Effective Number of Relativistic Degrees of Freedom

The early Universe was a furnace of relativistic particles, and its energy density set the pace of everything that happened next — how fast space expanded, when the weak interactions switched off, how much helium was forged. Tracking every photon, quark, lepton and gauge boson separately would be hopeless, so cosmologists compress the entire relativistic particle content into a single dimensionless number, $g_*(T)$ , the effective number of relativistic degrees of freedom. It is the count that multiplies the blackbody law for one bosonic state to give the total radiation energy density: $\rho_{\rm rad} = \tfrac{\pi^2}{30}\,g_*(T)\,T^4$ . The construction rests on three ideas from equilibrium thermodynamics, developed in this chapter's earlier articles and recalled here by name. First, a relativistic boson contributes $\tfrac{\pi^2}{30}g\,T^4$ and a fermion exactly $\tfrac{7}{8}$ of that, a ratio fixed by Bose–Einstein versus Fermi–Dirac statistics. Second, each species is weighted by its internal states $g$ — spins, colours, particle and antiparticle. Third, a species that has decoupled from the photon bath carries its own temperature $T_i$ , and enters $g_*$ scaled by $(T_i/T)^4$ . A companion count $g_{*s}(T)$ , weighted by $(T_i/T)^3$ , governs entropy and is what conservation of comoving entropy actually constrains. As the Universe cools, the temperature sweeps down through the mass thresholds of the particle spectrum. Each time a species becomes non-relativistic it annihilates, dumps its entropy into the surviving plasma, and $g_*$ steps down — from $106.75$ across the whole Standard Model to $3.36$ today, when only photons and three cold neutrino backgrounds remain. Getting these steps right, especially the subtle heating of photons over neutrinos at electron–positron annihilation, is what pins down the primordial helium yield and the $1.95\ \mathrm{K}$ temperature of the relic neutrino sea.

Key takeaways

The effective number of relativistic degrees of freedom $g_*(T)$ is a temperature-dependent count of the species that contribute to the radiation energy density, defined by $\rho_{\rm rad} = \tfrac{\pi^2}{30}g_*(T)\,T^4$ .
Bosons enter with weight $1$ and fermions with weight $7/8$ , each times their internal degrees of freedom (spin, colour, particle and antiparticle).
Two counts exist: $g_*$ for energy ( $\rho \propto g_* T^4$ ) and $g_{*s}$ for entropy ( $s \propto g_{*s} T^3$ ); they coincide only while every species shares one temperature.
$g_*$ falls in steps as species become non-relativistic — from $106.75$ in the full Standard Model, through $10.75$ at nucleosynthesis, to $3.36$ ( $g_{*s}=3.91$ ) today.
The step at $e^+e^-$ annihilation heats photons over the already-decoupled neutrinos, giving $T_\nu/T_\gamma=(4/11)^{1/3}\approx0.714$ and hence $T_\nu\approx1.95\ \mathrm{K}$ .
Through the Friedmann equation $H\propto\sqrt{g_*}\,T^2$ , this one number sets the expansion rate that fixes weak freeze-out, the neutron-to-proton ratio, and the helium abundance.
Key idea: collapse the entire relativistic particle content into one number — $\rho_{\rm rad}=\tfrac{\pi^2}{30}g_*(T)\,T^4$ with bosons weighted 1, fermions weighted 7/8, and each decoupled species scaled by $(T_i/T)^4$ .

The core insight

Collapse the entire relativistic particle content into one number: $\rho_{\rm rad} = \frac{\pi^2}{30} g_*(T)\, T^4$ , with bosons weighted 1, fermions weighted $7/8$ , and each decoupled species scaled by $(T_i/T)^4$ . Because species drop out as the Universe cools, $g_*$ is a descending staircase in temperature.

Where this lives today

The effective degree-of-freedom count is now a precision tool. Its most active frontier is $N_{\rm eff}$ , the effective number of neutrino species, defined through $\rho_{\rm rad}=\rho_\gamma[1+\tfrac{7}{8}(4/11)^{4/3}N_{\rm eff}]$ . The Standard Model predicts $N_{\rm eff}=3.044$ — slightly above three because neutrino decoupling is not quite complete when $e^+e^-$ annihilation begins, so a little annihilation entropy leaks into the neutrinos and the finite-temperature QED correction shifts the count. Planck's measurement of the cosmic microwave background gives $N_{\rm eff}=2.99\pm0.17$ , in striking agreement and tight enough to constrain hypothetical extra light species (sterile neutrinos, thermalized axions, dark radiation) at the level of a fraction of one degree of freedom. The same $g_*(T)$ enters every calculation of a thermal relic's abundance: a WIMP freezing out at $T_f\sim m_\chi/20$ has a relic density set by $H(T_f)\propto\sqrt{g_*(T_f)}$ , so precise dark-matter predictions demand $g_*$ across the QCD transition, where lattice results have replaced the old sharp-step approximation with a smooth crossover. Next-generation CMB surveys (CMB-S4, the Simons Observatory) aim to measure $N_{\rm eff}$ to $\pm0.03$ — sharp enough to detect, or exclude, any particle that was ever in thermal equilibrium down to the temperature of the QCD epoch.

The physical picture

Think of $g_*$ as the number of independent channels available to hold radiation energy at a given temperature. Every photon polarization, every quark colour, every neutrino helicity is a channel, and each open channel stores energy $\propto T^4$ . When the plasma cools past a particle's rest mass, that channel closes: the species is Boltzmann-suppressed, annihilates or decays, and hands its energy and entropy to the survivors. Like burners switching off under a row of pots, the total heat is redistributed among fewer and fewer species, and $g_*(T)$ steps downward as the clock runs forward. The first figure is that staircase — the actual history of the count from the electroweak epoch to today.

Figure

The staircase of g_*(T): the relativistic count falls from 106.75 above the electroweak scale, through 10.75 at nucleosynthesis (photons, e⁺e⁻ and three neutrinos), to 3.36 today as species annihilate and dump their entropy into the survivors

The fermion factor $7/8$ that weights each closed channel is not a fudge: it is the exact ratio of the Fermi–Dirac to the Bose–Einstein energy integral, $\int x^3\,\mathrm{d}x/(e^x+1)$ over $\int x^3\,\mathrm{d}x/(e^x-1)$ . And the reason two counts are needed — $g_*$ for energy, $g_{*s}$ for entropy — is that once a species decouples and cools below the photon temperature, energy density weights its coldness by $(T_i/T)^4$ while entropy weights it by $(T_i/T)^3$ . The single most consequential step in the staircase is electron–positron annihilation. Neutrinos have just decoupled, so when the $e^+e^-$ pairs annihilate their entropy pours only into the photons, heating the photon bath above the neutrino bath by a factor that the second figure makes concrete.

Figure

Electron–positron annihilation reheats photons but not the already-decoupled neutrinos: comoving entropy conservation takes g_*s from 11/2 to 2 in the photon sector, so T_γ/T_ν = (11/4)^(1/3) = 1.401 and the relic neutrinos end up colder at T_ν/T_γ = 0.714

The core equations

Everything begins with the energy density of a single bath of relativistic particles. The Stefan–Boltzmann law for one bosonic degree of freedom becomes the whole radiation budget once we insert the count $g_*$ .

\rho_{\rm rad} = \frac{\pi^2}{30}\, g_*(T)\, T^4 \qquad (1)

where: $\rho_{\rm rad}$ — total energy density of all relativistic species; $g_*(T)$ — effective number of relativistic degrees of freedom for energy; $T$ — photon (plasma) temperature. In natural units $\hbar=c=k_B=1$ ; a larger $g_*$ at fixed $T$ means more energy and, through Friedmann, faster expansion. The count itself is a weighted sum over species. Bosons enter directly, fermions with the statistical factor $7/8$ , and any species not sharing the photon temperature is rescaled by the fourth power of its own temperature.

g_*(T) = \sum_{i\in\text{bosons}} g_i \left(\frac{T_i}{T}\right)^4 + \frac{7}{8}\sum_{i\in\text{fermions}} g_i \left(\frac{T_i}{T}\right)^4 \qquad (2)

where: $g_i$ — internal degrees of freedom of species $i$ (spin states × particle/antiparticle × colour); $T_i$ — its temperature, equal to $T$ unless it has decoupled; $7/8$ — the Fermi–Dirac suppression. Read it as: count the states, weight fermions by $7/8$ , and discount any cold relic by $(T_i/T)^4$ . Entropy obeys a parallel bookkeeping, and it is the one that is actually conserved in a comoving volume, which makes it the workhorse for tracking decoupled relics.

s = \frac{2\pi^2}{45}\, g_{*s}(T)\, T^3, \qquad g_{*s}(T) = \sum_{i\in\text{bosons}} g_i \left(\frac{T_i}{T}\right)^3 + \frac{7}{8}\sum_{i\in\text{fermions}} g_i \left(\frac{T_i}{T}\right)^3 \qquad (3)

where: $s$ — entropy density of the plasma; $g_{*s}(T)$ — effective degrees of freedom for entropy. The single change from (2) is the power: $(T_i/T)^3$ rather than $(T_i/T)^4$ . That one exponent is the entire reason $g_*\neq g_{*s}$ once neutrinos run cold. The internal count $g_i$ is pure state-counting. A worked tally of the common species anchors every later sum.

g_\gamma = 2,\quad g_{e^\pm}=4,\quad g_\nu^{(\text{per flavour})}=2,\quad g_{\rm quark}^{(\text{per flavour})}=12 \qquad (4)

where: photon — 2 polarizations; electron — 2 spins × 2 for $e^-,e^+$ ; one neutrino flavour — 1 helicity × 2 for $\nu,\bar\nu$ in the massless-left-handed Standard Model; one quark flavour — 2 spin × 2 antiparticle × 3 colour. Read each as (spin) × (particle/antiparticle) × (colour), with colour absent for colourless species. The Friedmann equation converts $\rho_{\rm rad}$ into an expansion rate, and $g_*$ rides along under a square root. This is the equation that makes $g_*$ physically decisive rather than mere bookkeeping.

H(T) = 1.66\, \sqrt{g_*}\;\frac{T^2}{M_{\rm Pl}} \qquad (5)

where: $H$ — Hubble rate; $M_{\rm Pl}=1.22\times10^{19}\ \mathrm{GeV}$ — the Planck mass; $1.66=\sqrt{8\pi^3/90}$ , the residue of combining Friedmann with (1). A universe with more relativistic species expands faster at the same temperature. Neutrino decoupling followed by $e^+e^-$ annihilation leaves the two backgrounds at different temperatures, captured in one ratio.

\frac{T_\nu}{T_\gamma} = \left(\frac{4}{11}\right)^{1/3} \approx 0.714 \qquad (6)

where: $T_\nu$ — neutrino temperature after decoupling; $T_\gamma$ — photon temperature after $e^+e^-$ annihilation. With today's $T_\gamma=2.725\ \mathrm{K}$ this gives $T_\nu\approx1.95\ \mathrm{K}$ — a background not yet directly detected. Finally, the two present-day counts follow by putting photons and three cold neutrino species into (2) and (3).

g_{*s}^{\rm today} = 2 + \frac{7}{8}\cdot 6 \cdot \frac{4}{11} = \frac{43}{11}\approx 3.91, \qquad g_{*}^{\rm today} = 2 + \frac{7}{8}\cdot 6 \cdot \left(\frac{4}{11}\right)^{4/3}\approx 3.36 \qquad (7)

where: $2$ — the photon; $\tfrac{7}{8}\cdot 6$ — three neutrino species (6 fermionic dof); $4/11=(T_\nu/T_\gamma)^3$ for entropy versus $(4/11)^{4/3}=(T_\nu/T_\gamma)^4$ for energy. The two differ precisely because entropy cubes the temperature ratio while energy raises it to the fourth.

Derivation AThe seven-eighths and the master formula

The goal here is to build equation (2) from first principles: to show why a relativistic fermion contributes exactly seven-eighths of what a boson does, and why the combination $\tfrac{\pi^2}{30}g_*T^4$ is the right compression of the whole plasma. We start from the momentum-space integral for the energy density of a single species in kinetic equilibrium, evaluate it separately for Bose–Einstein and Fermi–Dirac statistics using two standard definite integrals, take their ratio, and then sum over all species with the appropriate temperature weights.

Step 1. For a relativistic species with $g$ internal states, energy $E\approx p$ , and phase-space occupation $f(p)=[e^{p/T}\mp1]^{-1}$ (upper sign bosons, lower fermions), the energy density is $\rho=\dfrac{g}{2\pi^2}\displaystyle\int_0^\infty\dfrac{p^3\,\mathrm{d}p}{e^{p/T}\mp1}$ . Substituting $x=p/T$ pulls out $T^4$ and leaves a pure number. Why: in the relativistic limit the only scale is $T$ , so dimensional analysis already forces $\rho\propto gT^4$ ; the statistics live entirely in the dimensionless integral.

Step 2. For bosons, $\displaystyle\int_0^\infty\frac{x^3\,\mathrm{d}x}{e^x-1}=\frac{\pi^4}{15}$ , giving $\rho_B=\dfrac{\pi^2}{30}g\,T^4$ . Why: this is the Stefan–Boltzmann law in disguise — for the photon's $g=2$ it reproduces $\rho_\gamma=\tfrac{\pi^2}{15}T^4$ , the familiar radiation density.

Step 3. For fermions, the alternating denominator gives $\displaystyle\int_0^\infty\frac{x^3\,\mathrm{d}x}{e^x+1}=\frac{7}{8}\cdot\frac{\pi^4}{15}$ , so $\rho_F=\dfrac{7}{8}\cdot\dfrac{\pi^2}{30}g\,T^4$ . The factor $7/8=1-2^{-3}$ comes from expanding $(e^x+1)^{-1}$ as an alternating series whose even terms subtract off one-eighth of the bosonic sum. Why: this single exact ratio is what lets us treat bosons and fermions on the same footing — a fermionic dof is just a bosonic dof discounted by seven-eighths.

Step 4. Sum over all species, giving each its own temperature factor to allow for decoupled relics, and factor out $\tfrac{\pi^2}{30}T^4$ . The bracket that remains is the definition (2): $g_*(T)=\sum_{\rm bos}g_i(T_i/T)^4+\tfrac{7}{8}\sum_{\rm ferm}g_i(T_i/T)^4$ . Why: this is the whole payoff — an entire zoo of particles collapses into one temperature-dependent number multiplying a universal blackbody expression. *Verified: the two definite integrals give $\pi^4/15$ and $\tfrac{7}{8}\pi^4/15$ ; photon $g=2$ reproduces $\rho_\gamma=\tfrac{\pi^2}{15}T^4$ ; the $7/8$ ratio is exact.*

Go deeper · Derivation B · 1 minElectron-positron annihilation and the four-elevenths

Derivation BElectron-positron annihilation and the four-elevenths

The aim is to derive the neutrino-to-photon temperature ratio $(4/11)^{1/3}$ , one of the sharpest predictions of hot big bang cosmology, using nothing but conservation of comoving entropy and the counting rules already established. The physical setting is a narrow window near $T\sim1\ \mathrm{MeV}$ : neutrinos have just fallen out of equilibrium and now free-stream, while photons and $e^+e^-$ remain tightly coupled. When the pairs annihilate, their entropy has nowhere to go but the photons, and the resulting asymmetry is permanent.

Step 1. Just before annihilation the photon-coupled bath contains photons ( $g=2$ ) and $e^\pm$ ( $g=4$ ), so its entropy weight is $g_{*s}^{\rm before}=2+\tfrac{7}{8}\cdot4=\tfrac{11}{2}$ . The neutrinos, already decoupled, are tracked as a separate bath and are deliberately excluded from this count. Why: only species in thermal contact share entropy, so the decoupled neutrinos must be bookkept independently — that separation is the crux of the whole effect.

Step 2. After annihilation the pairs are gone and only photons remain in the coupled bath, so $g_{*s}^{\rm after}=2$ . Comoving entropy conservation for that bath reads $g_{*s}^{\rm before}(aT)^3_{\rm before}=g_{*s}^{\rm after}(aT_\gamma)^3_{\rm after}$ . Why: entropy per comoving volume $S\propto g_{*s}(aT)^3$ is conserved through the smooth annihilation, so a falling $g_{*s}$ must be compensated by a rising $aT_\gamma$ — the photons are heated.

Step 3. Solve for the photon heating: $\left(aT_\gamma\right)^3_{\rm after}=\tfrac{11}{2}/2\;\left(aT\right)^3_{\rm before}=\tfrac{11}{4}(aT)^3_{\rm before}$ . The neutrinos meanwhile keep $aT_\nu=\text{const}$ , and since they had $aT_\nu=aT$ before decoupling, $(aT_\gamma/aT_\nu)^3=11/4$ . Why: the $a^3$ factors cancel the expansion common to both baths, isolating the pure heating that annihilation imparts to photons alone.

Step 4. Take the cube root: $T_\gamma/T_\nu=(11/4)^{1/3}\approx1.401$ , equivalently $T_\nu/T_\gamma=(4/11)^{1/3}\approx0.714$ . With $T_\gamma=2.725\ \mathrm{K}$ today, $T_\nu\approx1.95\ \mathrm{K}$ . Why: this is a falsifiable number — it fixes the cosmic neutrino background temperature and, through $(4/11)^{4/3}$ , its contribution to today's $g_*=3.36$ . *Verified: $(11/4)^{1/3}=1.401$ ; $(4/11)^{1/3}=0.714$ ; $0.714\times2.725\ \mathrm{K}=1.95\ \mathrm{K}$ ; matches Ryden Eq. 5.53 and Dodelson Ch. 3.*

Go deeper · Derivation C · 1 minCounting the full Standard Model

Derivation CCounting the full Standard Model

The objective is to reproduce the canonical maximum $g_*=106.75$ by counting every Standard-Model particle at a temperature above the electroweak scale, where all of them are relativistic and share a single temperature so that $g_*=g_{*s}$ . This is the anchor value at the top of the staircase, and doing the sum carefully is the best way to internalize the state-counting rules — the traps are the gluon colour factor, the massive gauge bosons' three polarizations, and the single helicity of Standard-Model neutrinos.

Step 1. Bosons. Gluons: $8$ colours × $2$ polarizations $=16$ . Massive $W^+,W^-,Z$ : $3$ each × $3$ polarizations (they have eaten Goldstone modes above the transition, so count $3$ ) $=9$ . Photon: $2$ . Higgs scalar: $1$ . Total $g_{\rm boson}=16+9+2+1=28$ . Why: gauge bosons are the subtle case — massless gluons and photons carry two transverse polarizations, but the massive weak bosons carry a third longitudinal mode.

Step 2. Fermions — quarks. Six flavours, each a colour triplet, spin- $\tfrac12$ with a distinct antiparticle: $g_{\rm quark}=6\times3\times2\times2=72$ . Why: colour triples the count for every quark, which is why the quark sector dominates the fermionic total.

Step 3. Fermions — leptons. Three charged leptons with two spins and antiparticles, $3\times2\times2=12$ ; three neutrinos with one helicity and antiparticles, $3\times1\times2=6$ . Total leptons $18$ , so $g_{\rm fermion}=72+18=90$ . Why: the neutrino's single helicity halves its count relative to a charged lepton — a Standard-Model feature that matters for $N_{\rm eff}$ .

Step 4. Combine with statistical weights: $g_*=g_{\rm boson}+\tfrac{7}{8}g_{\rm fermion}=28+\tfrac{7}{8}\cdot90=28+78.75=106.75$ . Why: this is the ceiling of the staircase; every later value is this number with channels progressively subtracted as the Universe cools. *Verified: bosonic $28$ and fermionic $90$ reproduce the canonical $g_*=106.75$ of Kolb & Turner Table 3.1.*

Limiting cases

The value of $g_*$ is entirely determined by which species are simultaneously relativistic ( $T\gtrsim m$ ) and in thermal contact. Reading the staircase off a few landmark temperatures:

Case	Result	Meaning
Full Standard Model, $T\gtrsim300\ \mathrm{GeV}$	$g_*=106.75$	Every quark, lepton, gauge boson and the Higgs is relativistic and shares one temperature; the maximum SM value, and $g_=g_{s}$ .
After QCD, before muon annihilation, $T\sim20$ – $100\ \mathrm{MeV}$	$g_*=14.25$	Only $\gamma$ , $e^\pm$ , $\mu^\pm$ and three $\nu$ remain: $2+\tfrac{7}{8}(4+4+6)=14.25$ .
Nucleosynthesis, $1\ \mathrm{MeV}\lesssim T\lesssim10\ \mathrm{MeV}$	$g_*=10.75$	Muons gone; $\gamma$ , $e^\pm$ and three $\nu$ : $2+\tfrac{7}{8}(4+6)=10.75$ . The BBN plateau.
Today, $T_0=2.725\ \mathrm{K}$	$g_=3.36,\ g_{s}=3.91$	Photons plus three cold neutrino backgrounds at $T_\nu=(4/11)^{1/3}T_\gamma$ ; energy and entropy counts now differ.

Common trapDo not conflate g_ with g_s

The two counts are not interchangeable. $g_*$ governs energy density and weights each decoupled species by $(T_i/T)^4$ ; $g_{*s}$ governs entropy density and weights by $(T_i/T)^3$ . They are equal only while every species shares a single temperature — which is true above $e^+e^-$ annihilation but false afterward, where $g_*=3.36$ while $g_{*s}=3.91$ . Two further traps: a species contributes only while relativistic, so a particle with $T\lesssim m$ is Boltzmann-suppressed ( $n\propto e^{-m/T}$ ) and silently drops out; and the neutrino temperature factor enters $g_*$ as $(4/11)^{4/3}$ , not $(4/11)$ — using the entropy power in the energy sum is a common slip that mis-states today's radiation budget.

Worked examples

1 · Photons and neutrinos today (fully worked)

Given only $T_\gamma=2.725\ \mathrm{K}$ and three Standard-Model neutrino species, compute $T_\nu$ , the present-day $g_{*s}$ and $g_*$ , and confirm they differ.

Step 1. Neutrino temperature. The relic neutrinos decoupled before $e^+e^-$ annihilation and were never reheated, so $T_\nu=(4/11)^{1/3}T_\gamma=0.7138\times2.725\ \mathrm{K}=\mathbf{1.945\ K}$ . This is the temperature of the cosmic neutrino background.

Step 2. Entropy count. Photons give $2$ ; three neutrino species carry $6$ fermionic dof at temperature ratio cubed, $(T_\nu/T_\gamma)^3=4/11$ . Hence $g_{*s}=2+\tfrac{7}{8}\cdot6\cdot\tfrac{4}{11}=2+\tfrac{21}{11}=\tfrac{43}{11}=\mathbf{3.909}$ .

Step 3. Energy count. Same species, but energy weights the temperature ratio to the fourth power: $(T_\nu/T_\gamma)^4=(4/11)^{4/3}=0.2595$ . Hence $g_*=2+\tfrac{7}{8}\cdot6\cdot0.2595=2+1.363=\mathbf{3.36}$ .

Step 4. Interpretation. The two counts differ, $g_{*s}=3.91$ versus $g_*=3.36$ , and they differ by exactly the ratio $(4/11)^{4/3}/(4/11)=(4/11)^{1/3}=0.714$ in the neutrino term — the fingerprint of the cube-versus-fourth-power distinction. T_ν = 1.95 K, g_s = 3.91, g_ = 3.36.

2 · The clock at nucleosynthesis (fully worked)

At $T=1\ \mathrm{MeV}$ the relativistic bath is $\gamma$ , $e^\pm$ and three neutrinos. Find $g_*$ , the Hubble rate, and the corresponding age of the Universe.

Step 1. Degrees of freedom. Photons $2$ , electrons/positrons $4$ , three neutrinos $6$ : $g_*=2+\tfrac{7}{8}(4+6)=2+8.75=\mathbf{10.75}$ .

Step 2. Hubble rate from equation (5). With $\sqrt{g_*}=\sqrt{10.75}=3.279$ , $T=10^{-3}\ \mathrm{GeV}$ so $T^2=10^{-6}\ \mathrm{GeV}^2$ , and $M_{\rm Pl}=1.22\times10^{19}\ \mathrm{GeV}$ :

H=1.66\times3.279\times\frac{10^{-6}}{1.22\times10^{19}}\ \mathrm{GeV}=4.46\times10^{-25}\ \mathrm{GeV}.

Converting with $1\ \mathrm{GeV}=1.519\times10^{24}\ \mathrm{s}^{-1}$ gives $H=\mathbf{0.68\ s^{-1}}$ .

Step 3. Age. In the radiation era $t=1/(2H)$ , so $t=1/(2\times0.68\ \mathrm{s}^{-1})=\mathbf{0.74\ s}$ — the celebrated "the Universe is about one second old at an MeV." A cross-check: $t\approx2.4/\sqrt{g_*}\,(T/\mathrm{MeV})^{-2}\ \mathrm{s}=2.4/3.279\approx0.73\ \mathrm{s}$ , consistent.

Step 4. Why it matters. Weak interactions freeze out when $\Gamma_{\rm weak}\sim H$ ; because $H\propto\sqrt{g_*}$ , any extra relativistic species raises $H$ , freezes the $n/p$ ratio earlier at a higher value, and boosts the helium yield. g_* = 10.75, H ≈ 0.68 s⁻¹, t ≈ 1 s.

3 · What one extra neutrino would do

Suppose a fourth light species thermalized like a neutrino, so $N_{\rm eff}=4$ instead of $3$ . By how much does $g_*$ rise at BBN, and the expansion rate?

Step 1. Each neutrino species adds $\tfrac{7}{8}\cdot2=1.75$ to the BBN-era $g_*$ , so $g_*$ rises from $10.75$ to $\mathbf{12.5}$ .

Step 2. The expansion rate scales as $\sqrt{g_*}$ : $\sqrt{12.5/10.75}=\sqrt{1.163}=\mathbf{1.078}$ , an $8\%$ faster expansion at fixed $T$ . That is enough to raise the primordial helium mass fraction by roughly $\Delta Y_p\approx0.013$ — a shift CMB and light-element data can see. One extra species: g_* → 12.5, expansion 8% faster.

Physics of the portrait

The effective number of relativistic degrees of freedom is not the discovery of any single person but the distilled residue of the hot big bang programme — the moment cosmology learned to treat the early Universe as a thermodynamic system whose particle content it could enumerate.

Origins · 1 mineffective relativistic degrees of freedom and how this was discovered

effective relativistic degrees of freedom

the temperature-dependent count g_(T) in ρ = (π²/30) g_ T⁴ · thermal history of the hot big bang

The idea that the Universe began hot and dense, and that its history could be read off from equilibrium thermodynamics, is the legacy of George Gamow and his students Ralph Alpher and Robert Herman in the late 1940s. Reasoning that a hot early phase must have cooked the light elements and left a relic radiation bath, Alpher and Herman predicted in 1948 a present-day temperature of a few kelvin — a number vindicated seventeen years later by Penzias and Wilson. The crucial refinement for $g_*$ came in 1953, when Alpher, Follin and Herman carried out the first genuinely detailed calculation of the thermal history, correctly accounting for neutrino decoupling and the reheating of photons at $e^+e^-$ annihilation that yields the $(4/11)^{1/3}$ ratio. The bookkeeping was sharpened over the following decades: Steigman, Schramm and Gunn showed in 1977 that the helium abundance limits the number of light neutrino species through its effect on the expansion rate, turning $g_*$ into a laboratory for particle physics; and Kolb and Turner's 1990 monograph codified $g_*(T)$ and $g_{*s}(T)$ in the form used ever since. What began as Gamow's intuition that the Universe is a fading fireball became a precise accounting device — one number, $g_*(T)$ , that carries the entire relativistic particle census through the first minutes of cosmic time.

History

1948

— Alpher and Herman predict a relic radiation background of a few kelvin

1953

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

1965

— Penzias and Wilson detect the 2.7 K cosmic microwave background

1977

— Steigman, Schramm and Gunn bound the neutrino species count from helium

1990

— Kolb and Turner codify g_(T) and g_s(T) in The Early Universe

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 5 · Dodelson, Modern Cosmology, 2nd ed., Ch. 3 · Kolb & Turner, The Early Universe, Ch. 3 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30.

Provenance. The bosonic/fermionic energy integrals and the exact $7/8$ ratio: Dodelson Ch. 3, Kolb & Turner Ch. 3. The Standard-Model count $g_*=106.75$ (bosonic $28$ , fermionic $90$ ): Kolb & Turner Table 3.1. The reheating derivation $(11/4)^{1/3}$ and $T_\nu=1.95\ \mathrm{K}$ : Ryden §5.5, tracing to Alpher, Follin & Herman 1953. Today's values $g_*=3.36$ , $g_{*s}=3.91$ : standard. The prefactor $1.66=\sqrt{8\pi^3/90}$ : Friedmann with radiation. The refined $N_{\rm eff}=3.044$ : modern finite-temperature calculations (superseding the older $3.046$ ). A verifier should recompute the two definite integrals, the Standard-Model state-counting sums, the entropy-conservation ratio, and $H(1\ \mathrm{MeV})\approx0.7\ \mathrm{s}^{-1}$ .

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Thermal History of the Universe · Entropy Conservation and Comoving Entropy. 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 $g_*$ staircase at any temperature you choose — pick an epoch and we will count the species and read off the expansion rate together.

Why collapse everything into one number instead of tracking each species?

Because in the radiation era every relativistic species obeys the same blackbody law, differing only by its statistical weight and temperature. Once you know that a boson contributes $\tfrac{\pi^2}{30}g\,T^4$ and a fermion $\tfrac78$ of that, the total energy density is just a weighted sum — and that weighted sum is $g_*(T)$ . Packaging it as one number lets you write the Friedmann equation, the age–temperature relation, and every relic-abundance calculation in a single compact form, with all the particle physics hidden inside one temperature-dependent coefficient. When a species goes non-relativistic you simply drop it from the sum; nothing else in the machinery changes.

Where exactly does the 7/8 come from?

From the difference between Fermi–Dirac and Bose–Einstein statistics in the energy integral. The energy density of a relativistic species is proportional to $\int_0^\infty x^3\,\mathrm{d}x/(e^x\mp1)$ , with a minus sign for bosons and a plus sign for fermions. The bosonic integral equals $\pi^4/15$ ; the fermionic one equals $\tfrac78\cdot\pi^4/15$ . The factor is exactly $7/8=1-2^{-3}$ , arising when you expand the fermionic denominator as an alternating geometric series — the alternating signs remove one-eighth of the bosonic total. It is a clean, exact number, not an approximation, and it applies to energy density; for number density the analogous factor is $3/4$ .

Why are there two counts, g_ and g_s?

Because energy density and entropy density scale differently with temperature: $\rho\propto T^4$ but $s\propto T^3$ . As long as every species shares a single temperature this distinction is invisible, and $g_*=g_{*s}$ . But once a species decouples and cools below the photon temperature — as neutrinos do — it enters the energy sum weighted by $(T_i/T)^4$ and the entropy sum weighted by $(T_i/T)^3$ . The colder relic is discounted more heavily in energy than in entropy, so the two counts separate. Today that gives $g_*=3.36$ and $g_{*s}=3.91$ . You use $g_{*s}$ whenever you invoke entropy conservation and $g_*$ whenever you need the energy density or expansion rate.

Why do photons end up hotter than neutrinos?

Timing. Neutrinos decouple from the plasma at about $1\ \mathrm{MeV}$ , just before the temperature drops below the electron mass. When the $e^+e^-$ pairs then annihilate, their energy and entropy can only be shared with particles still in thermal contact — the photons. The neutrinos, already free-streaming, receive nothing. Conservation of comoving entropy in the photon sector then requires the photon temperature to rise relative to the neutrinos by exactly $(11/4)^{1/3}\approx1.401$ . That is why the cosmic neutrino background sits at $1.95\ \mathrm{K}$ today while the photon background is $2.725\ \mathrm{K}$ — a $40\%$ temperature gap frozen in during the first few seconds.

How does g_* affect the primordial helium abundance?

Through the expansion rate. The Friedmann equation gives $H\propto\sqrt{g_*}\,T^2$ , so a larger $g_*$ means faster expansion at a given temperature. The neutron-to-proton ratio freezes out when the weak interaction rate falls below $H$ ; a faster expansion freezes it out earlier, at a higher temperature, leaving more neutrons. Since essentially every surviving neutron ends up in $^4$ He, more neutrons means more helium. This is why the primordial helium mass fraction $Y_p$ is such a sensitive probe of any extra relativistic species: adding one thermalized neutrino raises $g_*$ from $10.75$ to $12.5$ , speeds expansion by $8\%$ , and measurably increases $Y_p$ .

What is N_eff and why is it 3.044 rather than 3?

$N_{\rm eff}$ is the effective number of neutrino species, defined so that the total relativistic energy beyond photons is $\tfrac78(4/11)^{4/3}N_{\rm eff}\rho_\gamma$ . Naively three Standard-Model neutrinos give $N_{\rm eff}=3$ . The refined value $3.044$ accounts for the fact that neutrino decoupling is not instantaneous: a small fraction of the $e^+e^-$ annihilation entropy leaks into the not-quite-decoupled neutrinos, warming them slightly, and finite-temperature QED corrections to the plasma add a further shift. The excess above three is therefore a real, calculable Standard-Model effect. CMB measurements ( $N_{\rm eff}=2.99\pm0.17$ from Planck) match it well and tightly constrain any extra dark radiation.