Early on, the universe was filled with hot plasma, and light could not travel far before scattering off a free electron. About 380,000 years after the Big Bang it had cooled to about 3,000 K; electrons and nuclei combined into neutral atoms, and the universe became transparent. That light has traveled freely ever since. Stretched by the expansion (see redshift and the scale factor), it arrives today as the cosmic microwave background, with the spectrum of the Planck function.
💡 Why it works
Every photon's wavelength stretches by the same factor, $1 + z$. A Planck spectrum at temperature $T$ stays a Planck spectrum when every wavelength is multiplied by the same factor: it becomes a Planck spectrum at $T/(1 + z)$, since only the combination $\lambda T$ appears in $x = hc/(\lambda kT)$. So the background is still a nearly perfect blackbody, just cooler: $$T(z) = T_0(1 + z), \qquad T_0 = 2.725\text{ K}.$$ Why 3,000 K and not the 158,000 K that matches hydrogen's 13.6 eV? There are over a billion photons for every atom, and even at 3,000 K the rare photons in the Wien tail of the spectrum outnumbered the atoms enough to keep hydrogen ionized.
🔑 Key idea
The CMB is a 2.725 K blackbody, peaking near 1.06 mm. It was released at $z \approx 1100$, when $T \approx 3000$ K. It fills all of space and arrives from every direction, the same to about one part in 100,000 once our own motion is accounted for.
The background radiation as the universe expands: 3,000 K at release, then 300 K, 30 K and 2.725 K today. The shape stays a blackbody; it slides to longer wavelengths and lower intensity.
Worked examples
✏️ Worked example: Where the CMB peaks today
Use Wien's law for $T = 2.725$ K.
$\lambda_{\max} = 2.898 \times 10^{-3}/2.725$ m.
Answer: $1.06 \times 10^{-3}$ m: about 1 mm, in the microwave band.
✏️ Worked example: The redshift of release
When was the background at 3,000 K?
$1 + z = 3000/2.725 = 1101$.
Answer: $z \approx 1100$. Its light then peaked near 970 nm, near-infrared, and the universe glowed like the surface of a red star.
✏️ Worked example: Our motion through the background
The background looks hotter ahead of us and cooler behind, by $\Delta T \approx T(v/c)$. Our speed relative to it is 370 km/s. Find $\Delta T$.
Answer: About 3.4 mK. Removing this dipole reveals the tiny variations, about $10^{-5}$, that seeded galaxies.
⚠️ Common mistake
Thinking the microwave background comes from one place in the sky, where the Big Bang happened. The Big Bang was not an explosion at a point in space; it happened everywhere, and the background was released everywhere at once. We see it coming from every direction because the light reaching us now left, from all around, at the time of release. Check: Arno Penzias and Robert Wilson found it in 1965 as a faint hiss their antenna picked up wherever they pointed it.
The COBE satellite (1990) measured its spectrum as a blackbody to better than one part in 10,000; the WMAP and Planck satellites mapped its tiny variations, which pin down the universe's age (13.8 billion years) and contents.
Practice
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