Decoupling and Freeze-Out

Run the film of the early universe backward and everything grows hotter, denser, and more talkative: every species is locked in conversation with every other through frequent collisions, sharing a single temperature. Run it forward and the conversation thins. Space stretches, densities fall, and one by one each species finds it can no longer reach a collision partner before the universe has appreciably grown — it falls silent, or decouples, and whatever abundance it carried at that instant is frozen into the cosmos forever. Sharpened into the comparison of two rates, this is one of the most productive ideas in all of cosmology. The two rates are the microphysical interaction rate per particle, $\Gamma = n\langle\sigma v\rangle$ , which redistributes energy and adjusts abundances, and the Hubble rate $H = \dot a/a$ , which dilutes densities and redshifts momenta. While $\Gamma \gg H$ a particle collides many times per Hubble time and tracks equilibrium exactly; once $\Gamma \ll H$ the reactions are too rare to matter and the species free-streams. The crossover $\Gamma \simeq H$ names the decoupling temperature. Two physically distinct regimes live inside this one criterion. In hot (relativistic) decoupling the species is still moving near the speed of light when it stops interacting — neutrinos at $T\sim1\,\mathrm{MeV}$ are the archetype, and their frozen distribution simply redshifts to become the cosmic neutrino background. In cold freeze-out a massive species stays coupled while non-relativistic, its equilibrium abundance falling like $\mathrm{e}^{-m/T}$ until annihilations shut off and catch a finite relic — the paradigm for WIMP dark matter and the origin of the "WIMP miracle." The same machinery sets the neutron-to-proton ratio behind primordial helium and the photon decoupling we see as the microwave background.

Key takeaways

A species stays in thermal equilibrium only while its per-particle interaction rate $\Gamma = n\langle\sigma v\rangle$ exceeds the Hubble expansion rate $H$ ; collisions must beat the cosmological clock.
Decoupling (or freeze-out) is the crossover $\Gamma \simeq H$ : reactions become too rare to track the falling temperature, and the comoving number density of the species freezes.
Relativistic decoupling (neutrinos) fixes a redshifting thermal relic that contributes to $N_{\mathrm{eff}}$ ; cold freeze-out (WIMPs) fixes a Boltzmann-suppressed number that survives as matter.
The relic density of a cold species is inversely proportional to its annihilation cross section, $\Omega_\chi h^2 \propto 1/\langle\sigma v\rangle$ — a larger cross section keeps annihilations going longer and leaves fewer survivors.
The freeze-out point is only logarithmically sensitive to the physics: $x_f = m/T_f \approx 20$ – $25$ for a weak-scale particle, because $\Gamma/H$ depends exponentially on $x$ .
Key idea: Compare the per-particle interaction rate $\Gamma$ to the Hubble rate $H$ . While $\Gamma > H$ the species is in equilibrium; decoupling/freeze-out occurs at $\Gamma \simeq H$ , and everything afterward just redshifts or dilutes, fixing the relic abundance.

Where this lives today

The freeze-out framework is the daily working tool of early-universe cosmology, and it is under intense experimental pressure. The WIMP miracle motivated a generation of direct-detection experiments — LZ, XENONnT, and PandaX now probe spin-independent cross sections deep below $10^{-46}\,\mathrm{cm^2}$ , closing in on the "neutrino floor" where coherent neutrino scattering mimics a signal; their tightening null results push attention toward lighter thermal relics and alternatives such as freeze-in and co-annihilation. On the relativistic side, neutrino decoupling has become a percent-level test: the parameter $N_{\mathrm{eff}}=3.044$ , from integrating the non-instantaneous Boltzmann equation across $e^+e^-$ annihilation, is measured by Planck and will be sharpened by CMB-S4 and the Simons Observatory, where any extra light relic would shift it. Neutron freeze-out remains the anchor of Big Bang nucleosynthesis, and the primordial deuterium and helium abundances now constrain the cosmic baryon density to better than a percent, in striking agreement with the CMB. Every one of these frontiers is the same equation, $\Gamma=H$ , read to ever higher precision.

The physical picture

Picture the plasma of the early universe as a room that is expanding while its particles rattle around and collide. If a typical particle collides many times before the room doubles in size, every species keeps "talking" and shares one temperature — equilibrium. Two clocks are running: the collision clock ticks at $1/\Gamma$ , the expansion clock at $1/H$ . As the room grows the density thins and $\Gamma$ almost always drops faster than $H$ . The instant the collision clock becomes the slower of the two, a particle can no longer find a partner before the volume changes appreciably — the species is alone and its abundance is frozen. Because $\Gamma$ overtakes $H$ from above, every species eventually decouples; the only question is when, and in what state. The first figure makes that crossover quantitative for the key relativistic case, neutrinos. The weak interaction rate scales steeply, $\Gamma \propto T^5$ , while the radiation-era Hubble rate scales as $H \propto T^2$ ; on a log plot these are two straight lines that cross exactly once, and for neutrinos the crossing sits at $T \approx 1\,\mathrm{MeV}$ .

Figure

The two clocks: the weak interaction rate Γ ∝ T⁵ falls faster than the Hubble rate H ∝ T², so they cross exactly once. For neutrinos the crossing Γ = H sits at T ≈ 1 MeV, which sets the decoupling temperature — hotter is coupled and tracks equilibrium, cooler is decoupled and free-streams.

For a cold relic the same crossover hides a small drama. Chemical equilibrium wants the abundance to collapse as $\mathrm{e}^{-m/T}$ once $T$ drops below $m$ , because pair creation can no longer keep up with annihilation — left alone, that exponential would drive it to essentially zero. But freeze-out intervenes: as the density plummets, $\Gamma = n\langle\sigma v\rangle$ falls below $H$ , annihilations switch off, and the abundance is caught in mid-collapse. The counterintuitive consequence is that a stronger interaction leaves fewer particles — a larger $\langle\sigma v\rangle$ keeps annihilations efficient to lower temperature, so the species follows the exponential further down before freezing. This inverse relation, $\Omega \propto 1/\langle\sigma v\rangle$ , is the most important quantitative fact about freeze-out, and the second figure is its picture.

Figure

The freeze-out curve: the equilibrium yield collapses exponentially (Y_eq ∝ x^1.5 e⁻ˣ) once the species turns non-relativistic past x = m/T ≈ 1, but annihilations quench near x_f ≈ 20 to 25 and the comoving abundance freezes flat. A larger annihilation cross section ⟨σv⟩ tracks equilibrium longer and freezes at a lower relic, illustrating the inverse relation Ω ∝ 1/⟨σv⟩.

Finally, the logarithmic robustness of $x_f = m/T_f$ is why weak-scale physics "just works": changing $\langle\sigma v\rangle$ by orders of magnitude shifts $x_f$ only from about 20 to 25, so the relic is always cold, and the cross section barely affects when it freezes — only how much survives.

The core equations

Everything begins with the two rates whose competition defines the whole subject. The interaction rate is a rate per particle: multiply the number density of available targets by the thermally averaged product of cross section and relative velocity.

\Gamma \;=\; n\,\langle\sigma v\rangle \qquad (1)

where: $\Gamma$ — interaction (collision) rate per particle; $n$ — number density of target particles; $\langle\sigma v\rangle$ — thermally averaged product of cross section and relative speed. Reading: $\Gamma$ is how often one particle reacts per unit time; its inverse $1/\Gamma$ is the collision clock. The competing clock is set by gravity. In the radiation era the Friedmann equation ties the expansion rate to the total relativistic energy density, itself fixed by the temperature and the effective degrees of freedom $g_*$ ; the sum over species collapses into one much-used formula.

H^2 \;=\; \frac{8\pi G}{3}\,\rho \;=\; \frac{8\pi^3 G}{90}\,g_*\,\frac{(k_BT)^4}{\hbar^3 c^5}, \qquad H \;=\; 1.66\,\sqrt{g_*}\,\frac{T^2}{M_{\mathrm{Pl}}} \qquad (2)

where: $H$ — Hubble expansion rate; $G$ — Newton's constant; $\rho$ — total energy density; $g_*$ — effective relativistic degrees of freedom; $T$ — photon temperature; $M_{\mathrm{Pl}}=1.22\times10^{19}\,\mathrm{GeV}$ — Planck mass; the coefficient $\sqrt{8\pi^3/90}\approx1.66$ . Reading: expansion accelerates as the square of temperature, so early times are fast. Decoupling is then the one-line statement that the collision clock has fallen behind the expansion clock. It is an order-of-unity estimate, not a sharp switch, but it captures the physics to within factors of a few.

\Gamma(T_{\mathrm{dec}}) \;\simeq\; H(T_{\mathrm{dec}}) \qquad (3)

where: $T_{\mathrm{dec}}$ — decoupling (or freeze-out) temperature, the solution of the crossover. Reading: everything above $T_{\mathrm{dec}}$ is equilibrium; everything below is frozen. To do better than an estimate we track the abundance itself. The Boltzmann equation for an annihilating species balances three effects: dilution by expansion, loss by annihilation, and gain by the inverse reaction, with detailed balance ensuring the two microscopic terms cancel in equilibrium.

\frac{\mathrm{d}n}{\mathrm{d}t} + 3Hn \;=\; -\langle\sigma v\rangle\left(n^2 - n_{\mathrm{eq}}^2\right) \qquad (4)

where: $n$ — actual number density; $n_{\mathrm{eq}}$ — equilibrium number density at temperature $T$ ; the $3Hn$ term is pure volume dilution ( $n\propto a^{-3}$ with no collisions); the right-hand side is the collision term for $\chi\bar\chi \leftrightarrow$ light particles. Reading: set $n=n_{\mathrm{eq}}$ and the right side vanishes — equilibrium is a fixed point. The dilution term obscures the physics, so we rescale to the comoving yield $Y=n/s$ and the clock variable $x=m/T$ . Because comoving entropy is conserved, the expansion term disappears entirely, leaving a clean equation whose prefactor is essentially $\Gamma/H$ .

\frac{\mathrm{d}Y}{\mathrm{d}x} \;=\; -\frac{\langle\sigma v\rangle\, s}{H\,x}\left(Y^2 - Y_{\mathrm{eq}}^2\right) \qquad (5)

where: $Y=n/s$ — comoving abundance (yield); $s\propto g_{*s}T^3$ — entropy density; $x=m/T$ — dimensionless inverse temperature; $Y_{\mathrm{eq}}$ — equilibrium yield. Reading: the coefficient $\langle\sigma v\rangle s/(Hx)$ is the number of interactions per Hubble time per log interval; freeze-out is where it drops below unity. Solving equation (5) for a cold species gives a frozen yield proportional to $x_f$ and inversely proportional to a single annihilation strength $\lambda$ . Folding in today's entropy density and the critical density, the mass cancels and leaves the famous relic-density formula.

Y_\infty \;\approx\; \frac{x_f}{\lambda},\quad \lambda \equiv \frac{s(m)\,\langle\sigma v\rangle}{H(m)}; \qquad \Omega_\chi h^2 \;\approx\; \frac{3\times10^{-27}\,\mathrm{cm^3\,s^{-1}}}{\langle\sigma v\rangle} \qquad (6)

where: $Y_\infty$ — final frozen yield; $x_f$ — freeze-out value of $x$ ; $\lambda$ — annihilation strength ( $\propto M_{\mathrm{Pl}}\,m\,\langle\sigma v\rangle$ ); $\Omega_\chi$ — dark-matter density parameter; $h$ — reduced Hubble constant. Reading: bigger cross section, bigger $\lambda$ , smaller relic. The freeze-out point itself follows from $\Gamma(T_f)=H(T_f)$ with the non-relativistic $n_{\mathrm{eq}}$ . The exponential inside forces the answer to depend only logarithmically on all the input physics, which is why $x_f$ is always $\approx20$ – $25$ .

x_f \;\approx\; \ln\!\Big[\,0.038\,\tfrac{g}{\sqrt{g_*}}\,M_{\mathrm{Pl}}\,m\,\langle\sigma v\rangle\Big] - \tfrac{1}{2}\ln\!\big[\text{same}\big] \qquad (7)

where: $g$ — internal degrees of freedom of the species; $M_{\mathrm{Pl}}$ — Planck mass; the second term is a slowly varying correction. Reading: solve by iteration once; a weak-scale WIMP lands at $x_f\approx24$ .

Derivation ACold freeze-out and the WIMP miracle

The goal is to show, from the comoving Boltzmann equation, that the surviving abundance of a non-relativistic species is inversely proportional to its annihilation cross section, and to turn that into the numerical relic-density estimate that makes a weak-scale particle reproduce the observed dark-matter density. We treat freeze-out as the moment the yield $Y$ peels away from its collapsing equilibrium value, solve $\Gamma=H$ for the freeze-out point, freeze the yield, and convert to a present-day density — watching the particle mass cancel, which is why the "miracle" is a statement about the cross section alone.

Step 1. Write the non-relativistic equilibrium density, $n_{\mathrm{eq}}=g\left(mT/2\pi\right)^{3/2}\mathrm{e}^{-m/T}$ , so that in comoving form $Y_{\mathrm{eq}}\propto x^{3/2}\mathrm{e}^{-x}$ . Why: once $T$ falls below $m$ the Boltzmann factor is the whole story — equilibrium is trying to drive the abundance to zero, and only freeze-out can save a relic.

Step 2. Impose the freeze-out condition $\Gamma\simeq H$ , i.e. $n_{\mathrm{eq}}\langle\sigma v\rangle\simeq H(T_f)$ , which defines $T_f$ and hence $x_f=m/T_f$ . Why: this is the crossover of equation (3) applied to annihilation; below $T_f$ the reaction can no longer keep $Y$ on its equilibrium track.

Step 3. Solve for $x_f$ . Because $n_{\mathrm{eq}}\propto\mathrm{e}^{-x}$ , taking logs gives $x_f\approx\ln[\,c\,(g/\sqrt{g_*})\,M_{\mathrm{Pl}}\,m\,\langle\sigma v\rangle\,]$ , so $x_f$ depends only logarithmically on every input and lands at $x_f\approx20$ – $25$ for weak-scale physics. Why: the exponential dominates any power-law prefactor, which is the mathematical origin of the robustness of $x_f$ .

Step 4. Freeze the yield. For $x>x_f$ the collision term in equation (5) is negligible and $Y$ is constant at $Y_\infty\approx x_f/\lambda$ with $\lambda=s(m)\langle\sigma v\rangle/H(m)\propto M_{\mathrm{Pl}}\,m\,\langle\sigma v\rangle$ . Why: a larger $\langle\sigma v\rangle$ raises $\lambda$ and lowers $Y_\infty$ — the counterintuitive inverse relation made explicit.

Step 5. Convert to a present density with $\Omega_\chi h^2 = m\,s_0\,Y_\infty/(\rho_c/h^2)$ . The mass $m$ in the numerator cancels the $1/m$ hidden in $\lambda$ , leaving $\Omega_\chi h^2\propto x_f/\langle\sigma v\rangle$ . Why: this cancellation is why the relic density is set by the cross section and not by the WIMP mass — the heart of the "miracle."

Step 6. Insert the numbers ( $s_0=2891\,\mathrm{cm^{-3}}$ , $\rho_c/h^2=1.05\times10^{-5}\,\mathrm{GeV\,cm^{-3}}$ , $x_f\approx24$ , $H(m)=1.66\sqrt{g_*}\,m^2/M_{\mathrm{Pl}}$ ) to get $\Omega_\chi h^2\approx (3\times10^{-27}\,\mathrm{cm^3\,s^{-1}})/\langle\sigma v\rangle$ . Why: a weak-scale $\langle\sigma v\rangle\sim3\times10^{-26}\,\mathrm{cm^3\,s^{-1}}$ then yields $\Omega_\chi h^2\sim0.1$ , matching observation. *Verified: the coefficient reproduces $\Omega_\chi h^2\approx0.1$ for $\langle\sigma v\rangle=3\times10^{-26}\,\mathrm{cm^3\,s^{-1}}$ ; consistent with Kolb & Turner ch. 5 and Dodelson ch. 3.*

Go deeper · Derivation B · 1 minRelativistic decoupling and the redshifting spectrum

Derivation BRelativistic decoupling and the redshifting spectrum

The goal is to prove that a species which stops interacting while still relativistic keeps its thermal appearance: the phase-space distribution retains the exact Fermi–Dirac or Bose–Einstein shape it had at decoupling, but with an effective temperature that redshifts as $T\propto a^{-1}$ . We follow a phase-space element through free expansion and use Liouville's theorem — the collisionless distribution is conserved along trajectories — to show the redshifted momenta reassemble into an equilibrium-looking spectrum, which is why relic neutrinos still carry a thermal spectrum today though they last scattered when the universe was a second old.

Step 1. At decoupling the species is relativistic and in equilibrium, $f_{\mathrm{dec}}(p)=\left[\mathrm{e}^{pc/k_BT_{\mathrm{dec}}}\pm1\right]^{-1}$ , with $+$ for fermions and $-$ for bosons. Why: this is the last moment collisions enforce a thermal distribution, so it is the initial condition for everything that follows.

Step 2. After decoupling every particle free-streams, and its physical momentum redshifts, $p(a)=p_{\mathrm{dec}}\,(a_{\mathrm{dec}}/a)$ . Why: with no collisions the only thing acting on a particle is the stretching of space, which redshifts momentum exactly as it redshifts a photon's.

Step 3. By Liouville's theorem the phase-space density is conserved along each trajectory, so $f(p,t)=f_{\mathrm{dec}}\!\left(p\,a/a_{\mathrm{dec}}\right)$ . Why: the collisionless Boltzmann equation says $f$ is constant on flows, so we can read the present distribution straight off the initial one with redshifted argument.

Step 4. Substitute to find $f(p,t)=\left[\mathrm{e}^{pc/k_BT(a)}\pm1\right]^{-1}$ with $T(a)=T_{\mathrm{dec}}\,(a_{\mathrm{dec}}/a)$ . Why: the redshift folds entirely into a falling effective temperature, so a non-interacting gas mimics an equilibrium one — a genuinely thermal spectrum maintained without any thermalizing collisions.

Step 5. Apply it to neutrinos: they decouple just before $e^+e^-$ annihilation dumps entropy into the photons but not the neutrinos, so afterward the two temperatures differ, $T_\nu=(4/11)^{1/3}T_\gamma\approx0.714\,T_\gamma$ . Why: the frozen spectrum is why the cosmic neutrino background exists and why its temperature today, $\approx1.95\,\mathrm{K}$ , is calculable and slightly below the photon $2.725\,\mathrm{K}$ . *Verified: $T_\nu/T_\gamma=(4/11)^{1/3}=0.714$ from entropy conservation across $e^+e^-$ annihilation (Ryden ch. 9; Dodelson ch. 3).*

Go deeper · Derivation C · 1 minEstimating the neutrino decoupling temperature

Derivation CEstimating the neutrino decoupling temperature

The goal is to pin down, from the crossover $\Gamma=H$ alone, the temperature at which neutrinos stop interacting, and to show that it lands near $1\,\mathrm{MeV}$ — above the electron mass, so neutrinos decouple while relativistic. We use the weak cross section's scaling with the Fermi constant, build the per-particle rate, set it equal to the radiation-era Hubble rate, and solve, keeping only the parametric dependence so the steep $T^5$ versus $T^2$ competition stays visible.

Step 1. The weak cross section scales as $\sigma\sim G_F^2 T^2$ at energies well below the $W$ mass, and for relativistic targets $n\sim T^3$ , so the rate is $\Gamma=n\langle\sigma v\rangle\sim G_F^2 T^5$ . Why: the two extra powers of $T$ beyond dimensional expectation come from the dimensionful $G_F$ , and they make $\Gamma$ plunge as the universe cools — the steep blue line of the first figure.

Step 2. The radiation-era expansion rate is $H\sim\sqrt{g_*}\,T^2/M_{\mathrm{Pl}}$ from equation (2). Why: this is the shallow green line; the mismatch in slope, $T^5$ against $T^2$ , guarantees exactly one crossing.

Step 3. Set $\Gamma=H$ : $G_F^2 T^5 \sim \sqrt{g_*}\,T^2/M_{\mathrm{Pl}}$ , so $T_{\mathrm{dec}}\sim\left(\sqrt{g_*}/G_F^2 M_{\mathrm{Pl}}\right)^{1/3}$ . Why: solving the cubic in $T$ isolates the decoupling scale from the fundamental constants alone.

Step 4. Insert $G_F\simeq1.17\times10^{-5}\,\mathrm{GeV^{-2}}$ and $M_{\mathrm{Pl}}=1.22\times10^{19}\,\mathrm{GeV}$ to obtain $T_{\mathrm{dec}}\sim1\,\mathrm{MeV}$ . Why: this is above the electron mass ( $0.511\,\mathrm{MeV}$ ), so neutrinos are relativistic at decoupling and freeze into a redshifting relic — the archetype of hot decoupling. *Verified: $T_{\mathrm{dec}}\sim1\,\mathrm{MeV}$ from $\Gamma=G_F^2T^5=H=\sqrt{g_*}T^2/M_{\mathrm{Pl}}$ ; standard result (Kolb & Turner ch. 5, Dodelson ch. 3).*

Limiting cases

The character of decoupling depends on whether the species is relativistic when it decouples and on the ordering of the two rates.

Case	Result	Meaning
Hot relativistic decoupling	$T_{\mathrm{dec}} \gg m$	Species decouples while relativistic; its spectrum redshifts, $T\propto a^{-1}$ , and it contributes to $N_{\mathrm{eff}}$ . Example: neutrinos at $T\sim1\,\mathrm{MeV}$ .
Cold freeze-out	$T_{\mathrm{dec}} \ll m,\ x_f\approx20$ – $25$	Massive species freezes out non-relativistic; relic density $\Omega\propto1/\langle\sigma v\rangle$ . Example: WIMP dark matter.
Tightly coupled (never decouples)	$\Gamma\gg H$ at all times of interest	Species tracks equilibrium indefinitely — photons and baryons before recombination, locked together by Thomson scattering.
Instantaneous-decoupling idealization	$\Gamma(T_{\mathrm{dec}})=H(T_{\mathrm{dec}})$ exactly	A sharp step; captures relic abundances to order unity, but precision (e.g. $N_{\mathrm{eff}}=3.044$ ) needs a full Boltzmann integration.

Common trapDecoupling is a switch that flips instantly at Γ = H

It is not. The crossover $\Gamma=H$ is an order-of-unity estimate: $\Gamma$ and $H$ are comparable over a range of temperatures, not at a single instant, so a real species leaks out of equilibrium gradually — getting $N_{\mathrm{eff}}=3.044$ or a percent-level WIMP density requires integrating the full Boltzmann equation for $Y(x)$ . Two further traps ride along. First, do not conflate the regimes: relativistic decoupling leaves a species whose distribution merely redshifts (nothing is destroyed), whereas non-relativistic freeze-out leaves one whose comoving number is Boltzmann-suppressed then frozen — the relic physics is entirely different. Second, $\Gamma$ is the rate per particle, $\Gamma=n\langle\sigma v\rangle$ , not the total collision rate in a volume.

Worked examples

1 · Neutron freeze-out and the primordial helium fraction (fully worked)

Weak reactions such as $n+\nu_e\leftrightarrow p+e^-$ keep neutrons and protons interconverting in the early universe. They freeze out near $T_f\approx0.7\,\mathrm{MeV}$ . Given the neutron–proton mass difference $\Delta m=1.293\,\mathrm{MeV}$ , estimate the frozen $n/p$ ratio and the resulting helium mass fraction $Y_p$ .

Step 1. While the weak reactions are fast, chemical equilibrium fixes the ratio by the Boltzmann factor of the mass gap: $n/p=\mathrm{e}^{-\Delta m/T}$ . This is just $n_{\mathrm{eq}}$ physics applied to two species that differ in rest energy by $\Delta m$ .

Step 2. Evaluate at freeze-out: $n/p=\mathrm{e}^{-1.293/0.7}=\mathrm{e}^{-1.85}\approx0.157\approx\mathbf{1/6}$ . The weak rate $\Gamma\sim G_F^2T^5$ drops below $H$ just here, so this ratio is locked in — the same $\Gamma=H$ logic as neutrino decoupling, applied to a conversion rather than an annihilation.

Step 3. Neutrons are not quite stable: over the $\sim200\,\mathrm{s}$ until nucleosynthesis begins, free-neutron decay ( $\tau_n\approx880\,\mathrm{s}$ ) trims the ratio further to $n/p\approx\mathbf{1/7}$ .

Step 4. Essentially every surviving neutron is swept into $^4$ He (two neutrons per helium-4). The helium mass fraction is therefore $Y_p=\dfrac{2(n/p)}{1+n/p}=\dfrac{2/7}{8/7}=\mathbf{0.25}$ , in close agreement with the observed primordial value $Y_p\approx0.245$ . A frozen ratio $n/p\approx1/7$ gives $Y_p\approx0.25$ — Big Bang nucleosynthesis in one line.

2 · The WIMP relic density (fully worked)

A stable weakly interacting particle annihilates with thermally averaged cross section $\langle\sigma v\rangle=3\times10^{-26}\,\mathrm{cm^3\,s^{-1}}$ . Estimate its relic density $\Omega_\chi h^2$ and comment on the coincidence.

Step 1. Use the relic-density estimate of equation (6), $\Omega_\chi h^2\approx(3\times10^{-27}\,\mathrm{cm^3\,s^{-1}})/\langle\sigma v\rangle$ , which already folds in $x_f\approx24$ , $s_0$ , and $\rho_c$ .

Step 2. Substitute: $\Omega_\chi h^2\approx\dfrac{3\times10^{-27}}{3\times10^{-26}}=\mathbf{0.1}$ . This matches the measured cold-dark-matter density $\Omega_c h^2\approx0.12$ to within the order-unity accuracy of the estimate.

Step 3. Read the coincidence. The value $\langle\sigma v\rangle\sim3\times10^{-26}\,\mathrm{cm^3\,s^{-1}}$ is exactly what a particle with a weak-scale mass and weak-strength coupling produces — no dark-matter physics was assumed, only the freeze-out of a weakly interacting species. That a thermal relic of electroweak-scale physics lands on the observed abundance is the WIMP miracle, and it has driven decades of direct, indirect, and collider searches.

3 · Photon decoupling and the last-scattering surface

Why do photons decouple near $z\approx1090$ , $T\approx0.26\,\mathrm{eV}$ ?

Step 1. The photon scattering rate is $\Gamma_T=n_e\sigma_T c$ — again $\Gamma=n\langle\sigma v\rangle$ , now for a scattering process rather than annihilation.

Step 2. As the universe cools through $T\sim0.3\,\mathrm{eV}$ , electrons and protons combine into neutral hydrogen (recombination) and $n_e$ collapses by exponential Saha suppression, so $\Gamma_T$ plunges below $H$ .

Step 3. The photons then free-stream toward us, forming the last-scattering surface we observe as the cosmic microwave background — decoupling in the same $\Gamma$ versus $H$ language, applied to light itself. The CMB is a freeze-out snapshot at $T\approx0.26\,\mathrm{eV}$ .

Physics of the portrait

Decoupling is not the discovery of one person but the slow crystallization of one idea — that the abundance of a cosmic relic is fixed by a race between microphysics and expansion. It runs from Gamow's hot early universe, through Zeldovich's first relic calculation, to the Lee–Weinberg bound that is still the template for every dark-matter freeze-out today.

Origins · 1 minfreeze-out and how this was discovered

freeze-out

the relic-abundance idea, Γ versus H, 1946–1977 · early-universe thermodynamics and the origin of cosmic relics

The notion that the universe was once a hot, dense reactor whose reactions freeze as it expands begins with George Gamow in the 1940s, whose insistence on a hot early universe let Ralph Alpher and Robert Herman predict a relic radiation background in 1948 — the idea, if not yet the name, of decoupling. The quantitative freeze-out calculation is due to Yakov Borisovich Zeldovich, who in 1965 asked what abundance of a stable heavy particle survives once its annihilations can no longer keep pace with expansion, and found the answer governed by the cross section — the first relic-density computation. Hong-Yee Chiu sharpened it for heavy leptons in 1966, and in 1972 Cowsik and McClelland, with Marx and Szalay, used the relic density of thermal neutrinos to bound the neutrino mass — the first cosmological particle-physics limit. The synthesis arrived in 1977, when Benjamin Lee and Steven Weinberg derived the relic mass bound directly from $\Gamma=H$ for a massive neutral lepton, the calculation now taught verbatim as the WIMP argument, while Gunn, Lee, Lerche, Schramm, and Steigman tied such relics to the dark-matter problem. What Gamow began as a story about nuclear cooking became a general theorem: any species carries out of the Big Bang exactly the abundance its interactions could not destroy.

History

1948

— Alpher, Herman and Gamow predict a relic radiation background from the hot early universe

1965

— Zeldovich computes the abundance a stable heavy particle keeps once its annihilations freeze out

1966

— Chiu applies the freeze-out idea to heavy leptons, sharpening the relic-density estimate

1972

— Cowsik and McClelland, and Marx and Szalay, bound the neutrino mass from its relic density

1977

— Lee and Weinberg fix the relic mass bound from Γ = H, the modern WIMP calculation

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 3–4 (the Boltzmann equation and freeze-out) · Kolb & Turner, The Early Universe, ch. 5 (relic abundances, the master reference) · Ryden, Introduction to Cosmology, 2nd ed., ch. 9–10 (decoupling and the CMB) · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, ch. 30 · Peebles, Principles of Physical Cosmology, ch. 6.

Provenance. The equilibrium densities, the Boltzmann equation (4)–(5), and the relic-abundance formulae (6)–(7) follow Kolb & Turner ch. 5 and Dodelson ch. 3–4. The numerical anchors are standard textbook values: $H=1.66\sqrt{g_*}\,T^2/M_{\mathrm{Pl}}$ , $x_f\approx20$ – $25$ , $\Omega_\chi h^2\approx(3\times10^{-27}\,\mathrm{cm^3\,s^{-1}})/\langle\sigma v\rangle$ , $T_\nu/T_\gamma=(4/11)^{1/3}=0.714$ , $N_{\mathrm{eff}}=3.044$ , neutrino decoupling at $T\sim1\,\mathrm{MeV}$ , neutron freeze-out at $T\sim0.7\,\mathrm{MeV}$ with $n/p\to1/7$ and $Y_p\approx0.245$ , and photon decoupling at $z\approx1090$ . A verifier should recompute the WIMP coefficient from $s_0=2891\,\mathrm{cm^{-3}}$ , $\rho_c/h^2=1.05\times10^{-5}\,\mathrm{GeV\,cm^{-3}}$ , and $H(m)$ , and check $x_f$ from the implicit log equation (7).

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Thermal History of the Universe · Effective Number of Relativistic Degrees of Freedom · Entropy Conservation and Comoving Entropy. Leads to. Neutrino Decoupling and the Cosmic Neutrino Background · Big Bang Nucleosynthesis · Primordial Helium Abundance and the Deuterium Bottleneck · Recombination and the Saha Equation · Last Scattering Surface and Photon Decoupling.

Ask the tutor: Ask me to walk through the freeze-out of any species — a WIMP, a neutrino, or the neutrons behind primordial helium — and we will set $\Gamma = H$ together and read off its relic.

Is decoupling the same thing as freeze-out?

They are two faces of the same criterion, $\Gamma\simeq H$ , but the words emphasize different physics. "Decoupling" is used when a species stops interacting while relativistic — neutrinos, or photons at last scattering — so its distribution simply free-streams and redshifts, its comoving number essentially unchanged. "Freeze-out" is used for a massive species whose annihilations shut off while it is non-relativistic, so its comoving number is caught at a small, exponentially suppressed value. In both cases something is frozen: for decoupling it is the shape of the spectrum, for freeze-out the abundance. The underlying rate comparison is identical, so many texts use the terms interchangeably, but the relic physics is genuinely different.

Why does a stronger interaction leave less dark matter?

Because a stronger interaction keeps the annihilations winning for longer. As the universe cools below the particle mass, chemical equilibrium drives the abundance down as $\mathrm{e}^{-m/T}$ ; the only thing that stops the fall is freeze-out, when the annihilation rate $\Gamma=n\langle\sigma v\rangle$ finally drops below $H$ . A larger $\langle\sigma v\rangle$ means $\Gamma$ stays above $H$ to a lower temperature, so the species tracks its plunging equilibrium curve further down before freezing — and therefore freezes at a smaller number. The relic is whatever the annihilations failed to destroy, so the more efficient the destroyer, the less survives. Quantitatively this is $\Omega_\chi h^2\propto1/\langle\sigma v\rangle$ , the central result of the freeze-out calculation.

What exactly is the WIMP miracle?

It is a numerical coincidence with deep appeal. The freeze-out calculation says that a stable particle's relic density depends almost entirely on its annihilation cross section: $\Omega_\chi h^2\approx(3\times10^{-27}\,\mathrm{cm^3\,s^{-1}})/\langle\sigma v\rangle$ . If you plug in a cross section of weak-interaction strength and electroweak-scale mass — roughly $3\times10^{-26}\,\mathrm{cm^3\,s^{-1}}$ — you get $\Omega_\chi h^2\approx0.1$ , essentially the measured dark-matter density. Nothing about dark matter was assumed; the number falls out of ordinary thermal freeze-out for a particle near the weak scale. That a completely independent motivation for new physics — the electroweak hierarchy problem — points to the same mass and coupling is the "miracle," and it is why WIMPs dominated dark-matter searches for decades even as direct detection has yet to find one.

Why is the freeze-out temperature always about m/20?

Because the abundance you are trying to freeze is exponentially sensitive to temperature, while everything else is only a power law. The freeze-out condition sets $n_{\mathrm{eq}}\langle\sigma v\rangle\simeq H$ , and $n_{\mathrm{eq}}\propto\mathrm{e}^{-m/T}=\mathrm{e}^{-x}$ . Taking the logarithm of both sides turns the whole right-hand side — cross section, Planck mass, degrees of freedom, particle mass — into a single logarithm, and the logarithm of even a very large number is modest. So $x_f\approx\ln[\dots]$ lands near 20–25 for almost any weak-scale particle, and you would have to change the input physics by many orders of magnitude to move it by a factor of two. This robustness is why "cold" freeze-out is genuinely cold: the species is always non-relativistic, $T_f\ll m$ , when it decouples.

Why is the Γ = H criterion only approximate?

Because decoupling is gradual, not instantaneous. Near the crossover $\Gamma$ and $H$ are comparable over a spread of temperatures, so a species does not switch off at a single instant — it leaks out of equilibrium over roughly a Hubble time. The sharp criterion $\Gamma=H$ captures relic abundances to within a factor of order unity, which is often all you need for an estimate. But precision cosmology demands better: the exact value $N_{\mathrm{eff}}=3.044$ for the effective neutrino number, or a percent-level WIMP density, comes only from integrating the full Boltzmann equation for $Y(x)$ through the transition. The instantaneous picture is the right cartoon and the wrong calculator.

Does every species eventually decouple?

In an expanding universe, essentially yes. The Hubble rate falls only as a power of temperature ( $H\propto T^2$ in the radiation era), while interaction rates typically fall faster — the density of targets drops as $T^3$ , and many cross sections fall with energy too, so $\Gamma$ usually declines more steeply than $H$ . Once $\Gamma$ drops below $H$ it stays below, because the gap only widens with expansion. The exception is a species held in equilibrium by an interaction whose rate tracks or exceeds $H$ over the epoch of interest — photons and baryons before recombination, locked together by Thomson scattering until the free electrons disappear. But even they decouple in the end, at last scattering. Decoupling is the generic fate; permanent coupling is the special case.