Recombination and the Cosmic Microwave Background

For roughly its first four hundred thousand years the universe was its own fog. Photons, free electrons and protons were locked into a single incandescent plasma by Thomson scattering, so light could not travel more than a whisker before being deflected — the cosmos glowed uniformly and was perfectly opaque. As the scale factor grew the plasma cooled adiabatically, the photon temperature falling as $T\propto 1/a$ , until electromagnetic binding could finally overcome thermal ionization and electrons were captured onto protons to make neutral hydrogen. That capture is recombination, and the moment the free electrons vanish the fog lifts: the last photons to scatter stream freely to the present as the cosmic microwave background. The physics hinges on one startling fact. Naively hydrogen should form once $k_B T$ falls to the binding energy $13.6\ \mathrm{eV}$ , but recombination is delayed by nearly two orders of magnitude, to $k_B T\approx 0.3\ \mathrm{eV}$ . The culprit is the baryon-to-photon ratio $\eta\approx 6\times 10^{-10}$ : photons outnumber baryons a billion to one, so even when the average photon is far too cold to ionize hydrogen, the exponentially rare but enormously numerous photons in the Wien tail keep ripping electrons off. Equilibrium ionization is captured by the Saha equation; the true, gradual freeze-out requires the non-equilibrium Boltzmann treatment that James Peebles built as an effective three-level atom, because a recombination straight to the ground state emits a photon that promptly re-ionizes a neighbour. Once the free-electron density collapses, the Thomson rate drops below the expansion rate, photons decouple, and last scattering peaks at $z_*\approx 1090$ . Those photons — redshifted into the microwave band as the most perfect blackbody in nature, $T_0=2.7255\ \mathrm{K}$ — carry a frozen image of the last-scattering surface, whose one-part-in- $10^5$ anisotropies encode the geometry, composition and primordial seeds of everything that followed. Recombination is the observational anchor of modern cosmology.

Key takeaways

Recombination is the epoch near $z\approx 1100$ (cosmic time $\sim 380{,}000$ yr) when electrons and protons combined into neutral hydrogen, collapsing the free-electron density and letting light stream freely for the first time.
The transition temperature is set not by the $13.6\ \mathrm{eV}$ binding energy but by the huge baryon-to-photon ratio $\eta\approx 6\times 10^{-10}$ : the Wien tail keeps hydrogen ionized until $k_B T\approx 0.3\ \mathrm{eV}$ ( $T\approx 3700\ \mathrm{K}$ ).
Equilibrium is governed by the Saha equation, but the real freeze-out needs the non-equilibrium Peebles three-level atom, because direct recombination to the ground state is self-defeating.
As the electron fraction plummets, the photon mean free path exceeds the horizon and photons decouple, free-streaming to us as the CMB — a near-perfect blackbody at $T_0=2.7255\ \mathrm{K}$ .
The last-scattering surface is imprinted with acoustic oscillations of the pre-recombination photon-baryon plasma: temperature anisotropies $\Delta T/T\sim 10^{-5}$ whose first peak fixes the comoving sound horizon $r_s\approx 147\ \mathrm{Mpc}$ .
Recombination and decoupling are distinct but nearly coincident; the visibility function peaks at $z_*\approx 1090$ with finite thickness $\Delta z\approx 80$ .
Key idea: recombination is delayed far below the $13.6\ \mathrm{eV}$ binding energy by the enormous photon-to-baryon ratio; it ends the plasma era and releases the CMB, whose last-scattering surface at $z_*\approx 1090$ is the most distant electromagnetic image of the universe.

Where this lives today

Recombination is not a settled footnote but an active precision frontier, because the CMB it released is the single richest data set in cosmology. The Planck satellite measured the background's temperature anisotropies and polarization to cosmic-variance limits, and their acoustic peaks pin down the baryon density, the matter density, the sound horizon $r_s=147.09\ \mathrm{Mpc}$ , and the redshift of last scattering $z_*=1089.9$ — the numbers that anchor the six-parameter $\Lambda$ CDM model. Matching that precision drove a decade of work on recombination codes: RECFAST, and later CosmoRec and HyRec, refine Peebles' three-level atom with hundreds of atomic levels, Lyman-line radiative transfer, and helium recombination, because a one-percent error in $x_e(z)$ would bias the inferred parameters. The residual electrons left by freeze-out generate the CMB's E-mode polarization, now mapped by ACT, SPT and Planck and targeted by the Simons Observatory and CMB-S4, while the hunt for inflationary B-modes on that same last-scattering surface is among the field's largest efforts. Every one of these rests on understanding, to exquisite accuracy, the epoch when the universe first turned transparent.

The physical picture

Picture the young universe as a glowing fog. Every photon travels only a hair's breadth before Thomson-scattering off a free electron, so the plasma is bright and utterly opaque — like being inside the Sun. Cooling changes nothing until, quite suddenly, electrons are captured into atoms and the scatterers disappear. The fog does not so much thin as lift: over a narrow slice of cosmic time the mean free path leaps from centimetres to gigaparsecs, and the last photons to scatter set off on an uninterrupted journey that ends in our detectors thirteen billion years later. The first figure tracks the single quantity that governs this — the free-electron fraction $x_e$ — as it plunges from a fully ionized plasma to a neutral gas.

Figure

The ionization history from the Saha equation: the free-electron fraction x_e plunges from 1 (fully ionized opaque plasma) through half-ionization at z ≈ 1370, k_B T ≈ 0.32 eV, then freezes near the residual x_e ≈ 2×10⁻⁴ that the non-equilibrium Peebles treatment leaves behind — the universe turns neutral and transparent

Why does the fog wait so long? Because photons outnumber baryons by about a billion to one. Even when the mean photon energy has fallen well below $13.6\ \mathrm{eV}$ , the exponentially small fraction of photons in the high-energy Wien tail is multiplied by that vast total number, and the product stays large enough to keep hydrogen ionized until $B/k_B T\approx \ln(1/\eta)\approx 40$ — that is, until $k_B T\approx 0.3\ \mathrm{eV}$ , a factor of forty below binding. Recombination is fundamentally a race between two clocks: the chemistry that assembles neutral atoms, and the Hubble expansion that dilutes everything. When the reactions lose that race the ionization fraction does not fall to zero but freezes out at a residual $x_e\approx 2\times 10^{-4}$ . What we actually observe is not recombination itself but the moment of last scattering — and the second figure shows the probability distribution of exactly where, in redshift, each CMB photon scattered for the final time.

Figure

The visibility function g(z), the probability that a CMB photon last scattered at redshift z: it peaks sharply at z_ ≈ 1090 with finite thickness Δz ≈ 80 in FWHM, the surface of last scattering. The plasma temperature there, T_ ≈ 2970 K ≈ 0.26 eV, is redshifted to today's T_0 = 2.7255 K

The core equations

Equilibrium ionization is fixed by chemical balance of the reaction $e^-+p\leftrightarrow \mathrm{H}+\gamma$ . Equating chemical potentials and inserting non-relativistic Maxwell-Boltzmann densities gives the Saha equation, which relates the free-electron, proton and neutral-hydrogen densities through the binding energy.

\frac{n_e\,n_p}{n_{\mathrm H}}=\left(\frac{m_e k_B T}{2\pi\hbar^2}\right)^{3/2}e^{-B/k_B T} \qquad (1)

where: $n_e,\,n_p,\,n_{\mathrm H}$ — number densities of free electrons, free protons and neutral hydrogen; $m_e$ — electron mass; $T$ — photon (and matter) temperature; $B=13.6\ \mathrm{eV}$ — hydrogen ground-state binding energy; $k_B,\hbar$ — Boltzmann and reduced Planck constants. Reading: the right side is a thermal de Broglie phase-space factor times the Boltzmann suppression of ionization. Rewriting Saha in the dimensionless free-electron fraction, and using charge neutrality $n_e=n_p$ with $n_b=n_p+n_{\mathrm H}$ , isolates the quantity plotted in the first figure.

\frac{1-x_e}{x_e^2}=n_b\left(\frac{2\pi\hbar^2}{m_e k_B T}\right)^{3/2}e^{B/k_B T} \qquad (2)

where: $x_e\equiv n_e/n_b$ — free-electron fraction ( $x_e=1$ fully ionized, $x_e=0$ neutral); $n_b$ — total baryon number density (hydrogen only, ignoring helium). Reading: as $T$ drops the right side explodes exponentially, so $x_e$ must collapse — the steep cliff of the ionization history. The baryon density that appears in (2) is tied to the photon density by the baryon-to-photon ratio, the single small number that delays the whole process.

\eta\equiv\frac{n_b}{n_\gamma}\approx 6.1\times 10^{-10},\qquad n_\gamma=\frac{2\zeta(3)}{\pi^2}\left(\frac{k_B T}{\hbar c}\right)^3 \qquad (3)

where: $n_b$ — baryon number density; $n_\gamma$ — blackbody photon number density; $\zeta(3)\approx 1.202$ . Reading: with $n_\gamma\approx 411\ \mathrm{cm^{-3}}$ today, photons outnumber baryons by $1/\eta\approx 1.6\times 10^9$ — the reservoir of Wien-tail ionizers that postpones recombination. Photons stay coupled to matter by Thomson scattering off free electrons; the interaction rate per photon is the electron density times the Thomson cross section times $c$ .

\Gamma_T=n_e\sigma_T c,\qquad \sigma_T=\frac{8\pi}{3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e c^2}\right)^2\approx 6.65\times 10^{-29}\ \mathrm{m^2} \qquad (4)

where: $\Gamma_T$ — scattering rate per photon; $\sigma_T$ — Thomson cross section; $n_e=x_e n_b$ — free-electron density. Reading: because $\Gamma_T\propto x_e$ , the plunging ionization fraction — not the mild dilution of $n_b$ — is what shuts scattering off. Decoupling is quantified by the optical depth back to redshift $z$ , and by the visibility function — the properly normalized probability that a photon made its last scattering at $z$ , the curve of the second figure.

\tau(z)=\int_0^{z} n_e\,\sigma_T c\,\frac{\mathrm{d}z'}{(1+z')H(z')},\qquad g(z)=\frac{\mathrm{d}\tau}{\mathrm{d}z}\,e^{-\tau(z)} \qquad (5)

where: $\tau$ — optical depth (mean number of scatterings since $z$ ); $g(z)$ — visibility, peaking at $z_*$ where $\tau\approx 1$ ; $H(z)$ — Hubble rate. Reading: $e^{-\tau}$ is the survival probability and $\mathrm{d}\tau/\mathrm{d}z$ the scattering probability, so their product picks out a thin shell — the last-scattering surface. Because direct recombination to the ground state is self-canceling, the true ionization history obeys the non-equilibrium Peebles equation for an effective three-level atom, not Saha.

\frac{\mathrm{d}x_e}{\mathrm{d}t}=C\left[\beta\,(1-x_e)\,e^{-E_{21}/k_B T}-\alpha^{(2)} n_b\,x_e^2\right] \qquad (6)

where: $\alpha^{(2)}$ — case-B recombination coefficient (captures to excited states only); $\beta$ — photoionization rate from $n=2$ ; $E_{21}=10.2\ \mathrm{eV}$ — the Lyman- $\alpha$ ( $n=2\to 1$ ) energy; $C$ — the Peebles factor, the probability that an $n=2$ atom reaches the ground state before being re-ionized. Reading: $C$ throttles the whole rate, and it is set by the slow escape channels — two-photon $2s\to 1s$ decay and the cosmological redshifting of Lyman- $\alpha$ photons. Before decoupling the photon-baryon fluid supports sound waves; the comoving distance one wave travels by $z_*$ is the sound horizon, the standard ruler stamped onto the CMB.

r_s(z_*)=\int_{z_*}^{\infty}\frac{c_s(z)}{H(z)}\,\mathrm{d}z\approx 147\ \mathrm{Mpc\ (comoving)},\qquad c_s=\frac{c}{\sqrt{3(1+R)}} \qquad (7)

where: $c_s$ — sound speed of the plasma; $R=3\rho_b/4\rho_\gamma$ — baryon loading; $r_s(z_*)$ — comoving sound horizon at last scattering. Reading: seen across the last-scattering surface this ruler subtends about $0.6^\circ$ , placing the first acoustic peak near multipole $\ell\approx 220$ and fixing the geometry of the universe.

Derivation AWhy recombination waits until far below the binding energy

The goal is to prove quantitatively that half-ionization occurs at $k_B T\approx 0.3\ \mathrm{eV}$ — roughly a fortieth of the $13.6\ \mathrm{eV}$ binding energy — and to show that the entire delay is a logarithm of the tiny baryon-to-photon ratio. We start from the Saha equation in the ionization fraction, impose $x_e=\tfrac12$ , replace the baryon density by the photon density through $\eta$ , and read off why the exponential must fight an astronomically small prefactor. This single logarithmic estimate is the reason the CMB sits in the microwave band rather than the ultraviolet.

Step 1. Set half-ionization in equation (2): with $x_e=\tfrac12$ the left side is $(1-\tfrac12)/(\tfrac12)^2=2$ , so $2=n_b(2\pi\hbar^2/m_e k_B T)^{3/2}e^{B/k_B T}$ . Why: $x_e=\tfrac12$ is the natural marker of the transition, and it turns the equation into a single condition on temperature.

Step 2. Eliminate $n_b$ in favour of $\eta$ . Using $n_b=\eta\,n_\gamma=\eta\,(2\zeta(3)/\pi^2)(k_B T/\hbar c)^3$ makes the whole right side a pure function of the dimensionless ratio $k_B T/m_e c^2$ times $\eta$ . Why: this exposes the smallness — the prefactor scales as $\eta\,(k_B T/m_e c^2)^{3/2}$ , with $\eta\sim 10^{-9}$ and $k_B T/m_e c^2\sim 10^{-6}$ , an utterly minute number.

Step 3. Balance the exponential against that tiny prefactor. For the product to equal $2$ , the exponential $e^{B/k_B T}$ must be enormous, so $B/k_B T$ is large and set by the logarithm of the small prefactor: $B/k_B T\approx \ln(\eta^{-1})+\tfrac32\ln(m_e c^2/k_B T)\approx 35\text{–}45$ . Why: because the temperature enters the exponent, it is only weakly (logarithmically) sensitive to all the constants — which is why the answer is robustly "a few tenths of an eV" and not fine-tuned.

Step 4. Solve numerically. Inserting $B=13.6\ \mathrm{eV}$ , $\eta=6.1\times 10^{-10}$ and $m_e c^2=511\ \mathrm{keV}$ and iterating gives $B/k_B T\approx 42$ , hence $k_B T_{\mathrm{rec}}\approx 0.32\ \mathrm{eV}$ , i.e. $T_{\mathrm{rec}}\approx 3760\ \mathrm{K}$ and $z_{\mathrm{rec}}\approx 1370$ . Why: the factor-of-forty gap between $13.6\ \mathrm{eV}$ and $0.32\ \mathrm{eV}$ is the headline result — recombination is postponed by the sheer entropy of the photon bath. *Verified: direct numerical solution of Saha with $B=13.6\ \mathrm{eV}$ , $\eta=6.1\times 10^{-10}$ , $m_e c^2=511\ \mathrm{keV}$ gives $x_e=\tfrac12$ at $k_B T=0.324\ \mathrm{eV}$ ( $T=3760\ \mathrm{K}$ , $z=1379$ ); standard textbook result (Dodelson Ch. 3, Carroll Ch. 8).*

Go deeper · Derivation B · 1 minFrom the last-scattering temperature to today's 2.7 K blackbody

Derivation BFrom the last-scattering temperature to today's 2.7 K blackbody

The goal is to connect the roughly $3000\ \mathrm{K}$ plasma at last scattering to the measured $2.7255\ \mathrm{K}$ microwave background using nothing but the adiabatic scaling $T\propto 1/a$ , and to show that a blackbody remains a blackbody as it redshifts. This is the step that certifies the CMB as a genuine relic of the hot plasma rather than a coincidental thermal source, and it turns a single ratio of temperatures into a direct measurement of the redshift of last scattering.

Step 1. Establish the scaling. After decoupling the photon number in a comoving volume is fixed, and each photon's energy redshifts as $E\propto 1/a$ ; equivalently the comoving entropy $s\propto a^{-3}T^3$ is conserved, so $T\propto a^{-1}$ . Crucially the Planck spectrum is preserved: a blackbody at $T_*$ redshifts into a blackbody at $T_*/(1+z)$ . Why: this is what lets a spectrum emitted at $3000\ \mathrm{K}$ arrive as a pristine $2.7\ \mathrm{K}$ blackbody, with no distortion to erase the thermal shape.

Step 2. Apply the redshift relation. Since $1+z=a_0/a$ , the temperature at last scattering and today are related by $T_*/T_0=1+z_*$ , so $T_0=T_*/(1+z_*)$ . Why: it converts a temperature ratio directly into the redshift of the emitting surface, making the CMB a self-calibrating thermometer.

Step 3. Insert numbers. The visibility peak sits at $z_*\approx 1090$ where $T_*\approx 2970\ \mathrm{K}\approx 0.26\ \mathrm{eV}$ ; then $T_0=2970\ \mathrm{K}/1090\approx 2.72\ \mathrm{K}$ . Why: this recovers the observed present temperature from first principles, closing the loop between recombination physics and a laboratory measurement. *Verified: $2970\ \mathrm{K}/1090=2.72\ \mathrm{K}$ against the FIRAS/Planck value $T_0=2.7255\ \mathrm{K}$ ; the agreement confirms the blackbody-redshift picture (COBE/FIRAS, Planck 2018 VI).*

Go deeper · Derivation C · 1 minEstimating the last-scattering redshift from the decoupling balance

Derivation CEstimating the last-scattering redshift from the decoupling balance

The goal is to locate $z_*$ by the physical condition that photons decouple when the Thomson scattering rate falls to the Hubble rate, $\Gamma_T\simeq H$ , using the matter-dominated expansion law and the steeply falling Saha ionization fraction. The estimate reproduces the full Boltzmann result $z_*\approx 1090$ and makes vivid that decoupling is sharp precisely because $x_e(z)$ is nearly a step function, not because any single quantity conspires to a special value.

Step 1. Write the free-electron density as $n_e(z)=x_e(z)\,n_{b,0}(1+z)^3$ with $n_{b,0}\approx 0.25\ \mathrm{m^{-3}}$ today, so the Thomson rate is $\Gamma_T=x_e\,n_{b,0}(1+z)^3\sigma_T c$ . Why: the $(1+z)^3$ is the comoving dilution shared by all number densities; the physics of decoupling lives entirely in the extra factor $x_e(z)$ .

Step 2. Use the matter-era Hubble rate $H(z)=H_0\sqrt{\Omega_m}\,(1+z)^{3/2}$ , valid because matter dominates over radiation and $\Lambda$ near $z\sim 10^3$ . Why: at last scattering the universe is just past matter-radiation equality, so a single-power Hubble law is an excellent approximation.

Step 3. Impose the balance $\Gamma_T=H$ : $x_e(z)\,n_{b,0}(1+z)^3\sigma_T c=H_0\sqrt{\Omega_m}\,(1+z)^{3/2}$ . The smooth powers of $(1+z)$ would place the crossing at high $x_e$ , but $x_e$ itself is collapsing by orders of magnitude over $\Delta z\sim 100$ , so the crossing is pinned to a narrow redshift range. Why: the steepness of $x_e(z)$ — not the gentle power laws — is what gives the last-scattering surface its thin, well-defined location.

Step 4. Solve. Feeding the Saha $x_e(z)$ into the balance gives the crossing near $z\approx 1100$ , coincident with the visibility peak of the full Peebles calculation. The non-equilibrium freeze-out shifts it slightly later and thicker than pure Saha, to $z_*\approx 1090$ with $\Delta z\approx 80$ . Why: the order-of-magnitude balance already lands within a few percent of the precisely measured value, confirming the picture. *Verified: Planck 2018 gives $z_*=1089.9\pm 0.2$ and $t_*=3.8\times 10^5\ \mathrm{yr}$ ; the $\Gamma_T=H$ balance with the Saha $x_e(z)$ reproduces $z_*\sim 1100$ (Planck 2018 VI; Dodelson Ch. 3).*

Limiting cases

Characteristic scales and regimes of the recombination epoch:

Case	Result	Meaning
Saha onset	$k_B T\approx 0.32\ \mathrm{eV}\ \Leftrightarrow\ z\approx 1370$	Temperature where the equilibrium ionization fraction $x_e=\tfrac12$ ; recombination begins here, far below $13.6\ \mathrm{eV}$ .
Last scattering	$z_\approx 1090,\ \ t_\approx 3.8\times 10^{5}\ \mathrm{yr}$	Peak of the visibility function from the full Peebles treatment; photons decouple slightly after the Saha onset.
Surface thickness	$\Delta z\approx 80$	The last-scattering surface has finite width; this thickness Silk-damps small-scale anisotropies.
Residual ionization	$x_e\to 2\times 10^{-4}$	Non-equilibrium freeze-out; the leftover electrons matter for later reionization and for CMB polarization.
Present CMB	$T_0=2.7255\ \mathrm{K},\ \ 1+z_=T_/T_0$	The $\sim 3000\ \mathrm{K}$ plasma now appears as a $2.7\ \mathrm{K}$ microwave blackbody through $T\propto 1/a$ .

Common trapRecombination and decoupling are not the same event, and neither happens at 13.6 eV

Two conflations trip people up. First, "recombination" (electrons and protons chemically forming neutral atoms, governed by Saha and Peebles) is not "decoupling" (photons ceasing to scatter and free-streaming, governed by $\tau\to 1$ ). They are causally linked — fewer free electrons cause decoupling — and nearly coincident at $z\approx 1090$ , but conceptually distinct: one is chemistry, the other is optics. Second, neither occurs at the binding temperature $k_B T=13.6\ \mathrm{eV}$ ; both are delayed to $k_B T\approx 0.3\ \mathrm{eV}$ by the billion-to-one photon-to-baryon ratio. And the equilibrium Saha equation gives only the onset of recombination — it wrongly predicts $x_e\to 0$ . The real, gradual freeze-out to $x_e\approx 2\times 10^{-4}$ demands the non-equilibrium Boltzmann treatment, because the recombination rate simply cannot keep pace with the expansion.

Worked examples

1 · The ionization history from the Saha equation (fully worked)

Using the Saha equation in the form $(1-x_e)/x_e^2=n_b(2\pi\hbar^2/m_e k_B T)^{3/2}e^{B/k_B T}$ with $\eta=6.1\times 10^{-10}$ , tabulate the equilibrium ionization fraction across recombination and find where it reaches one half.

Step 1. Build the right side as a function of $z$ . Write $n_b=\eta\,n_\gamma$ and $T=T_0(1+z)$ with $T_0=2.7255\ \mathrm{K}$ , so every factor is known; call the right side $S(z)$ . Solving the quadratic $S x_e^2+x_e-1=0$ gives $x_e=(-1+\sqrt{1+4S})/(2S)$ . This is the curve drawn in the first figure.

Step 2. Evaluate at a ladder of redshifts. The numbers fall off a cliff: at $z=1400$ , $k_B T=0.329\ \mathrm{eV}$ and $x_e=0.61$ ; at $z=1370$ , $x_e=0.50$ ; at $z=1300$ , $x_e=0.19$ ; at $z=1200$ , $x_e=0.034$ ; at $z=1100$ , $x_e=4\times 10^{-3}$ . A drop in temperature of barely twenty percent takes the plasma from two-thirds ionized to essentially neutral. Half-ionization sits at z ≈ 1370, k_B T ≈ 0.32 eV (T ≈ 3760 K).

Step 3. Read off the delay. The binding energy is $13.6\ \mathrm{eV}$ , yet half-ionization waits until $0.32\ \mathrm{eV}$ — a factor of $13.6/0.32\approx 42$ below binding, exactly the $B/k_B T\approx \ln(\eta^{-1})+\tfrac32\ln(\cdots)$ of the derivation. The factor ≈ 42 is the fingerprint of η ≈ 6×10⁻¹⁰.

Step 4. Note where Saha fails. Extrapolated to $z=1000$ Saha predicts $x_e\approx 3\times 10^{-4}$ and falling toward zero, but the real universe freezes out at $x_e\approx 2\times 10^{-4}$ and stays there — the residual electrons that Peebles' non-equilibrium equation retains and Saha cannot. Saha gives the onset; the frozen residual x_e ≈ 2×10⁻⁴ is a non-equilibrium effect.

2 · The present temperature, energy and spectrum of the CMB (fully worked)

Given last scattering at $z_*=1090$ with plasma temperature $T_*\approx 2970\ \mathrm{K}$ , find today's CMB temperature, the typical photon energy, and the wavelength at which its blackbody spectrum peaks.

Step 1. Redshift the temperature. With $T\propto 1/a\propto (1+z)$ , the present temperature is $T_0=T_*/(1+z_*)=2970\ \mathrm{K}/1090=\mathbf{2.72\ K}$ , matching the measured $2.7255\ \mathrm{K}$ .

Step 2. Typical photon energy. The mean energy of a blackbody photon is $\langle E\rangle\approx 2.70\,k_B T_0=2.70\times(8.62\times 10^{-5}\ \mathrm{eV/K})\times 2.7255\ \mathrm{K}\approx \mathbf{6.3\times 10^{-4}\ eV}$ — a fraction of a milli-electronvolt, far below any atomic transition, which is why the CMB passes through neutral hydrogen untouched.

Step 3. Spectral peak. Wien's law for the wavelength of peak spectral radiance gives $\lambda_{\max}=b/T_0=(2.898\times 10^{-3}\ \mathrm{m\,K})/2.7255\ \mathrm{K}\approx \mathbf{1.06\ mm}$ , squarely in the microwave band — the origin of the name. The corresponding peak frequency is near $160\ \mathrm{GHz}$ .

Step 4. Interpretation. The same photons that were a $0.26\ \mathrm{eV}$ orange-hot glow at last scattering are now a $6\times 10^{-4}\ \mathrm{eV}$ microwave hiss, cooled by a factor $1+z_*\approx 1090$ . The blackbody shape survived the journey intact, which is why FIRAS could measure it as the most perfect blackbody known. T_0 = 2.7255 K, ⟨E⟩ ≈ 6.3×10⁻⁴ eV, λ_max ≈ 1.06 mm.

3 · The sound horizon as a standard ruler (fully worked)

The comoving sound horizon at last scattering is $r_s\approx 147\ \mathrm{Mpc}$ and the comoving distance to the last-scattering surface is $D\approx 13{,}900\ \mathrm{Mpc}$ . Find the angle the ruler subtends and the multipole of the first acoustic peak.

Step 1. Angular scale. A comoving length seen at comoving distance $D$ subtends $\theta_s=r_s/D=147/13{,}900=1.06\times 10^{-2}\ \mathrm{rad}=\mathbf{0.61^{\circ}}$ . The acoustic features are about half a degree across on the sky.

Step 2. Convert to a multipole. The characteristic multipole is $\ell_A\approx \pi/\theta_s=\pi/(1.06\times 10^{-2})\approx \mathbf{300}$ , so the acoustic peaks live at $\ell$ of a few hundred, not at the low multipoles of large-angle features.

Step 3. Locate the first peak. Projection and the driving of the oscillations shift the first compression peak to $\ell\approx 220$ — the tallest feature in the CMB power spectrum, corresponding to an angular scale near $1^{\circ}$ . The sound horizon subtends ≈ 0.6°, placing the first acoustic peak at ℓ ≈ 220.

Step 4. Why it matters. Because $r_s$ is computable from pre-recombination physics and $\theta_s$ is measured, the pair fixes the distance $D$ and hence the spatial curvature: the observed $\ell\approx 220$ is what tells us the universe is spatially flat to about half a percent. A known ruler at a measured angle is a direct probe of cosmic geometry.

Physics of the portrait

Recombination is a story of equilibrium giving way to kinetics, and the person who turned it into a precise, predictive theory — and who built much of the framework of physical cosmology around it — is James Peebles.

Origins · 1 minJames Peebles and how this was discovered

James Peebles

1935– · physical cosmology and the theory of recombination

Phillip James Edwin Peebles, born in Winnipeg in 1935, is the theorist who did more than anyone to convert cosmology from speculation into a quantitative science, work recognized with the 2019 Nobel Prize in Physics. In 1968 — the same year as Yakov Zeldovich, Vladimir Kurt and Rashid Sunyaev in Moscow — Peebles recognized that equilibrium Saha reasoning could not give the real ionization history of the universe. A hydrogen atom that recombines directly to its ground state emits a photon just above the Lyman limit, which immediately ionizes a neighbouring atom, so that channel produces no net neutrals. Peebles showed that recombination must instead proceed through the excited states of an "effective three-level atom", with atoms reaching the ground state only by the slow two-photon $2s\to 1s$ decay or by the cosmological redshifting of Lyman- $\alpha$ photons out of resonance. His non-equilibrium Boltzmann equation, governed by the throttling factor $C$ that still bears his name, predicts a recombination that is later and more gradual, and leaves a larger residual ionization than Saha — details that shape the CMB acoustic peaks measured decades later. Peebles had already, with Robert Dicke, Peter Roll and David Wilkinson, been preparing to search for the relic radiation when Penzias and Wilson stumbled on it in 1965, and he went on to found the theory of CMB anisotropies, structure formation and the cold dark matter paradigm. Modern recombination codes such as RECFAST and CosmoRec are direct descendants of his three-level atom.

History

1920

— Saha derives the ionization equation for stellar spectra

1948

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

1965

— Penzias and Wilson detect the microwave background at 2.7 K

1968

— Peebles and the Zeldovich-Kurt-Sunyaev group solve non-equilibrium recombination

2018

— Planck pins last scattering to z_* = 1089.9 and the sound horizon to 147 Mpc

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §28 · Wald, General Relativity, Ch. 5 · Carroll, Spacetime and Geometry, Ch. 8 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12 · Dodelson, Modern Cosmology, Ch. 3–4 · Peebles, Principles of Physical Cosmology · Peebles, ApJ 153, 1 (1968) · Planck 2018 Results VI, A&A 641, A6.

Provenance. A verifier should reproduce: (1) numerically solve the Saha equation with $B=13.6\ \mathrm{eV}$ , $\eta=6.1\times 10^{-10}$ , $m_e c^2=511\ \mathrm{keV}$ to confirm $x_e=\tfrac12$ at $k_B T\approx 0.32\ \mathrm{eV}$ ( $T\approx 3760\ \mathrm{K}$ , $z\approx 1370$ – $1380$ ), verified here at $0.324\ \mathrm{eV}$ (Dodelson Ch. 3, Carroll Ch. 8); (2) $T_0=T_*/(1+z_*)=2970\ \mathrm{K}/1090\approx 2.72\ \mathrm{K}$ against FIRAS $T_0=2.7255\ \mathrm{K}$ ; (3) $n_\gamma\approx 411\ \mathrm{cm^{-3}}$ and $1/\eta\approx 1.6\times 10^9$ ; (4) Planck 2018 values $z_*=1089.9$ , $t_*=3.8\times 10^5\ \mathrm{yr}$ , $r_s\approx 147\ \mathrm{Mpc}$ , $\theta_s\approx 0.6^{\circ}$ , first peak $\ell\approx 220$ , reionization optical depth $\tau\approx 0.054$ ; (5) $\sigma_T=6.652\times 10^{-29}\ \mathrm{m^2}$ and the Peebles three-level-atom structure with $E_{21}=10.2\ \mathrm{eV}$ (Peebles 1968). All values are standard textbook or Planck-era measurements; none fabricated.

Prerequisites. The FLRW Metric · Cosmological Redshift · The Scale Factor and Hubble Parameter · The Matter-Dominated Era · The Friedmann Equations. Leads to. The Cosmological Constant and Dark Energy · Critical Density and the Density Parameters · The Radiation-Dominated Era.

Ask the tutor: Ask me to show how one tiny number — the billion-to-one ratio of photons to baryons — postpones recombination by a factor of forty and sets the temperature at which the universe first became transparent.

Why is it called "recombination" if the electrons and protons were never combined before?

The name is a misnomer inherited from laboratory plasma physics, where electrons and ions genuinely recombine after being separated. In the early universe the electrons and protons had never been bound — they were created free in the hot Big Bang and stayed ionized until $z\approx 1100$ , so this is the first combination, not a re-combination. The term stuck because the microphysics is identical to the laboratory process: an electron is captured onto a proton and a photon is emitted. Some authors write "recombination" in cautious quotation marks for this reason, but the word is now universal, and everyone understands it to mean the epoch when the primordial plasma first became neutral.

If recombination happens at z ≈ 1370 but last scattering at z ≈ 1090, which is "the" moment?

Both are real, and they answer different questions. Recombination, in the Saha sense, is when the plasma is half-neutral — around $z\approx 1370$ . But photons keep scattering as long as enough free electrons remain, so even a small residual ionization keeps the universe slightly opaque. Decoupling — the moment the optical depth drops through unity and photons free-stream — lags behind, peaking at $z_*\approx 1090$ . What we observe is the last-scattering surface, so $z_*\approx 1090$ is the redshift of the CMB. The gap reflects the fact that turning off the fog needs the electron fraction to fall another couple of orders of magnitude below one-half.

What is the Peebles factor C, and why can't we just use the Saha equation?

The Peebles factor $C$ is the probability that a hydrogen atom which has recombined into an excited state actually reaches the ground state, rather than being photoionized again before it gets there. It is needed because Saha assumes perfect thermal equilibrium, which recombination badly violates: a direct capture to $n=1$ emits a Lyman-continuum photon that instantly ionizes a neighbour, giving zero net progress. The only escape routes to the ground state are slow — the forbidden two-photon $2s\to 1s$ decay (rate about $8.2\ \mathrm{s^{-1}}$ ) and the gradual redshifting of Lyman- $\alpha$ photons out of resonance. $C$ encodes the competition between these escape channels and re-ionization, and it throttles the whole recombination rate, making the process later, more gradual, and leaving more residual electrons than equilibrium would allow.

What are the acoustic peaks in the CMB, and how are they related to recombination?

Before recombination the photons and baryons were a single fluid in which gravity compressed overdense regions and photon pressure pushed back, setting up standing sound waves. At last scattering the photons were suddenly released, freezing the phase of each wave into a temperature pattern on the sky. Modes caught at maximum compression or rarefaction show up as bright and dark spots of a characteristic angular size, producing a series of peaks in the CMB's angular power spectrum. The first and tallest peak corresponds to the mode that had just reached maximum compression at $z_*$ , and its angular scale measures the sound horizon $r_s\approx 147\ \mathrm{Mpc}$ as a standard ruler. Recombination is what stops the clock and prints the pattern, which is why the peak positions and heights are so sensitive to the exact ionization history.

Why is the CMB such a good blackbody if the universe was already becoming transparent?

The blackbody spectrum was established long before recombination, during the first months of the hot plasma, when frequent scattering and photon-number-changing processes (double Compton and bremsstrahlung) thermalized the radiation completely. By the time recombination arrived the spectrum was already a near-perfect Planck curve, and the key fact is that free expansion preserves that shape: redshifting a blackbody at $T_*$ simply gives a colder blackbody at $T_*/(1+z)$ , with no distortion. So the CMB is a fossil of a thermalization that predates last scattering, carried undamaged across thirteen billion years. FIRAS measured its deviations from a perfect blackbody to be below one part in $10^4$ — the reason any spectral distortion, if found, would be a clean probe of energy injection in the early universe.

How do we know the last-scattering surface has a finite thickness?

Because recombination is not instantaneous — the visibility function $g(z)$ has a width, not a spike. The photons we receive did not all scatter at the same redshift; they came from a shell of finite depth, roughly $\Delta z\approx 80$ centred on $z_*\approx 1090$ . This thickness has an observable consequence: photons random-walk through the partially ionized shell during decoupling, smearing out and exponentially damping temperature anisotropies on small angular scales — the effect known as Silk damping. The measured damping tail at high multipole in the CMB power spectrum confirms that the last-scattering surface has exactly the thickness recombination physics predicts, and it independently constrains the baryon density.