The Surface of Last Scattering

A wall of fog at the edge of time

Stand at night and look in any direction. Past the stars, past the galaxies, past the most distant quasars, there is a point where the universe simply stops being transparent. Not because there is nothing further out, but because everything further out is buried inside a glowing fog. That fog is the surface of last scattering — the moment, roughly 380,000 years after the Big Bang, when the universe cooled enough for light to escape the soup of matter it had been trapped in since the beginning.

The analogy that works best is the Sun. You cannot see into the Sun's interior because its plasma is opaque; what you see is the photosphere, the thin shell where photons last scattered before flying to your eye. The early universe was one enormous photosphere. Everywhere was hot, ionised, and opaque. Then, almost simultaneously across all of space, it switched off — electrons paired with protons, the fog lifted, and the light that had been bouncing around for hundreds of thousands of years streamed out in straight lines. We are still bathed in that light. The surface of last scattering is, quite literally, the oldest thing it is physically possible to see with a telescope.

Crucially, it is not a place. It is a time that looks like a place because light takes time to travel. The whole universe recombined at once, but only the light from a shell exactly one universe-age away has had time to reach us right now. That shell is centred on us — and on every other observer too, each of whom sees their own surface of last scattering. We are inside the sphere, not outside it, looking back at the surface of our own past.

The physics: from plasma to transparency

For its first few hundred thousand years the universe was a tightly coupled plasma of photons, free electrons, and bare nuclei (mostly protons, with about 24% helium by mass laid down during big bang nucleosynthesis). Photons could not travel freely because free electrons scatter light extremely efficiently through Thomson scattering, with a cross-section

σ_T = (8π/3)(e²/m_e c²)² ≈ 6.652 × 10⁻²⁵ cm²

The photon mean free path is set by the free-electron density n_e and that cross-section, λ = 1/(n_e σ_T). While electrons were free, λ was tiny compared to the size of the cosmos, so a photon took an enormous number of random-walk steps and matter and radiation behaved as a single fluid.

As the universe expanded, its temperature fell as T ∝ (1 + z). When T dropped to about 3000 K, electrons and protons combined into neutral hydrogen — the epoch of recombination. The equilibrium ionisation fraction is governed by the Saha equation,

      n_e n_p     (2π m_e k_B T)^(3/2)      −13.6 eV
      ───────  =  ─────────────────── exp ( ─────────── )
       n_H               h³                    k_B T

Because neutral atoms barely scatter the cooling microwave photons, the free-electron density n_e collapsed by orders of magnitude over a narrow span of redshift. The photon mean free path shot up until it exceeded the Hubble radius itself: light decoupled from matter. The probability that a given photon has scattered since some redshift is captured by the optical depth τ(z); the surface of last scattering is the locus where the visibility function g(z) = e^(−τ) dτ/dz peaks — the redshift at which a photon arriving today is most likely to have had its final scatter.

The key numbers

The Planck satellite's 2018 cosmological analysis pins these quantities tightly. The values below are the modern, measured benchmarks.

Quantity	Value	Note
Redshift of last scattering, z*	1089.80 ± 0.21	Visibility-function peak (Planck 2018)
Cosmic age at z*	≈ 372,000 yr	Often quoted as "380,000 years"
Plasma temperature then	≈ 2970 K	= T₀(1 + z*) = 2.725 × 1090.8
CMB temperature today, T₀	2.7255 ± 0.0006 K	FIRAS / COBE blackbody fit
Redshift width of surface, Δz	≈ 80 (FWHM)	Finite duration of recombination
Comoving radius to surface	≈ 13.9 Gpc ≈ 45.5 Gly	Comoving distance to z*
Comoving thickness of shell	≈ 19 Mpc	From Δz ≈ 80
Photon-to-baryon ratio, η⁻¹	≈ 1.6 × 10⁹	Why recombination is delayed
RMS temperature fluctuation	≈ 18 µK (ΔT/T ~ 10⁻⁵)	The acoustic ripples

The single most counter-intuitive entry is the comoving radius. The light has been travelling for 13.8 billion years, so a naive answer would be "13.8 billion light-years away." But the space that emitted it has been carried outward by cosmic expansion the whole time. Account for that and the matter that released our CMB is now about 45.5 billion light-years away in comoving distance — the practical edge of the observable universe.

How we observe and measure it

We do not infer the surface of last scattering indirectly; we photograph it. Every CMB experiment is, in effect, a camera pointed at this shell.

• Penzias & Wilson, 1965. A 4080 MHz horn antenna at Bell Labs registered an excess antenna temperature of about 3.5 K that would not go away no matter where they pointed. It was the surface of last scattering, redshifted into microwaves.
• COBE (1989–1993). The FIRAS instrument showed the spectrum is a near-perfect blackbody at 2.725 K — the best blackbody ever measured in nature — and the DMR instrument found the first temperature anisotropies at the 10⁻⁵ level, earning Mather and Smoot the 2006 Nobel Prize.
• WMAP (2001–2010). Mapped the anisotropies at sub-degree resolution, located the first acoustic peak at ℓ ≈ 220, and established the now-standard 13.7-billion-year, flat ΛCDM cosmology.
• Planck (2009–2013). The European Space Agency's mission measured the temperature power spectrum to multipoles ℓ ≈ 2500 and the polarisation spectrum, delivering the precision parameters quoted above. COBE observed from low-Earth orbit above the atmosphere, while WMAP and Planck operated from the much quieter Sun–Earth L2 point.

Measuring the angle subtended by features on the surface against their known physical size turns the CMB into a standard ruler. The acoustic peaks are the signature of sound waves in the photon-baryon fluid; their angular scale fixes the geometry of the universe. The fact that the first peak sits exactly at ℓ ≈ 220 is the headline evidence that space is spatially flat to within about half a percent.

Worked example: where is the surface today?

Let us turn z* into a present-day distance. The temperature scaling gives the plasma temperature directly:

T(z*) = T₀ (1 + z*) = 2.725 K × 1090.8 ≈ 2972 K

To get the distance we integrate the comoving line element in a flat ΛCDM universe:

D_C = (c/H₀) ∫₀^z*  dz / E(z),    E(z) = √(Ω_m(1+z)³ + Ω_Λ)

With the Planck values H₀ = 67.4 km/s/Mpc (so the Hubble distance c/H₀ ≈ 4450 Mpc), Ω_m = 0.315, Ω_Λ = 0.685, the integral evaluates to about 3.13. Therefore

D_C ≈ 4450 Mpc × 3.13 ≈ 13,900 Mpc ≈ 13.9 Gpc
    ≈ 45.5 billion light-years

Now the angular-diameter distance, which tells us the apparent size of features on the surface, is D_A = D_C/(1 + z*) ≈ 13,900/1090.8 ≈ 12.7 Mpc. This is why the sound horizon — comoving size r_s ≈ 145 Mpc (a physical ~133 kpc at recombination) — subtends an angle θ ≈ r_s/D_C ≈ 145/13,900 ≈ 0.0104 rad ≈ 0.6°. That sets the acoustic scale ℓ_A ≈ π/θ ≈ 300; the first acoustic peak is observed slightly below it, at ℓ ≈ 220, because of the way the oscillations project and are driven. Every step here is built from the single measured number z* = 1090 plus the cosmological parameters.

Discovery and the people who decoded it

The theoretical groundwork came first. In 1948 Ralph Alpher and Robert Herman, working from George Gamow's hot Big Bang, predicted that the universe should be filled with relic radiation now cooled to "about 5 K." The prediction was largely forgotten. In 1965 Arno Penzias and Robert Wilson at Bell Labs stumbled on the signal while chasing antenna noise; Robert Dicke, Jim Peebles, Peter Roll, and David Wilkinson at Princeton, who had been building an experiment to look for exactly this, supplied the interpretation in a companion paper. Penzias and Wilson received the 1978 Nobel Prize.

The detailed physics of recombination and the resulting anisotropies was worked out by Yakov Zeldovich, Rashid Sunyaev, and Peebles in the late 1960s and 1970s; the predicted acoustic peaks are sometimes called Sakharov oscillations after Andrei Sakharov's 1965 anticipation of them. Jim Peebles received the 2019 Nobel Prize for this body of theoretical cosmology. The era of precision came with the three great satellites: COBE (Mather and Smoot, Nobel 2006) proved the blackbody and found the ripples; WMAP turned the ripples into parameters; and Planck (2018) reduced the headline uncertainties to the sub-percent level that anchors the standard model of cosmology today.

Related surfaces and phenomena

• The cosmic microwave background. The radiation emitted by the surface, redshifted into the microwave band and observed across the whole sky. The surface is the source; the CMB is what arrives.
• Recombination vs. decoupling. Two distinct events that nearly coincide. Recombination is when the ionisation fraction crosses ~50% (z ≈ 1300–1400 in the Saha treatment, lower with the full Peebles network). Decoupling/last scattering is when the optical depth to today reaches unity (z* ≈ 1090). The matter–radiation thermal coupling (the "drag epoch") ends slightly later still, at z_drag ≈ 1060, which is the redshift relevant to baryon acoustic oscillations.
• The cosmic neutrino background. A still-earlier last-scattering-like surface, where neutrinos decoupled about one second after the Big Bang at T ≈ 10¹⁰ K. It pervades space at ~1.95 K today but is fiendishly hard to detect directly.
• The reionisation surface. Hundreds of millions of years later the first stars and quasars re-ionised the neutral hydrogen, giving photons a second, much weaker chance to scatter. This adds a small optical depth (τ ≈ 0.054) and imprints a large-angle polarisation bump that Planck used to date reionisation to z ≈ 7–8.
• The gravitational-wave "last scattering." Primordial gravitational waves never scattered at all; their effective horizon reaches back to inflation. Detecting their B-mode imprint on the CMB polarisation would let us see physically "through" the photon surface to ~10⁻³² s.

The ripples: sound frozen on the surface

The surface is not perfectly uniform. Its temperature varies by about one part in 100,000 — ±18 microkelvin — and those variations are not random noise. Before decoupling, the photon-baryon fluid sloshed: gravity pulled matter into dark-matter potential wells, radiation pressure pushed back, and the competition set up standing sound waves. The speed of those waves was relativistic, about c/√3 ≈ 0.58c. At the instant of last scattering the oscillations froze, leaving a snapshot of compressions (hot spots) and rarefactions (cold spots) at characteristic wavelengths.

Decompose that snapshot into spherical harmonics and you get the famous CMB angular power spectrum. The first acoustic peak at ℓ ≈ 220 corresponds to the mode that had just reached maximum compression; subsequent peaks at ℓ ≈ 540 and ℓ ≈ 810 are higher harmonics. The relative heights of odd and even peaks weigh the baryons against the dark matter; the peak positions fix the geometry; the damping tail at high ℓ measures the diffusion of photons out of the fluctuations during the finite thickness of the surface. From this one curve we read that the universe is flat, 13.8 billion years old, and made of roughly 5% baryons, 27% dark matter, and 68% dark energy.

Common misconceptions and subtleties

• "It is the edge of the universe." No. It is the edge of the observable universe for light, and only for us. Space and matter extend far beyond it; the plasma that emitted our CMB is now ordinary galaxies, just too far away for their light to have reached us yet.
• "Recombination happens at 13.6 eV." It does not. The huge photon-to-baryon ratio (~1.6 billion to one) means the Wien tail of the radiation keeps hydrogen ionised until the temperature falls to ~0.26 eV (3000 K), almost 50 times lower than the naive ionisation energy.
• "The CMB is the Big Bang itself." It is not the bang; it is a baby photo taken 380,000 years afterward. The actual hot, dense beginning is hidden behind the opaque plasma and cannot be seen in light at all.
• "The surface is infinitely thin." It has a real comoving thickness of about 19 Mpc, because recombination took finite time (Δz ≈ 80). This blurring, combined with photon diffusion, produces Silk damping that erases small-scale structure.
• "Different observers see the same sphere." Each observer sits at the centre of their own last-scattering sphere. The CMB you measure and the CMB measured in the Andromeda galaxy come from overlapping but not identical shells.
• "The CMB photons are still being made." They were all released in one brief episode and have simply been redshifting ever since — from visible/near-infrared at 3000 K down to 2.725 K microwaves today, a stretch by a factor of ~1090.