Last Scattering Surface and Photon Decoupling

The cosmic microwave background is the oldest light in the Universe, and the moment it was set free is the sharpest boundary in cosmology. For its first 380,000 years the Universe was an opaque plasma: free electrons scattered photons through Thomson scattering so relentlessly that the mean free path was minute, and the whole medium glowed as a featureless fog. A photon could no more cross the Universe than cross the interior of the Sun — deflected, thermalized, trapped. Then the plasma cooled through roughly $3000\,\mathrm{K}$ , electrons combined with protons into neutral hydrogen, the free-electron density collapsed, and the fog lifted almost all at once. The last photons to scatter off the vanishing electrons streamed away untouched and are, after $13.8$ billion years of redshifting, the microwaves we detect today. The geometric locus of those final scatterings — one point along every line of sight, at comoving radius $\approx14\,\mathrm{Gpc}$ — is the last scattering surface, the photosphere of the observable Universe. The event that created it is photon decoupling: the freeze-out of Thomson scattering, when the interaction rate per photon fell below the expansion rate. Crucially, decoupling is neither the same as recombination nor the same as reaching the hydrogen binding energy — it is a race between two clocks, the scattering clock $1/\Gamma$ and the expansion clock $1/H$ , won only after recombination has drained the electrons to a residual fraction of about one percent. This article assembles the rate condition, the optical depth, and the visibility function that together fix when, how sharply, and from where the CMB was released.

Key takeaways

The last scattering surface is the set of points from which each CMB photon we detect made its final Thomson scatter — the effective photosphere of the observable Universe, at comoving radius $\approx14\,\mathrm{Gpc}$ .
Photon decoupling is the freeze-out of Thomson scattering: it occurs when the rate per photon $\Gamma=n_e\sigma_T c$ drops below the Hubble rate $H$ , at $z_{\mathrm{dec}}\approx1090$ , $T\approx0.26\,\mathrm{eV}\approx2970\,\mathrm{K}$ , cosmic age $t\approx3.8\times10^{5}\,\mathrm{yr}$ .
Decoupling follows recombination but is not identical to it: the plasma is $50\%$ ionized already at $z\approx1370$ , yet stays opaque until the free-electron fraction falls to $x_e\sim10^{-2}$ .
Recombination is not instantaneous, so the surface has finite thickness $\Delta z\approx80$ , set by the width of the visibility function $g(z)=-\mathrm{d}e^{-\tau}/\mathrm{d}z$ — a comoving depth of $\approx18\,\mathrm{Mpc}$ that blurs small-scale anisotropies.
The reason hydrogen stays ionized until $T\ll13.6\,\mathrm{eV}$ is the enormous photon-to-baryon ratio $\eta^{-1}\sim1.6\times10^{9}$ : the rare Wien-tail photons keep ionizing until $T$ falls to about $1/40$ of the binding energy.
Key idea: Decoupling is the freeze-out of Thomson scattering — it happens when the rate per photon $\Gamma=n_e\sigma_T c$ falls below the expansion rate $H$ , triggered by the recombination-driven collapse of the free-electron density $n_e=x_e\,n_{\mathrm{H}}$ .

The core insight

Decoupling is the freeze-out of Thomson scattering: it happens when the interaction rate per photon $\Gamma=n_e\sigma_T c$ falls below the expansion rate $H$ , which is triggered by the recombination-driven collapse of the free-electron density $n_e=x_e\,n_{\mathrm{H}}$ .

Where this lives today

The last scattering surface is the working backdrop of precision cosmology. Every CMB experiment — from COBE and WMAP to Planck, ACT, and the South Pole Telescope — measures the temperature and polarization pattern of this single surface, and the cosmological parameter fit is, at bottom, a model of the physics on and behind it. The comoving sound horizon at decoupling, $r_s\approx145\,\mathrm{Mpc}$ , subtends $\theta_s\approx0.6^\circ$ at the surface (acoustic scale $\ell_A\approx300$ ); the first temperature peak at $\ell\approx220$ is the sharpest standard ruler in cosmology, pinning the spatial curvature to near-flatness. Polarization adds a second handle: the quadrupole seen by electrons during the finite-thickness scattering epoch generates the $E$ -mode pattern, while late reionization by the first stars ( $z\sim6\text{–}10$ ) rescatters a few percent of the photons, adding an optical depth $\tau\approx0.054$ and a second bump to the visibility function. The frontier is the primordial $B$ -mode — the imprint of inflationary gravitational waves on this same surface — pursued by BICEP/Keck, the Simons Observatory, and CMB-S4, along with spectral-distortion probes of whether the recombination history was disturbed by decaying dark matter or primordial black holes.

The physical picture

Imagine a fog-filled room lit from within. While the water droplets — here, the free electrons — are dense, light scatters at every step and the room glows uniformly. Let the fog clear and, at the instant it thins to nothing, the last light that scattered off the last droplets flies straight to your eye, forming a bright shell at a fixed distance. That shell is the last scattering surface, and its redshifted glow is the CMB. The transition is fast because recombination destroys the electron supply exponentially, so the scattering rate plummets far faster than the gently declining expansion rate. The threshold is about rates, not about temperature reaching the binding energy. Even a whisper of residual ionization keeps the plasma opaque while $n_e\sigma_T c > H$ , so photons are released only once recombination has removed all but about one electron in a hundred. And because there are more than a billion photons per baryon, hydrogen is held ionized by the rare energetic tail of the blackbody spectrum until $T$ has fallen to a fortieth of $13.6\,\mathrm{eV}$ — a direct fingerprint of the tiny baryon-to-photon ratio $\eta$ .

Figure

From opaque plasma to transparent Universe: while free electrons are abundant a photon random-walks with a tiny mean free path, but once recombination drops the ionization to x_e ≈ 10⁻² the Thomson rate Γ = n_e σ_T c falls below the expansion rate H and the photon streams freely to us

Last scattering is a surface in our past light cone, not a physical wall in space. Every observer sees their own last scattering surface at the same comoving radius; the CMB is one relic seen from a billion vantage points, like the edge of a fog seen by many at once. The second figure shows the quantity that actually defines the surface: the Thomson visibility function, sharply peaked in redshift even though the ionization fraction crosses one-half far earlier.

Figure

The visibility function peaks at z ≈ 1090 while the ionization fraction collapses: g(z) = −d e^(−τ)/dz has width Δz ≈ 80 (FWHM), yet Saha ionization already passes x_e = 0.5 far earlier at z ≈ 1370 — recombination precedes decoupling

The core equations

The decoupling condition is the whole story in one line: photons are released when a photon can expect fewer than one further scatter for the rest of cosmic history — when the scattering rate per photon crosses below the expansion rate.

\Gamma \equiv n_e\,\sigma_T\,c \;\simeq\; H(z_{\mathrm{dec}}) \qquad (1)

where: $\Gamma$ — Thomson scattering rate per photon; $n_e$ — free-electron number density; $\sigma_T=6.652\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross section; $H(z)$ — Hubble rate; $c$ — speed of light. Reading it: $\Gamma>H$ means many scatters per Hubble time (opaque); $\Gamma<H$ means no further scatter ever (transparent); the crossover marks release. The rate condition is only an order-of-magnitude locator. The exact statement is built from the Thomson optical depth — the expected number of scatters between redshift $z$ and today.

\tau(z) = \int_0^z n_e(z')\,\sigma_T\,c\,\frac{\mathrm{d}t}{\mathrm{d}z'}\,\mathrm{d}z' = \int_0^z \frac{n_e(z')\,\sigma_T\,c}{(1+z')\,H(z')}\,\mathrm{d}z' \qquad (2)

where: $\tau$ — Thomson optical depth from us to redshift $z$ ; $n_e(z')$ — free-electron density; $\mathrm{d}t/\mathrm{d}z'=-1/[(1+z')H(z')]$ — proper time per unit redshift. Reading it: $e^{-\tau(z)}$ is the probability a photon from $z$ reaches us unscattered, and $\tau=1$ marks the edge of transparency. The redshift distribution of last scatters — the surface's true profile — is the visibility function, the probability density that a detected photon made its final scatter at $z$ .

g(z) = -\frac{\mathrm{d}\,e^{-\tau}}{\mathrm{d}z} = \frac{\mathrm{d}\tau}{\mathrm{d}z}\,e^{-\tau},\qquad \int_0^\infty g(z)\,\mathrm{d}z = 1 \qquad (3)

where: $g(z)$ — probability density of last scattering at $z$ ; $\mathrm{d}\tau/\mathrm{d}z$ — differential opacity (favours high $z$ ); $e^{-\tau}$ — survival factor (favours low $z$ ). Reading it: the product peaks where the two competing factors balance, defining $z_{\mathrm{dec}}$ by its peak and the surface thickness by its width. The electron supply that drives all three is the ionization fraction times the baryon density, with helium held aside because it recombines earlier.

n_e = x_e\,n_{\mathrm{H}} = x_e\,(1-Y_p)\,n_{\mathrm{b}} = x_e\,(1-Y_p)\,\eta\,n_\gamma \qquad (4)

where: $x_e=n_e/n_{\mathrm{H}}$ — ionization fraction; $n_{\mathrm{b}}$ — baryon number density; $Y_p\approx0.245$ — primordial helium mass fraction; $\eta\approx6.1\times10^{-10}$ — baryon-to-photon ratio; $n_\gamma\propto T^3$ — photon number density. Reading it: because $\eta$ is tiny, $n_e$ is a billionfold below $n_\gamma$ , and the exponential fall of $x_e$ is what makes $\Gamma$ collapse. Near decoupling the Universe is matter-dominated, so the expansion rate has a simple scaling that fixes the rate condition's right side.

H(z) \simeq H_0\,\Omega_m^{1/2}\,(1+z)^{3/2} \qquad (5)

where: $H_0\approx67\text{–}70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ — present Hubble constant; $\Omega_m\approx0.31$ — matter density parameter; $(1+z)^{3/2}$ — matter-era scaling. Reading it: at $z_{\mathrm{dec}}\approx1090$ this gives $H\approx4.4\times10^{-14}\,\mathrm{s^{-1}}$ , the clock the scattering rate must beat. The temperature at last scattering follows from adiabatic redshifting of the present blackbody.

T_{\mathrm{dec}} = T_0\,(1+z_{\mathrm{dec}}) \approx 2.7255\,\mathrm{K}\times 1091 \approx 2970\,\mathrm{K} \approx 0.26\,\mathrm{eV} \qquad (6)

where: $T_0=2.7255\,\mathrm{K}$ — present CMB temperature (FIRAS); $z_{\mathrm{dec}}\approx1090$ — decoupling redshift. Reading it: this is far below $B/k_B=1.58\times10^{5}\,\mathrm{K}$ , the ionization temperature — the gap is the work of $\eta$ . The equilibrium ionization that locates recombination is the Saha equation, valid until the reaction falls out of equilibrium.

\frac{x_e^2}{1-x_e} = \frac{1}{n_{\mathrm{b}}}\left(\frac{m_e k_B T}{2\pi\hbar^2}\right)^{3/2} e^{-B/k_B T} \qquad (7)

where: $B=13.6\,\mathrm{eV}$ — hydrogen binding energy; $m_e$ — electron mass; the prefactor $\propto1/n_{\mathrm{b}}\propto1/\eta$ is enormous. Reading it: the huge prefactor forces the exponential to be tiny, so $x_e=0.5$ is reached only at $k_BT\approx0.32\,\mathrm{eV}$ , i.e. $z\approx1370$ . Finally, because recombination is not instantaneous, the surface has a comoving depth set by the visibility width.

\Delta r_{\mathrm{comov}} \approx \frac{c\,\Delta z}{H(z_{\mathrm{dec}})} \approx 18\,\mathrm{Mpc},\qquad \Delta z \approx 80 \qquad (8)

where: $\Delta z$ — FWHM of $g(z)$ ; $\Delta r_{\mathrm{comov}}$ — comoving thickness (comoving distance element $\mathrm{d}\chi=c\,\mathrm{d}z/H$ ). Reading it: this is a comoving depth; the proper thickness is smaller by $(1+z)\approx10^{3}$ , only about $16\,\mathrm{kpc}$ .

Derivation ALocating photon decoupling from the rate balance

The goal is to turn the qualitative rate condition $\Gamma\simeq H$ into an actual redshift and to show that decoupling is delayed until the ionization fraction has fallen to a residual percent. We assemble four ingredients — the Thomson rate through the free-electron density, today's baryon density from $\eta$ , the matter-dominated Hubble rate, and the Saha ionization fraction — and demand that the scattering clock and the expansion clock strike together. The exponential steepness of Saha's $x_e(z)$ is what pins $z_{\mathrm{dec}}$ despite the crude criterion.

Step 1. Write the scattering rate through the free-electron density, $\Gamma=n_e\sigma_T c=x_e\,(1-Y_p)\,n_{\mathrm{b},0}\,(1+z)^3\,\sigma_T c$ , so that all redshift dependence sits in the $(1+z)^3$ dilution and the ionization fraction $x_e(z)$ . Why: it separates the smooth geometric dilution of electrons from the exponential chemistry of recombination — and the exponential is what triggers decoupling.

Step 2. Fix today's baryon density from the measured $\eta$ and the blackbody photon density, $n_{\mathrm{b},0}=\eta\,n_{\gamma,0}=6.1\times10^{-10}\times411\,\mathrm{cm^{-3}}\approx2.5\times10^{-7}\,\mathrm{cm^{-3}}$ . Why: this anchors the normalization of $\Gamma$ to two well-measured numbers — $\eta$ and $T_0$ — with no free parameters.

Step 3. Evaluate the matter-dominated expansion rate at $z\sim1090$ , $H(z)=H_0\,\Omega_m^{1/2}(1+z)^{3/2}$ with $H_0\approx2.17\times10^{-18}\,\mathrm{s^{-1}}$ and $\Omega_m^{1/2}\approx0.56$ , giving $H(z_{\mathrm{dec}})\approx4.4\times10^{-14}\,\mathrm{s^{-1}}$ . Why: the target the scattering rate must fall to is set by cosmology alone, and it is small — a photon at decoupling scatters roughly once per Hubble time, not billions of times.

Step 4. Impose $\Gamma=H$ and solve for the residual ionization: with $n_{\mathrm{H}}(z_{\mathrm{dec}})=(1-Y_p)n_{\mathrm{b},0}(1+z)^3\approx246\,\mathrm{cm^{-3}}$ and $\sigma_T c\approx2.0\times10^{-14}\,\mathrm{cm^{3}\,s^{-1}}$ , the balance $x_e\,n_{\mathrm{H}}\,\sigma_T c=H$ gives $x_e\approx9\times10^{-3}$ . Why: this is the punchline — decoupling waits not for $x_e\to0$ but for $x_e\sim10^{-2}$ , which is why it lags $50\%$ recombination.

Step 5. Feed that residual $x_e$ back through Saha's exponential $x_e(z)$ : because $x_e$ falls by orders of magnitude over $\Delta z\sim100$ , the condition $x_e(z_{\mathrm{dec}})\approx10^{-2}$ is met at $z_{\mathrm{dec}}\approx1090$ , giving $T_{\mathrm{dec}}\approx0.26\,\mathrm{eV}$ and cosmic age $t\approx3.8\times10^{5}\,\mathrm{yr}$ . Why: the steepness of Saha makes the answer insensitive to the crude $\Gamma=H$ threshold — any reasonable choice lands near $z\approx1090$ , agreeing with the full Boltzmann calculation.

Go deeper · Derivation B · 1 minWhy decoupling waits until far below the binding energy

Derivation BWhy decoupling waits until far below the binding energy

The goal is to explain the most counter-intuitive number in early-Universe cosmology: hydrogen, with a binding energy of $13.6\,\mathrm{eV}$ , does not recombine until the temperature has fallen to about $0.3\,\mathrm{eV}$ , forty times lower. The culprit is the baryon-to-photon ratio $\eta\sim10^{-9}$ , which enters Saha as a gigantic prefactor $1/\eta$ . We set the ionization fraction to one-half, express the baryon density through $\eta$ , and read off the temperature at which the exponential is forced down far enough to balance that prefactor.

Step 1. Set $x_e=1/2$ in the Saha equation, so the left side is $x_e^2/(1-x_e)=1/2$ , and write the baryon density through the photon density, $n_{\mathrm{b}}=\eta\,n_\gamma=\eta\,\dfrac{2\zeta(3)}{\pi^2}\left(\dfrac{k_BT}{\hbar c}\right)^3$ with $\zeta(3)\approx1.202$ . Why: routing $n_{\mathrm{b}}$ through $\eta$ exposes where the small number lives — it multiplies the phase-space factor and controls the whole balance.

Step 2. Form the dimensionless combination that must equal order unity, $\dfrac{1}{n_{\mathrm{b}}}\left(\dfrac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2}e^{-B/k_BT}\sim\dfrac{1}{\eta}\left(\dfrac{m_ec^2}{k_BT}\right)^{3/2}e^{-B/k_BT}$ . Why: stripping the expression to its scaling shows the contest plainly — a huge algebraic prefactor $\eta^{-1}(m_ec^2/k_BT)^{3/2}\sim10^{18}$ against a decaying exponential.

Step 3. Balance the two by taking logarithms: $B/k_BT\approx\ln\!\left[\eta^{-1}(m_ec^2/k_BT)^{3/2}\right]\approx\ln(10^{18})\approx42$ , so $k_BT\approx B/42\approx0.32\,\mathrm{eV}$ . Why: because $\ln(10^{18})$ is only about $42$ , the temperature must drop to a fortieth of the binding energy — the whole delay is the logarithm of the inverse baryon-to-photon ratio.

Step 4. Convert to redshift, $z=T/T_0-1\approx3700/2.73-1\approx1370$ , the epoch of $50\%$ recombination, comfortably before the $z_{\mathrm{dec}}\approx1090$ of decoupling. Why: it confirms recombination precedes decoupling — the rare Wien-tail photons, one in $10^{9}$ , keep ionizing until $T$ falls this far.

Go deeper · Derivation C · 1 minThe visibility function and the thickness of the surface

Derivation CThe visibility function and the thickness of the surface

The goal is to show that "the surface of last scattering" is a genuine, computable object with a definite thickness, and that the redshift of last scattering is the peak of a probability distribution, not a single instant. Starting from the optical depth, we build the visibility function as a normalized probability density, argue that its peak is where growing opacity and falling survival trade off, and turn its redshift width into a comoving depth relevant to the CMB power spectrum.

Step 1. Interpret $e^{-\tau(z)}$ as the probability a photon reaches today from $z$ without scattering; then the probability its last scatter fell in $[z,z+\mathrm{d}z]$ is $g(z)\,\mathrm{d}z=e^{-\tau}\,\mathrm{d}\tau=-\mathrm{d}(e^{-\tau})$ . Why: this makes $g(z)$ an automatically normalized probability density — integrating over all $z$ gives $e^{-\tau(0)}-e^{-\tau(\infty)}=1$ .

Step 2. Locate the peak by competition: $\mathrm{d}\tau/\mathrm{d}z\propto n_e$ rises steeply toward higher $z$ (more electrons), while $e^{-\tau}$ falls toward higher $z$ (more intervening scatterers), so their product is sharply peaked where opacity is building fastest yet transparency still holds — this defines $z_{\mathrm{dec}}\approx1090$ , essentially where $\tau\approx1$ . Why: the peak is not put in by hand but emerges from two opposing exponential trends, which is why the surface is thin even though recombination spans a wider range in $x_e$ .

Step 3. Read the FWHM of $g(z)$ as the redshift thickness, $\Delta z\approx80$ , and convert to a comoving depth using the comoving distance element $\mathrm{d}\chi=c\,\mathrm{d}z/H(z)$ : $\Delta r_{\mathrm{comov}}\approx c\,\Delta z/H(z_{\mathrm{dec}})\approx(3\times10^{5})(80)/(1.34\times10^{6})\approx18\,\mathrm{Mpc}$ . Why: fluctuations smaller than this depth get averaged along the line of sight and smeared, so it sets a floor on the angular sharpness of the CMB, alongside Silk diffusion damping.

Limiting cases

The physics interpolates between four regimes as the Universe cools through recombination:

Case	Result	Meaning
Deep plasma era, $T\gg1\,\mathrm{eV}$	$\Gamma=n_e\sigma_T c \gg H$	Fully ionized, $x_e\approx1$ ; photons and baryons locked into one tightly coupled fluid, mean free path $\ell\ll$ horizon.
Recombination onset, $T\sim0.3\,\mathrm{eV}$	$n_e\propto x_e\,n_{\mathrm{H}}$ falls exponentially (Saha)	Free electrons vanish; $\Gamma$ drops far faster than $H$ , driving decoupling within a narrow redshift interval.
Decoupling, $z\approx1090$	$\Gamma(z_{\mathrm{dec}})=H(z_{\mathrm{dec}}),\ \tau\approx1$	Optical depth reaches unity; the visibility function peaks; photons make their last scatter and free-stream.
After decoupling, $z<1090$	$\ell_\gamma>c/H,\quad T_\gamma=T_0(1+z)$	Universe transparent; the blackbody spectrum is preserved by adiabatic redshifting, cooling to $2.7255\,\mathrm{K}$ today.

Common trapRecombination, decoupling, and last scattering are the same event

They are three distinct milestones that merely lie close in redshift, and conflating them is the classic error. Recombination is the chemistry — protons and electrons binding into neutral hydrogen — with the plasma $50\%$ recombined already at $z\approx1370$ ( $T\approx0.32\,\mathrm{eV}$ ). Decoupling is the optics — the Thomson rate falling below the Hubble rate, $\Gamma=H$ — which happens later, at $z\approx1090$ , because the plasma stays opaque until $x_e\sim10^{-2}$ . Last scattering is the statistics — the final scatter of each detected photon — whose redshift distribution is the visibility function $g(z)$ , peaked at decoupling with width $\Delta z\approx80$ . A second trap: the decoupling temperature $\approx3000\,\mathrm{K}$ is nowhere near the naive ionization temperature $B/k_B\approx1.6\times10^{5}\,\mathrm{K}$ — the factor-of-fifty gap is entirely the work of the enormous photon-to-baryon ratio.

Worked examples

1 · The residual ionization at decoupling (fully worked)

Show that at $z_{\mathrm{dec}}=1090$ the plasma is still about one percent ionized when photons decouple. Use $n_{\gamma,0}=411\,\mathrm{cm^{-3}}$ , $\eta=6.1\times10^{-10}$ , $Y_p=0.245$ , $\sigma_T=6.652\times10^{-25}\,\mathrm{cm^2}$ , $H_0=67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $\Omega_m=0.31$ .

Step 1. Present hydrogen density: $n_{\mathrm{H},0}=(1-Y_p)\,\eta\,n_{\gamma,0}=0.755\times6.1\times10^{-10}\times411=1.89\times10^{-7}\,\mathrm{cm^{-3}}$ . Scale to decoupling by $(1+z)^3=1091^3=1.30\times10^{9}$ : $n_{\mathrm{H}}(z_{\mathrm{dec}})=\mathbf{246\ cm^{-3}}$ .

Step 2. Expansion rate at decoupling: $H_0=67/(3.086\times10^{19})=2.17\times10^{-18}\,\mathrm{s^{-1}}$ , and $H(z_{\mathrm{dec}})=H_0\,\Omega_m^{1/2}(1+z)^{3/2}=2.17\times10^{-18}\times0.557\times1091^{1.5}=\mathbf{4.4\times10^{-14}\ s^{-1}}$ .

Step 3. Impose $\Gamma=H$ with $\Gamma=x_e\,n_{\mathrm{H}}\,\sigma_T c$ and $\sigma_T c=6.652\times10^{-25}\times3.0\times10^{10}=2.0\times10^{-14}\,\mathrm{cm^{3}\,s^{-1}}$ : solving $x_e=\dfrac{H}{n_{\mathrm{H}}\,\sigma_T c}=\dfrac{4.4\times10^{-14}}{246\times2.0\times10^{-14}}$ gives $x_e\approx\mathbf{9\times10^{-3}}$ .

Step 4. Interpret: the plasma is still nearly one percent ionized at last scattering, yet that is dilute enough that a typical photon will not scatter again before the Universe expands its remaining electrons away. Photons decouple at $x_e\sim10^{-2}$ , not at $x_e=0$ — decoupling lags recombination.

2 · Thickness of the last scattering surface

The visibility function has FWHM $\Delta z\approx80$ . Find the comoving depth of the surface, and comment on its observable effect.

Step 1. The comoving distance element is $\mathrm{d}\chi=c\,\mathrm{d}z/H(z)$ , so the comoving thickness is $\Delta r_{\mathrm{comov}}=c\,\Delta z/H(z_{\mathrm{dec}})$ . In convenient units, $H(z_{\mathrm{dec}})=67\times0.557\times1091^{1.5}=1.34\times10^{6}\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Step 2. Insert $c=3.0\times10^{5}\,\mathrm{km\,s^{-1}}$ and $\Delta z=80$ : $\Delta r_{\mathrm{comov}}=\dfrac{(3.0\times10^{5})(80)}{1.34\times10^{6}}\approx\mathbf{18\ Mpc}$ comoving. The proper thickness is smaller by $(1+z)\approx1091$ , only $\Delta r_{\mathrm{prop}}\approx\mathbf{16\ kpc}$ .

Step 3. Consequence: fluctuations with comoving wavelength shorter than $\sim18\,\mathrm{Mpc}$ are averaged along the line of sight and suppressed. Together with Silk (diffusion) damping this cuts off the CMB acoustic peaks at high multipole. A surface $\sim18\,\mathrm{Mpc}$ deep blurs anisotropies below the damping scale, $\ell\gtrsim1000$ .

Physics of the portrait

The last scattering surface is where a prediction, a serendipitous discovery, and a hard calculation converge: a relic radiation guessed from nuclear physics, stumbled upon as noise, then computed atom by atom.

Origins · 1 minlast scattering surface and how this was discovered

last scattering surface

the relic photosphere at z ≈ 1090, released when Thomson scattering froze out · hot big bang cosmology and the CMB

The idea that the Universe has a photosphere begins with George Gamow's hot big bang and the 1948 estimate by Ralph Alpher and Robert Herman that a relic radiation of "about $5\,\mathrm{K}$ " should survive — a prediction so far ahead of the instruments that it was forgotten for fifteen years. In 1965 Arno Penzias and Robert Wilson, calibrating a horn antenna at Bell Labs, found an irremovable microwave excess of about $3\,\mathrm{K}$ ; in the companion paper Robert Dicke, Jim Peebles, Peter Roll, and David Wilkinson identified it as the cooled fireball, and the last scattering surface became an observed object. The theoretical surface was sharpened the same year: Peebles, and independently Yakov Zeldovich, Vladimir Kurt, and Rashid Sunyaev, showed in 1968 that equilibrium Saha is not enough — recombination stalls because ground-state captures make fresh ionizing photons, so the electron fraction freezes out instead through the two-photon decay of the $2s$ state and the redshifting of Lyman- $\alpha$ . This "effective three-level atom" gave the residual $x_e(z)$ that fixes the visibility function, and it is the ancestor of every modern recombination code (RECFAST, HyRec, CosmoRec). In 1970 Sunyaev and Zeldovich predicted the acoustic peaks the plasma would imprint on the surface — the standard ruler WMAP and Planck would resolve. Peebles received the 2019 Nobel Prize for the framework in which the last scattering surface is a precision cosmological tool.

History

1948

— Alpher and Herman predict a surviving relic radiation of a few kelvin from the hot early Universe

1965

— Penzias and Wilson detect the microwave background; Dicke, Peebles, Roll, and Wilkinson interpret it as the cooled fireball

1968

— Peebles, and Zeldovich, Kurt, and Sunyaev, compute nonequilibrium recombination and the residual ionization fraction

1970

— Sunyaev and Zeldovich predict acoustic peaks imprinted on the surface of last scattering

2018

— Planck fixes the decoupling redshift at z ≈ 1090 and the surface thickness at Δz ≈ 80

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 8–9 (recombination and the CMB) · Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 3–4 (recombination, Boltzmann equation, visibility) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 (thermal history) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29–30 · Peebles, ApJ 153, 1 (1968); Zeldovich, Kurt & Sunyaev (1968) — nonequilibrium recombination. The numerical values used throughout are standard Planck-era results: $z_{\mathrm{dec}}\approx1090$ , $T_{\mathrm{dec}}\approx0.26\,\mathrm{eV}\approx2970\,\mathrm{K}$ , $t\approx3.8\times10^{5}\,\mathrm{yr}$ , $\Delta z\approx80$ giving $\Delta r_{\mathrm{comov}}\approx18\,\mathrm{Mpc}$ , $\sigma_T=6.652\times10^{-25}\,\mathrm{cm^2}$ , $T_0=2.7255\,\mathrm{K}$ , $n_{\gamma,0}=411\,\mathrm{cm^{-3}}$ , $\eta\approx6.1\times10^{-10}$ , $Y_p\approx0.245$ , $r_s\approx145\,\mathrm{Mpc}$ , first acoustic peak $\ell\approx220$ , reionization optical depth $\tau\approx0.054$ . Key checks: the residual $x_e\approx9\times10^{-3}$ follows from $\Gamma=H$ ; the comoving thickness $c\,\Delta z/H\approx18\,\mathrm{Mpc}$ from the visibility FWHM; and the Saha half-ionization temperature $k_BT\approx B/\ln(\eta^{-1})\approx0.32\,\mathrm{eV}$ . Note in particular that the comoving depth is $c\,\Delta z/H(z)$ , not $c\,\Delta z/[(1+z)H(z)]$ — the latter is the far smaller proper depth.

Prerequisites. Recombination and the Saha Equation · Decoupling and Freeze-Out · Thermal History of the Universe · Equilibrium Thermodynamics of the Early Universe. Leads to. Big Bang Nucleosynthesis · Baryon Asymmetry and Baryogenesis.

Ask the tutor: Ask me to walk through how the visibility function $g(z)$ is built from the optical depth, or how a change in the baryon density $\eta$ moves both recombination and the peaks of the CMB spectrum.

Is recombination the same thing as decoupling?

No — they are close in redshift but physically distinct. Recombination is the chemistry of electrons and protons binding into neutral hydrogen; by Saha the plasma is half-recombined already at $z\approx1370$ . Decoupling is the optics: photons keep scattering until the Thomson rate $\Gamma=n_e\sigma_T c$ falls below the Hubble rate $H$ , at $z\approx1090$ , when the ionization fraction has dropped to about $10^{-2}$ . The gap exists because even a small residual electron density keeps the plasma opaque — the photons wait until almost all the electrons are gone, not merely until most atoms have formed.

Why does the Universe stay ionized until only 3000 K, not 160,000 K?

Because there are roughly a billion photons for every baryon. The naive expectation, that hydrogen ionizes around $T\sim B/k_B\approx1.6\times10^{5}\,\mathrm{K}$ , ignores that even when the typical photon is far below $13.6\,\mathrm{eV}$ , the exponentially rare Wien-tail photons are still numerous enough to keep ionizing — simply because there are so many photons. Saha makes this quantitative: the balance sits where $B/k_BT\approx\ln(\eta^{-1})\approx42$ , so recombination is delayed until $T$ falls to about a fortieth of the binding-energy temperature, near $3000\,\mathrm{K}$ .

What exactly is the visibility function?

It is the probability distribution, in redshift, of where a detected CMB photon made its final scatter. Formally $g(z)=-\mathrm{d}e^{-\tau}/\mathrm{d}z=(\mathrm{d}\tau/\mathrm{d}z)\,e^{-\tau}$ , the product of the differential opacity (favouring high redshift, where electrons are dense) and the survival probability (favouring low redshift). It is sharply peaked near $z\approx1090$ with width $\Delta z\approx80$ : that peak is the decoupling redshift, and the width is the physical thickness of the surface. It is normalized to unity because every photon scattered for the last time somewhere.

Why does the last scattering surface have a thickness at all?

Because recombination is not instantaneous. If all the electrons vanished at one instant, every photon would have last scattered at the same redshift and the surface would be infinitely thin. Instead the ionization fraction falls smoothly, so photons make their last scatter over a spread $\Delta z\approx80$ , a comoving depth of about $18\,\mathrm{Mpc}$ . This matters observationally: fluctuations smaller than that depth get averaged along the line of sight and washed out, contributing — alongside Silk diffusion damping — to the fall-off of the CMB power spectrum at high multipole.

Is the last scattering surface a real place I could travel to?

Not in the way a wall is. It is a surface in our past light cone: for every direction we look, there is one point at comoving radius $\approx14\,\mathrm{Gpc}$ from which that direction's CMB photon was last scattered. An observer sitting there today would see nothing special — ordinary galaxies — and would have their own last scattering surface drawn around themselves at the same comoving radius. The surface is defined by our vantage point and the finite speed of light, not by anything physically distinguished about that location.

How does a larger baryon-to-photon ratio shift decoupling?

A larger $\eta$ means more baryons — and more electrons — per photon, with two effects. First, recombination occurs at higher temperature (higher redshift), because the Saha balance $B/k_BT\approx\ln(\eta^{-1})$ moves up as $\eta^{-1}$ shrinks. Second, at fixed ionization fraction the electron density $n_e$ is larger, so $\Gamma$ is higher and the plasma needs further recombination before $\Gamma$ drops below $H$ . Both push decoupling slightly earlier and reshape the visibility function — one reason the CMB acoustic peak positions are sensitive to the baryon density.