/a$ .
When a species becomes non-relativistic and annihilates, it hands its entropy to the still-coupled plasma, so the coupled temperature falls more slowly than $1/a$ — a reheating relative to any already-decoupled species.
The classic case is $e^+e^-$ annihilation, which heats photons but not the decoupled neutrinos, fixing $T_\nu/T_\gamma=(4/11)^{1/3}\approx0.714$ and hence $T_\nu\approx1.945\ \mathrm{K}$ today.
The same constant $s\,a^3$ normalizes every comoving abundance: baryon-to-photon $\eta$ , baryon-to-entropy $n_b/s$ , and dark-matter freeze-out yields are all referred to it.
Key idea: adiabatic expansion conserves entropy per comoving volume, $s\,a^3=\frac{2\pi^2}{45}g_{*s}T^3a^3=\text{const}$ , so $T$ redshifts as $1/a$ only when $g_{*s}$ is constant; each annihilation transfers entropy to the surviving plasma and reheats it.

The core insight

Adiabatic expansion conserves entropy per comoving volume: $s\,a^3=\frac{2\pi^2}{45}g_{*s}T^3a^3=\text{const}$ . Whenever a species annihilates it transfers its entropy to the remaining coupled plasma, so the photon temperature redshifts as $1/a$ only while $g_{*s}$ is fixed and falls more slowly across every annihilation threshold.

Where this lives today

Entropy conservation is not a museum piece; it is the load-bearing assumption of every precision-cosmology pipeline. The prediction $N_\mathrm{eff}=3.044$ , built on $T_\nu/T_\gamma=(4/11)^{1/3}$ plus the small non-instantaneous-decoupling correction, is one of the sharpest tests of the standard model of cosmology: Planck combined with BBN measures $N_\mathrm{eff}=2.99\pm0.17$ , and future surveys such as the Simons Observatory and CMB-S4 aim for $\sigma(N_\mathrm{eff})\sim0.03$ , tight enough to detect a single extra light species or a percent-level departure in the neutrino sector. The same $s\,a^3=\text{const}$ underlies every relic-abundance calculation — WIMP dark-matter freeze-out, axion and sterile-neutrino production, and the dilution of any relic by a late entropy injection — because it is what makes the comoving yield $n_X/s$ a fixed, redshift-independent number. And the boldest frontier is direct: the PTOLEMY experiment aims to capture relic neutrinos from the predicted $336\ \mathrm{cm}^{-3}$ sea via neutrino capture on tritium, which would be the first laboratory confirmation of a background whose temperature and density are pure consequences of the entropy bookkeeping laid out here.

The physical picture

Picture a comoving box — a coordinate cube whose physical edge grows as $a(t)$ . As long as the particles inside collide fast enough, the gas relaxes to equilibrium at every instant: the expansion is quasi-static and reversible, so no entropy is created. The number of microscopic arrangements consistent with the box's macrostate — its entropy — is frozen even though the box swells and everything in it cools. The bookkeeping is cleanest in $s\,a^3$ rather than in $T$ or $n$ alone, because $s\,a^3$ is a genuine constant of the adiabatic motion, whereas $T$ and $n$ each carry the extra wrinkle of a changing $g_{*s}$ . The wrinkle is where the physics lives. When a massive species annihilates, its rest energy and entropy do not vanish — they are handed to the particles it is still colliding with, giving them a boost so their temperature drops more slowly than $1/a$ . Any species that had already stopped interacting misses the windfall and simply keeps cooling as $1/a$ , ending up colder. That is exactly why the cosmic neutrino background is predicted to sit below the CMB: neutrinos left the plasma just before $e^+e^-$ annihilation reheated the photons.

Figure

Comoving entropy is conserved while g star s tracks which species are coupled. Across e plus e minus annihilation g star s in the coupled sector drops from 11/2 to 2, so a T cubed rises by 11/4 and photons are reheated by a factor 1.401 relative to the decoupled neutrinos, fixing T nu over T gamma equal to 0.714

The index of the whole thermal history is the function $g_{*s}(T)$ : it starts near $106.75$ in the full Standard-Model plasma above the electroweak scale, steps down each time a species annihilates or a transition removes degrees of freedom, and settles at $g_{*s,0}=43/11\approx3.909$ once only photons and the three cooled neutrino flavors remain relativistic. Every downward step is a place where the coupled temperature is reheated relative to $1/a$ , and the neutrino step is simply the last and most observable one. The second figure is that staircase — the master map a cosmologist reads to convert a temperature into a scale factor and back.

Figure

The effective entropy degrees of freedom g star s of T fall in steps from 106.75 in the full Standard Model plasma through the QCD transition near 150 MeV down to 10.75 before e plus e minus annihilation and finally to 43/11 equal to 3.909 today. Each step marks a species leaving the coupled plasma and reheating what remains

The core equations

Entropy conservation is a chain of five statements: a thermodynamic definition of entropy density, its relativistic form, the conservation law itself, the temperature-scale-factor relation it implies, and the counting rule for $g_{*s}$ . We build them in order, then apply the chain to the neutrinos. The starting point is the equilibrium definition of entropy density for a fluid with negligible chemical potential. It follows from the first law applied to a comoving volume where all the heat is carried by entropy.

s=\frac{\rho+p}{T} \qquad (1)

where: $s$ — entropy density (entropy per unit physical volume); $\rho$ — energy density; $p$ — pressure; $T$ — temperature. Reading: at $\mu=0$ the entropy density is fixed entirely by the energy density, pressure, and temperature — no independent particle-number bookkeeping is needed. For a gas of relativistic species the equation of state is $p=\rho/3$ , and inserting the standard relativistic energy density collapses (1) to a clean cubic in temperature.

s=\frac{2\pi^2}{45}\,g_{*s}\,T^3 \qquad (2)

where: $g_{*s}$ — effective entropy degrees of freedom; $T$ — photon (plasma) temperature. Reading: entropy density scales as $g_{*s}T^3$ , one power of $T$ below the energy density's $g_*T^4$ — the difference in power is what makes $g_{*s}$ and $g_*$ diverge once temperatures split. The conservation law is the physical heart of the article: in adiabatic expansion the entropy in a comoving volume is a constant of the motion.

\frac{\mathrm{d}}{\mathrm{d}t}\!\left(s\,a^3\right)=0 \quad\Longrightarrow\quad s\,a^3=\text{const} \qquad (3)

where: $a$ — cosmological scale factor; $s\,a^3$ — entropy inside a comoving volume. Reading: this is not an approximation bolted on after the fact — Derivation A shows it is the fluid equation itself, rewritten. Combining (2) and (3) gives the working relation that replaces the textbook $T\propto1/a$ with something exact across annihilation thresholds.

g_{*s}(T)\,T^3\,a^3=\text{const} \qquad (4)

where: $g_{*s}(T)$ — the staircase function of the previous section. Reading: $T\propto g_{*s}^{-1/3}a^{-1}$ , so the temperature falls faster than $1/a$ only in the trivial sense that $g_{*s}$ is fixed; whenever $g_{*s}$ drops, $aT$ jumps up to compensate. The counting rule assembles $g_{*s}$ from the internal degrees of freedom of every relativistic species, each weighted by the cube of its own temperature, with the familiar $7/8$ penalty on fermions.

g_{*s}=\sum_{i=\text{bosons}}g_i\!\left(\frac{T_i}{T}\right)^{\!3}+\frac{7}{8}\sum_{i=\text{fermions}}g_i\!\left(\frac{T_i}{T}\right)^{\!3} \qquad (5)

where: $g_i$ — internal degrees of freedom of species $i$ (spin, color, particle/antiparticle); $T_i$ — temperature of species $i$ , which need not equal the photon $T$ . Reading: the cube of the temperature ratio is the whole difference from the energy version $g_*$ , which uses the fourth power. Applying the chain across $e^+e^-$ annihilation, with neutrinos already decoupled, yields the field's most quoted number.

\frac{T_\nu}{T_\gamma}=\left(\frac{4}{11}\right)^{\!1/3}\approx0.7138 \qquad (6)

where: $T_\nu$ — cosmic neutrino background temperature; $T_\gamma$ — photon (CMB) temperature. Reading: derived in full below, this ratio is a pure consequence of entropy conservation and the ordering of decoupling before annihilation. Anchoring to the measured photon temperature gives the present-day thermometer readings of the two backgrounds.

T_\gamma=2.7255\ \mathrm{K}\;\Rightarrow\;T_\nu\approx1.945\ \mathrm{K} \qquad (7)

where: $2.7255\ \mathrm{K}$ — the FIRAS/COBE-measured CMB temperature. Reading: the neutrino background is a firm prediction at $1.945\ \mathrm{K}$ , colder than the CMB by exactly the factor $(4/11)^{1/3}$ . Finally, because $s\,a^3$ and any conserved number $N\,a^3$ both dilute as $a^{-3}$ , their ratio is a redshift-independent constant — the master normalizer for relic abundances.

\frac{n_b}{s}=\text{const}\approx0.86\times10^{-10}\left(\frac{\eta}{6\times10^{-10}}\right) \qquad (8)

where: $n_b$ — net baryon number density; $\eta=n_b/n_\gamma$ — baryon-to-photon ratio; $s$ — entropy density. Reading: $n_b/s$ is more fundamental than $\eta$ because $s$ is conserved while $n_\gamma$ jumps at every annihilation.

Derivation AComoving entropy conservation is the fluid equation in disguise

The goal is to prove that the statement $s\,a^3=\text{const}$ is not an extra postulate but is exactly the covariant energy-conservation (fluid) equation of a homogeneous universe, rewritten with the equilibrium relation $s=(\rho+p)/T$ . We start from the $00$ -component of $\nabla_\mu T^{\mu\nu}=0$ for a perfect fluid, substitute the thermodynamic identities valid at vanishing chemical potential, and integrate to expose the conserved comoving entropy. This tells us that whenever the plasma is in equilibrium, no separate entropy equation is needed — conservation of energy already guarantees it.

Step 1. Write the fluid equation, $\dot\rho+3\frac{\dot a}{a}(\rho+p)=0$ , the $00$ -component of $\nabla_\mu T^{\mu\nu}=0$ for a homogeneous perfect fluid in an FRW background. Why: this is the only dynamical content of energy conservation in a homogeneous universe — everything else must follow from it.

Step 2. Insert the equilibrium relation $\rho+p=Ts$ from equation (1), giving $\dot\rho+3\frac{\dot a}{a}Ts=0$ . Why: this is the single place where thermodynamics enters; without equilibrium the substitution fails and entropy can be produced.

Step 3. Use the Gibbs-Duhem identity at $\mu=0$ : since $\rho=Ts-p$ and $\mathrm{d}p=s\,\mathrm{d}T$ , differentiating gives $\mathrm{d}\rho=T\,\mathrm{d}s$ , hence $\dot\rho=T\dot s$ . Why: it converts the energy-density rate into an entropy-density rate, which is what we want to track.

Step 4. Combine to get $T\dot s+3\frac{\dot a}{a}Ts=0$ , and divide by $T$ to obtain $\dot s+3\frac{\dot a}{a}s=0$ . Why: dividing out $T$ leaves a pure continuity equation for entropy density, identical in form to number conservation.

Step 5. Integrate $\dot s/s=-3\dot a/a$ to find $s\propto a^{-3}$ , i.e. $s\,a^3=\text{const}$ . Why: the entropy density dilutes exactly like a conserved number density, so the comoving entropy is fixed — the result we set out to prove.

Go deeper · Derivation B · 1 minThe neutrino-to-photon temperature ratio from entropy conservation

Derivation BThe neutrino-to-photon temperature ratio from entropy conservation

The goal is to derive $T_\nu/T_\gamma=(4/11)^{1/3}$ by applying comoving entropy conservation separately to two sectors that stop talking to each other just before $e^+e^-$ annihilation: the coupled photon-electron-positron plasma, and the decoupled neutrinos. The key physical input is timing — neutrinos leave equilibrium at $T\sim1\ \mathrm{MeV}$ , slightly above the electron mass $m_e\approx0.511\ \mathrm{MeV}$ , so the entropy released when pairs annihilate flows only into photons. We count $g_{*s}$ before and after in the coupled sector, hold $aT_\nu$ fixed for the neutrinos, and divide the two conservation statements.

Step 1. Count the coupled degrees of freedom before annihilation: photons contribute $g=2$ , and $e^\pm$ contribute $\frac{7}{8}(2+2)=\frac{7}{2}$ , so $g_{*s}^{\text{before}}=2+\frac{7}{2}=\frac{11}{2}$ . Why: only species still in equilibrium with photons share in the entropy transfer, and just before annihilation those are photons and the electron-positron pairs.

Step 2. After the pairs annihilate, only photons remain relativistic in the coupled sector, so $g_{*s}^{\text{after}}=2$ . Why: the electrons and positrons have become non-relativistic and disappeared, leaving photons to absorb all their entropy.

Step 3. Apply $g_{*s}(aT)^3=\text{const}$ to the coupled sector: $\frac{11}{2}(aT)^3_{\text{before}}=2\,(aT_\gamma)^3_{\text{after}}$ . Why: this is equation (4) written across the transition for the sector that stays in equilibrium.

Step 4. For the decoupled neutrinos, entropy conservation in their own sector gives $aT_\nu=\text{const}$ ; since $T_\nu=T$ just before annihilation, $(aT_\nu)_{\text{after}}=(aT)_{\text{before}}$ . Why: having decoupled, neutrinos redshift undisturbed as $1/a$ and simply remember the common temperature they last shared.

Step 5. Divide the photon relation by $(aT_\nu)^3=(aT)^3_{\text{before}}$ to get $\left(T_\gamma/T_\nu\right)^3=\frac{11}{4}$ , hence $T_\nu/T_\gamma=(4/11)^{1/3}\approx0.7138$ . Why: the ratio is entirely fixed by the drop in $g_{*s}$ from $11/2$ to $2$ , an integer-counting result independent of any measured scale. *Note: $(4/11)^{1/3}=0.71377$ , and with $T_\gamma=2.7255\ \mathrm{K}$ this gives $T_\nu=1.9454\ \mathrm{K}$ .*

Go deeper · Derivation C · 1 minThe relic neutrino number density today

Derivation CThe relic neutrino number density today

The goal is to turn the temperature ratio into a countable prediction: how many relic neutrinos fill each cubic centimetre of space now. Neutrinos froze out while relativistic and keep a relativistic Fermi-Dirac spectrum at temperature $T_\nu=(4/11)^{1/3}T_\gamma$ , so we can evaluate their number density directly and compare it with the photon number density that FIRAS fixes. The subtlety worth stating carefully is the degrees-of-freedom count — each flavor contributes one neutrino helicity plus one antineutrino helicity — because it is the most common place to be off by a factor of two.

Step 1. Photon number density: $n_\gamma=\frac{2\zeta(3)}{\pi^2}T_\gamma^3\approx411\ \mathrm{cm}^{-3}$ with $\zeta(3)\approx1.202$ evaluated at $T_\gamma=2.7255\ \mathrm{K}$ . Why: photons are the reference bosonic gas of $g=2$ ; every relic abundance is naturally quoted per photon.

Step 2. Neutrino number density per flavor, counting $\nu$ and $\bar\nu$ : fermions carry a $3/4$ factor relative to bosons, and one flavor supplies $g=2$ states, so $n_\nu+n_{\bar\nu}=\frac{3}{4}\cdot\frac{2\zeta(3)}{\pi^2}\,T_\nu^3$ . Why: the $3/4$ is the Fermi-Dirac versus Bose-Einstein number ratio, and the two states are the single neutrino helicity plus its antiparticle.

Step 3. Take the ratio and insert the temperature relation: $\dfrac{n_\nu+n_{\bar\nu}}{n_\gamma}=\dfrac{3}{4}\left(\dfrac{T_\nu}{T_\gamma}\right)^{3}=\dfrac{3}{4}\cdot\dfrac{4}{11}=\dfrac{3}{11}$ . Why: the temperature cube is exactly $4/11$ by the previous derivation, so the $g_{*s}$ counting collapses the ratio to a rational number.

Step 4. Evaluate: $n_\nu+n_{\bar\nu}=\frac{3}{11}(411)\approx112\ \mathrm{cm}^{-3}$ per flavor, and summing three flavors gives $\approx336\ \mathrm{cm}^{-3}$ of relic neutrinos plus antineutrinos. Why: this is the concrete, if still undetected, prediction — a neutrino sea denser than the photon sea by number in every direction. *Note: $\frac{3}{11}\times411=112\ \mathrm{cm}^{-3}$ ; three flavors give $336\ \mathrm{cm}^{-3}$ . The per-flavor value is $\frac{3}{11}n_\gamma$ , not $\frac{6}{11}n_\gamma$ .*

Limiting cases

Entropy conservation wears different faces in different regimes; the table collects the four that matter.

Case	Result	Meaning
All species one temperature	$g_{s}=g_=\sum_{\text{bos}}g_i+\tfrac{7}{8}\sum_{\text{ferm}}g_i$	Before any decoupling the entropy and energy counts coincide and $T\propto1/a$ exactly.
A species decoupled at $T_i\neq T$	$g_{*s}=\sum_{\text{bos}}g_i\big(\tfrac{T_i}{T}\big)^3+\tfrac{7}{8}\sum_{\text{ferm}}g_i\big(\tfrac{T_i}{T}\big)^3$	Each decoupled species is weighted by the cube of its own temperature; now $g_{s}\neq g_$ (which uses the fourth power).
Radiation era today	$g_{*s}\to\tfrac{43}{11}\approx3.909$	Photons plus three neutrino flavors at $T_\nu=(4/11)^{1/3}T_\gamma$ : $2+\tfrac{7}{8}\cdot6\cdot\tfrac{4}{11}=43/11$ , while $g_*\approx3.36$ .
Entropy-production epoch	$\dfrac{\mathrm{d}(s\,a^3)}{\mathrm{d}t}\ge0$	First-order phase transitions, out-of-equilibrium decays, or inflationary reheating inject entropy and dilute pre-existing relics.

Common trapConfusing g star s with g star

The single most common error is to treat $g_{*s}$ (entropy degrees of freedom) and $g_*$ (energy-density degrees of freedom) as interchangeable. They are numerically equal only while every relativistic species shares one temperature. The instant a species decouples and cools at its own $T_i$ , the two counts split: $g_*$ weights that species by $(T_i/T)^4$ but $g_{*s}$ weights it by $(T_i/T)^3$ . Today the gap is real and measurable — $g_{*s,0}=43/11\approx3.909$ while $g_{*,0}\approx3.36$ . Using $g_*$ where $g_{*s}$ belongs corrupts the $T_\nu/T_\gamma$ ratio, the relic abundance normalization, and the expansion rate during nucleosynthesis. Rule of thumb: anything conserved in a comoving volume (entropy, temperature ratios, relic yields) uses $g_{*s}$ ; anything about the expansion rate through $\rho$ uses $g_*$ .

Worked examples

1 · The cosmic neutrino background today (fully worked)

Given the measured CMB temperature $T_\gamma=2.7255\ \mathrm{K}$ , predict the neutrino background temperature and its number density, and compare with the photon number density.

Step 1. Temperature. Entropy conservation across $e^+e^-$ annihilation fixes $T_\nu/T_\gamma=(4/11)^{1/3}=0.7138$ , so $T_\nu=0.7138\times2.7255=\mathbf{1.945\ K}$ . This is colder than the CMB by exactly the factor set by the coupled-sector drop $g_{*s}:11/2\to2$ .

Step 2. Photon number density. $n_\gamma=\frac{2\zeta(3)}{\pi^2}T_\gamma^3=\mathbf{411\ cm^{-3}}$ with $\zeta(3)=1.202$ — the reference against which the neutrino sea is measured.

Step 3. Neutrino number density. Per flavor, counting $\nu$ and $\bar\nu$ , $n_\nu+n_{\bar\nu}=\frac{3}{4}\left(\frac{T_\nu}{T_\gamma}\right)^3 n_\gamma=\frac{3}{4}\cdot\frac{4}{11}\cdot411=\frac{3}{11}\cdot411=\mathbf{112\ cm^{-3}}$ . Summing three flavors gives $\mathbf{\approx336\ cm^{-3}}$ of relic neutrinos and antineutrinos — a background denser by number than the CMB photons, yet still undetected because each carries only $\sim10^{-4}\ \mathrm{eV}$ of energy.

2 · Effective degrees of freedom and the radiation budget after annihilation (fully worked)

After $e^+e^-$ annihilation, compute the entropy and energy degrees of freedom, and the neutrino contribution to the radiation energy density.

Step 1. Entropy count. Photons give $g=2$ ; three neutrino flavors ( $\nu+\bar\nu$ , so $g=6$ fermionic states) at $T_\nu=(4/11)^{1/3}T_\gamma$ contribute $\frac{7}{8}\cdot6\cdot\left(\frac{T_\nu}{T_\gamma}\right)^3=\frac{7}{8}\cdot6\cdot\frac{4}{11}=\frac{21}{11}$ . Hence $g_{*s,0}=2+\frac{21}{11}=\mathbf{43/11=3.909}$ .

Step 2. Energy count. The same neutrinos enter $g_*$ with the fourth power: $\frac{7}{8}\cdot6\cdot\left(\frac{4}{11}\right)^{4/3}=\frac{7}{8}\cdot6\cdot0.2596=1.363$ , so $g_{*,0}=2+1.363=\mathbf{3.36}$ . The split between $3.909$ and $3.36$ is the concrete signature of neutrinos and photons no longer sharing a temperature.

Step 3. Radiation budget. The neutrino energy density is $\rho_\nu=N_\mathrm{eff}\cdot\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\rho_\gamma$ . Instantaneous decoupling gives $N_\mathrm{eff}=3$ ; the tiny reheating that leaks into neutrinos during annihilation nudges this to the standard $\mathbf{N_{eff}=3.044}$ . This one number, fixed by entropy conservation, is what CMB and BBN observations test for hidden relativistic species.

3 · Baryon-to-entropy as a conserved clock

Show that $n_b/s$ is constant and evaluate it for $\eta=6\times10^{-10}$ .

Step 1. Both $n_b$ (a conserved number) and $s$ dilute as $a^{-3}$ away from entropy-production epochs, so their ratio $n_b/s$ is independent of redshift — unlike $\eta=n_b/n_\gamma$ , which steps whenever $n_\gamma$ is reheated.

Step 2. Using $s_0=\frac{2\pi^2}{45}g_{*s,0}T_\gamma^3\approx2890\ \mathrm{cm}^{-3}$ and $s_0/n_\gamma\approx7.04$ , we get $n_b/s=\eta\,(n_\gamma/s)=\frac{6\times10^{-10}}{7.04}=\mathbf{0.85\times10^{-10}}$ , matching equation (8). This is the truly conserved measure of the matter-antimatter asymmetry.

Physics of the portrait

Comoving entropy conservation is a piece of relativistic thermodynamics before it is a piece of cosmology, and it entered the subject through one book.

Origins · 1 minRichard Tolman and how this was discovered

Richard Tolman

1881–1948 · relativistic thermodynamics and cosmology

Richard Chace Tolman was the physical chemist turned mathematical physicist who welded thermodynamics onto general relativity. Trained at MIT and steeped in statistical mechanics, he spent the 1920s and 1930s at Caltech asking what the laws of heat become in a curved, expanding spacetime. His 1934 treatise Relativity, Thermodynamics and Cosmology is where the modern statement lives: for matter and radiation in equilibrium, an expanding Friedmann universe evolves reversibly and adiabatically, so the entropy in a comoving volume is conserved and a black-body gas cools as $T\propto1/a$ . Tolman worked out the relativistic form of the second law, the behavior of radiation in expanding models, and the conditions under which entropy could instead be produced — the same caveats that today flag reheating and first-order transitions. He did this decades before the hot big bang was taken seriously; the framework sat ready when George Gamow, Ralph Alpher, and Robert Herman revived a hot early universe in the late 1940s and needed exactly Tolman's adiabatic accounting to track temperature against expansion. When Alpher, Follin, and Herman computed neutrino decoupling in 1953 and found the photon-neutrino temperature offset, they were applying Tolman's conservation law across an annihilation threshold. The clean modern derivation — $g_{*s}T^3a^3=\text{const}$ feeding $T_\nu/T_\gamma=(4/11)^{1/3}$ — is Tolman's thermodynamics wearing the language of particle degrees of freedom.

History

1934

— Tolman formulates reversible adiabatic expansion, fixing entropy conservation and T ∝ 1/a for radiation

1948

— Alpher, Herman predict a relic radiation background of order a few kelvin from the hot early universe

1953

— Alpher, Follin, Herman compute neutrino decoupling and the photon-neutrino temperature offset

1965

— Penzias and Wilson detect the 3 K cosmic microwave background

1966

— Peebles computes primordial helium using the correct thermal history and neutrino temperature

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 5 (Thermal History) · Dodelson, Modern Cosmology, 2nd ed., ch. 3 · Kolb and Turner, The Early Universe, ch. 3 · Weinberg, Cosmology, ch. 3 · Carroll and Ostlie, An Introduction to Modern Astrophysics, ch. 30 · Mo, van den Bosch, and White, Galaxy Formation and Evolution, ch. 3. The key numerical anchors for this article are: $g_{*s}^{\text{before}}=11/2$ , $g_{*s}^{\text{after}}=2$ ; $T_\nu/T_\gamma=(4/11)^{1/3}=0.7138$ , giving $T_\nu=1.945\ \mathrm{K}$ from $T_\gamma=2.7255\ \mathrm{K}$ ; $g_{*s,0}=43/11=3.909$ , $g_{*,0}=3.36$ ; $n_\gamma=411\ \mathrm{cm}^{-3}$ ; per-flavor relic $n_\nu+n_{\bar\nu}=\tfrac{3}{11}n_\gamma=112\ \mathrm{cm}^{-3}$ , all-flavor $336\ \mathrm{cm}^{-3}$ ; $s_0\approx2890\ \mathrm{cm}^{-3}$ ; and $n_b/s\approx0.86\times10^{-10}$ at $\eta=6\times10^{-10}$ .

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Thermal History of the Universe · Effective Number of Relativistic Degrees of Freedom. Leads to. Electron-Positron Annihilation and Photon Reheating · Neutrino Decoupling and the Cosmic Neutrino Background · Big Bang Nucleosynthesis · Decoupling and Freeze-Out.

Ask the tutor: Ask me to walk through the $g_{*s}:11/2\to2$ counting and watch the number $(4/11)^{1/3}$ fall out of pure entropy bookkeeping.

Why track entropy rather than just temperature or energy

Because $s\,a^3$ is a genuine constant of the adiabatic expansion, whereas $T$ , $\rho$ , and $n$ each carry the extra complication of a changing $g_{*s}$ . If you track temperature, you must remember that $T\propto1/a$ only holds between annihilation thresholds and jumps at each one. If you track energy, it redshifts differently for relativistic and non-relativistic species. Entropy sidesteps all of it: writing entropy before equals entropy after across any transition pins down temperature ratios and abundances in one line. It is the conserved quantity that makes the whole thermal history calculable without integrating the messy microphysics of every reaction.

Why are neutrinos colder than photons if both started in the same plasma

Timing. Neutrinos decouple at $T\sim1\ \mathrm{MeV}$ , just above the electron mass. Very shortly after, the temperature falls below $m_e\approx0.511\ \mathrm{MeV}$ and electron-positron pairs annihilate, releasing their energy and entropy. But by then neutrinos are already free-streaming and no longer collide with the plasma, so that entropy flows only into photons, electrons, and positrons that remain coupled. The photons are reheated by a factor $(11/4)^{1/3}\approx1.40$ relative to the $1/a$ scaling, while neutrinos coast along at the old rate. The result is the fixed offset $T_\nu/T_\gamma=(4/11)^{1/3}$ , frozen in for all later times.

Where does the factor 11/4 actually come from

It is a degrees-of-freedom ratio, nothing more. Just before annihilation the coupled sector contains photons ( $g=2$ ) and $e^\pm$ ( $\frac{7}{8}\times4=\frac{7}{2}$ ), giving $g_{*s}=\frac{11}{2}$ . Afterward only photons remain, $g_{*s}=2$ . Comoving entropy conservation, $g_{*s}(aT)^3=\text{const}$ , then demands $(aT_\gamma)^3$ grow by the ratio $\frac{11/2}{2}=\frac{11}{4}$ . Since the decoupled neutrinos kept $aT_\nu$ fixed at its pre-annihilation value, dividing gives $(T_\gamma/T_\nu)^3=11/4$ . The 11 is $2+\frac{7}{2}$ doubled, and the 4 is $2$ doubled — pure integer counting of spin and particle-antiparticle states.

Is the entropy really conserved exactly, or only approximately

Almost exactly, but not perfectly. Conservation assumes the expansion is adiabatic and reversible — that all relevant reactions keep pace with the Hubble rate so no net entropy is produced. This holds superbly during smooth radiation-dominated expansion. It fails, deliberately, during out-of-equilibrium episodes: reheating at the end of inflation, a strongly first-order phase transition, or the decay of a massive metastable particle, each of which injects entropy and dilutes any relic present beforehand. Even the neutrino case has a tiny correction: decoupling is not perfectly instantaneous, so a sliver of annihilation entropy leaks into neutrinos, which is precisely why $N_\mathrm{eff}=3.044$ rather than exactly 3.

What is the difference between g star and g star s in one sentence

Both count relativistic degrees of freedom with a $7/8$ penalty on fermions, but $g_*$ weights each species by $(T_i/T)^4$ because it feeds the energy density $\rho\propto g_*T^4$ , while $g_{*s}$ weights each by $(T_i/T)^3$ because it feeds the entropy density $s\propto g_{*s}T^3$ ; they agree when all temperatures are equal and diverge (today $3.36$ versus $3.909$ ) once neutrinos and photons part ways.

Why is n_b over s preferred to the baryon-to-photon ratio

Because $s$ is conserved in a comoving volume while $n_\gamma$ is not. Every time a species annihilates and reheats the photons, $n_\gamma$ jumps upward, so $\eta=n_b/n_\gamma$ actually changes across those thresholds — the value quoted today is smaller than the value during, say, nucleosynthesis by the accumulated reheating. The entropy density absorbs those jumps automatically, so $n_b/s$ stays genuinely constant from baryogenesis to now. Theorists quote $n_b/s\approx0.86\times10^{-10}$ ; the observational $\eta\approx6\times10^{-10}$ is the same physics referred to the present photon count.