advanced19 min studyLesson 853 in Physics

Entropy conservation and relativistic degrees of freedom

Track energy and entropy separately when species become massive or leave a thermal bath.

Before you begin

On this page

The question for today

What you’ll understand

Distinguish the energy count $g_*$ from the entropy count $g_{*s}$ .
Use comoving entropy conservation across an annihilation threshold.
Count relativistic species with unequal temperatures and declared internal multiplicities.

Temperature is an incomplete cosmic clock when the particle content changes. A bath that loses a massive species can cool more slowly than inverse scale factor because the remaining particles share its entropy. The useful bookkeeping quantities are effective degrees of freedom for energy and for entropy, which need not coincide. We use natural units, negligible chemical potentials, and thermal Bose or Fermi distributions for each counted relativistic species. These assumptions make simple counting possible. Near a mass threshold, through an interacting transition, or for a distorted distribution, the corresponding thermodynamic integrals must replace a sharp integer count.

Two weighted counts, two different moments

An internal multiplicity counts independently occupied spin, polarization, particle-antiparticle, color, and flavor states as appropriate. It must not silently double-count an antiparticle already included in a listed species. The Fermi energy and entropy integrals carry a factor of seven eighths relative to bosons with the same temperature and multiplicity. That factor is not the three-quarters ratio appearing in relativistic number densities; the integrals weight different powers of momentum.

Define all effective counts relative to a reference temperature, here the photon temperature. A colder decoupled relativistic sector contributes its fourth temperature-ratio power to energy and its third power to entropy. Therefore the same additional particle can alter the expansion rate and the entropy-temperature conversion by different amounts. Equality of the counts is a useful ideal-gas limit when all species share a temperature, not a definition that remains valid across every cosmic epoch.

\rho_r=\frac{\pi^2}{30}g_*T_\gamma^4,\quad s=\frac{2\pi^2}{45}g_{*s}T_\gamma^3,\quad g_* =\sum_Bg_i r_i^4+\frac78\sum_Fg_i r_i^4,\quad g_{*s}=\sum_Bg_i r_i^3+\frac78\sum_Fg_i r_i^3

Here

r_i=T_i/T_\gamma

and

g_i

is an internal multiplicity. The explicit sums apply to ultrarelativistic thermal species with negligible chemical potentials; the density definitions can also define effective noninteger counts more generally.↔ Scroll wide equations horizontally.

Conserved entropy does not require constant photon number

For a closed sector undergoing reversible expansion without significant entropy production, entropy in a comoving volume is constant. Number-changing reactions can remain active while this statement holds. Electron-positron annihilation in an equilibrated electromagnetic bath is the central example: photon number is not conserved, yet the comoving entropy is approximately conserved. Holding photon number fixed at the same time would impose an additional, incorrect constraint on the temperature evolution.

Apply the conservation law to the sector that actually exchanges heat. A decoupled sector keeps its own entropy separately, provided no late decay or other injection reheats it. Applying a single common-temperature count to both sectors after decoupling incorrectly sends the annihilation entropy into particles that no longer interact. Across a continuous threshold, the gradual change in the entropy count modifies the temperature derivative even when the total evolution remains adiabatic.

S=s a^3=\text{constant},\qquad aT_\gamma g_{*s}^{1/3}=\text{constant},\qquad \frac{T_2}{T_1}=\frac{a_1}{a_2}\left(\frac{g_{*s,1}}{g_{*s,2}}\right)^{1/3}

The same closed sector must be used at both endpoints. The last expression assumes the stated entropy-density definition and no intervening entropy production or injection.↔ Scroll wide equations horizontally.

Thresholds change the clock as well as the particle inventory

In a flat radiation-dominated background, the energy count enters Friedmann's equation, yielding $H=1.66\sqrt{g_*}\,T^2/M_{\rm Pl}$ when the radiation is characterized by reference temperature $T$ . Here the unreduced Planck mass is $M_{\rm Pl}=G^{-1/2}$ , unlike the reduced mass used in scalar-field cosmology. At constant counts, the age is $t=0.301M_{\rm Pl}/(\sqrt{g_*}T^2)$ . Variable counts require solving the expansion and entropy relations together.

A threshold table gives asymptotic regimes, rather than an exact staircase in nature. Species with mass comparable to temperature are partially populated, and their pressure is no longer exactly one third of their energy density. Interactions can also modify the equation of state. Entropy release from an out-of-equilibrium massive-particle decay is a different phenomenon: comoving entropy can then increase, so an adiabatic threshold formula cannot be applied unchanged.

Work it through

Cooling across an idealized electron threshold

Consider only an isolated electromagnetic bath. Initially photons and relativistic electrons and positrons share a temperature; finally only photons contribute appreciably to its entropy. Assume reversible evolution, zero chemical potentials, and a scale-factor increase of two between the chosen endpoints. Find the final-to-initial photon-temperature ratio.

Count two photon polarizations and four electron-plus-positron spin states. The initial entropy count is $2+(7/8)4=11/2$ . The final count is $2$ . Exclude any already decoupled neutrinos from both electromagnetic-sector counts.
Conserve the electromagnetic entropy in the expanding comoving volume. The temperature ratio is $\tfrac12[(11/2)/2]^{1/3}=\tfrac12(11/4)^{1/3}$ . The volume growth contributes the factor one half, while the loss of populated states contributes a compensating cube root.
Evaluate the cube root, approximately $1.401$ , to obtain a temperature ratio near $0.7005$ . The bath has cooled overall, but is warmer than the ratio $0.5$ predicted by inverse-scale-factor cooling with an unchanged entropy count.

The result

The photon temperature falls to approximately $0.7005$ of its initial value. Calling this relative heating does not mean that temperature rises between these stipulated endpoints; it means cooling is slower than in a freely redshifting bath.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

See why an entropy count multiplies a cube

Reset and set both controls to coefficient $a=1$ , position $x=1$ ; the signed area under $y=ax^2$ is $1/3$ . Reset and set both controls to $a=1$ , $x=2$ ; that area is $8/3$ . The cubic scaling illustrates the temperature power in a fixed-count entropy density. The lab is a polynomial integral, with no fermion weighting, mass threshold, or expanding bath.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAt fixed relativistic entropy count, doubling temperature multiplies entropy density by eight; changing the count requires separate physics.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

For thermal relativistic distributions, energy scales as

T_i^4

and entropy as

T_i^3

. Different sector temperatures therefore produce different effective counts.

Your turn

Try it yourself

A photon bath with two polarizations has temperature $T$ . Add one decoupled ultrarelativistic thermal fermion species with total internal multiplicity two and temperature $T/2$ . Neglect chemical potentials and interactions. Compute both effective counts relative to the photon temperature.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Include the fermion factor and use different powers of the temperature ratio.

Compare with the solution

The fermion adds $(7/8)2(1/2)^4=7/64$ to the energy count, so $g_*=135/64=2.109375$ . It adds $(7/8)2(1/2)^3=7/32$ to the entropy count, giving $g_{*s}=71/32=2.21875$ . The two-state multiplicity is already complete and must not receive another particle-antiparticle factor. The colder sector contributes proportionally less energy than entropy relative to the reference bath.

Keep the idea

Take it with you

Declare internal multiplicities and sector temperatures before counting states.
Conserve entropy separately in sectors that no longer exchange heat.
The energy count determines expansion, while the entropy count determines adiabatic cooling.

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