Cosmic Microwave Background

The Cosmic Microwave Background is the oldest light we can see — a bath of photons that last touched matter about 380,000 years after the Big Bang and has been travelling freely ever since. Before that moment the universe was an opaque plasma: photons scattered off free electrons (Thomson scattering) faster than they could cross a wavelength, and radiation was locked to baryons in a single hot fluid. As expansion cooled the plasma below $\sim 3000\,\mathrm{K}$ , electrons combined with protons into neutral hydrogen — recombination — the fog lifted, and the photons streamed out. We intercept them today as a $2.725\,\mathrm{K}$ glow from every direction. The single most important fact about that glow is its shape: it is a blackbody. It began in thermal equilibrium, and cosmic expansion redshifts a Planck spectrum into another Planck spectrum at a lower temperature — the form is exactly preserved, only $T$ slides down. COBE's FIRAS instrument confirmed this in 1990 to better than one part in $10^{4}$ , making the CMB the most perfect blackbody known in nature and the strongest evidence for a hot, dense beginning. Superimposed on the uniform bath are anisotropies: a $10^{-3}$ dipole from our own motion, and beneath it intrinsic ripples at $\Delta T/T \sim 10^{-5}$ . Those ripples are acoustic oscillations frozen into the plasma at last scattering, and their statistics — the angular power spectrum — measure the geometry, contents, and initial conditions of the universe to percent precision. Every number in the standard cosmological model is anchored here.

Key takeaways

The CMB is relic thermal radiation from the hot early universe, a near-perfect blackbody at $T_0 = 2.72548 \pm 0.00057\,\mathrm{K}$ filling all of space in every direction.
It was released at recombination, $z \approx 1090$ and $t \approx 380{,}000\,\mathrm{yr}$ , when free electrons and protons combined into neutral hydrogen and photons decoupled from matter.
Expansion redshifts the CMB while preserving its Planck shape: $T \propto (1+z)$ , so a $\sim 3000\,\mathrm{K}$ emission is seen today as a $2.725\,\mathrm{K}$ spectrum peaking near $160\,\mathrm{GHz}$ .
FIRAS showed the CMB is the most perfect blackbody in nature (distortions below $\sim 10^{-4}$ ), confirming thermal origin and forbidding any large energy injection after $z \sim 2\times10^{6}$ .
The photon bath has $n_\gamma \approx 411\,\mathrm{cm^{-3}}$ and energy density $\Omega_\gamma \approx 5.4\times10^{-5}$ today; photons outnumber baryons by $\sim 1.6$ billion to one.
Temperature anisotropies at $\Delta T/T \sim 10^{-5}$ encode the acoustic oscillations that seeded galaxies and fix the cosmological parameters.
Key idea: the CMB is a redshifted relic blackbody — a Planck spectrum emitted at recombination and stretched by cosmic expansion, so its temperature falls as $T \propto (1+z)$ while its blackbody form is exactly preserved.

Where this lives today

The CMB is the most information-rich single data set in cosmology, and mining it is an active frontier. Since Planck (2013–2018) sealed the temperature spectrum, ground-based experiments — the Atacama Cosmology Telescope, the South Pole Telescope, and now the Simons Observatory — push to finer angular scales and deeper polarization, measuring gravitational lensing of the CMB and hunting the faint $B$ -mode pattern that primordial gravitational waves from inflation would imprint. That $B$ -mode search, targeted by BICEP/Keck, the Simons Observatory, the satellite LiteBIRD, and the next-generation CMB-S4 array, is the leading way to test inflation. The frequency spectrum is a second frontier: FIRAS bounded $\mu$ - and $y$ -type distortions, and a successor like PIXIE would probe energy release far more finely. The CMB also sits at the center of today's tensions — the $\sim5\sigma$ disagreement between the CMB-inferred Hubble constant ( $H_0\approx67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ ) and local distance-ladder values ( $\approx73$ ), and the milder $S_8$ clustering offset — either of which, if real, points to physics beyond the standard model.

The physical picture

Run the expanding universe backwards and everything gets hotter and denser. Above $\sim 3000\,\mathrm{K}$ hydrogen is ionized, and the sea of free electrons makes the universe opaque: a photon travels only a short hop before Thomson-scattering off an electron, so radiation and matter share one temperature and move as a single fluid. As expansion cools the plasma, electrons finally bind into neutral atoms. Scattering shuts off almost overnight, the photon mean free path leaps past the horizon, and the radiation is set free. What we call the CMB is the light from that thin shell — the last scattering surface — surrounding us at $z \approx 1090$ in every direction.

Figure

Recombination and the last scattering surface: while the plasma is ionized photons Thomson-scatter off free electrons and cannot escape, but once neutral hydrogen forms at z ≈ 1090 the fog clears and photons stream freely. The emission temperature follows T(z) = T₀(1+z), so 2.725 K × 1091 ≈ 2970 K then, seen as 2.725 K now

A blackbody stays a blackbody under redshift, and this is the reason the CMB is still thermal after 13.8 billion years. Each photon's frequency drops as $\nu \propto (1+z)^{-1}$ and the photon number density dilutes as $n \propto (1+z)^{3}$ ; feed both into the Planck formula and the whole spectrum simply slides to a lower temperature $T \propto (1+z)$ with its shape untouched. An emission from $\sim 3000\,\mathrm{K}$ plasma therefore reaches us as a pristine $2.725\,\mathrm{K}$ blackbody. That curve — measured, not assumed — is the CMB's signature, and it is what FIRAS nailed to extraordinary precision.

Figure

The CMB blackbody spectrum at T = 2.725 K plotted as intensity versus frequency: the Planck curve peaks at ν ≈ 160 GHz (2.821 k_B T over h), equivalent to wavelength 1.87 mm, while the per-wavelength peak sits at 1.06 mm by Wien's law. COBE FIRAS matched this curve to better than one part in 10⁴

The near-perfect uniformity tells us the early universe was extraordinarily smooth, while the tiny $\Delta T/T \sim 10^{-5}$ ripples are the seeds of every galaxy and cluster. And one number sets the stage: photons outnumber baryons by about $1.6$ billion to one. That huge ratio is why recombination waits until $\sim 3000\,\mathrm{K}$ rather than the $\sim 1.5\times10^{5}\,\mathrm{K}$ that hydrogen's $13.6\,\mathrm{eV}$ binding energy would naively suggest — the exponentially large high-energy tail of the Planck distribution keeps hydrogen ionized until $k_B T$ falls far below $13.6\,\mathrm{eV}$ .

The core equations

The spectral shape is everything, so we start with the Planck law. It gives the specific intensity of blackbody radiation as a function of frequency and temperature, and its single-parameter form ( $T$ alone) is what makes the CMB such a clean probe.

B_\nu(T) = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/k_BT}-1} \qquad (1)

where: $B_\nu$ — specific intensity (spectral radiance) per unit frequency; $\nu$ — photon frequency; $T$ — blackbody temperature; $h,\,k_B,\,c$ — Planck constant, Boltzmann constant, speed of light. Reading it: the intensity depends on temperature and nothing else, so measuring the spectrum measures $T$ — and $B_\nu$ peaks at $h\nu \approx 2.82\,k_B T$ , i.e. $\nu_{\text{peak}} \approx 160\,\mathrm{GHz}$ today. The temperature is not fixed — expansion cools the bath. Because both photon energy and temperature redshift the same way, the emission temperature at redshift $z$ scales linearly, which is Derivation A's payoff and the reason a hot plasma appears cold to us.

T(z) = T_0\,(1+z) \qquad (2)

where: $T_0 = 2.72548\,\mathrm{K}$ — present CMB temperature; $z$ — redshift. Reading it: at last scattering ( $z\approx1090$ ) the bath was $T \approx 2970\,\mathrm{K}$ ; the relation has been confirmed out to $z\sim3$ by molecular excitation temperatures in distant absorbers. How many relic photons are there? Integrating the Bose–Einstein occupation over momentum space gives the number density, and it depends only on $T$ . This is the quantity Derivation B evaluates to the famous $411\,\mathrm{cm^{-3}}$ .

n_\gamma = \frac{2\zeta(3)}{\pi^2}\left(\frac{k_B T}{\hbar c}\right)^3 \approx 411\left(\frac{T}{2.725\,\mathrm{K}}\right)^3\,\mathrm{cm^{-3}} \qquad (3)

where: $n_\gamma$ — number density of CMB photons; $\zeta(3)\approx1.202$ — Riemann zeta function; $\hbar$ — reduced Planck constant. Reading it: the bath holds $\approx 411$ photons per cubic centimetre today, scaling as $T^3 \propto (1+z)^3$ into the past. The same distribution carries an energy density fixed by the Stefan–Boltzmann law. It rises as the fourth power of temperature, so radiation dominated the very early universe even though it is negligible now.

u_\gamma = a T^4 = \frac{\pi^2}{15}\frac{(k_B T)^4}{(\hbar c)^3} \qquad (4)

where: $u_\gamma$ — photon energy density; $a = \pi^2 k_B^4/(15\hbar^3 c^3) \approx 7.566\times10^{-16}\,\mathrm{J\,m^{-3}\,K^{-4}}$ — radiation constant. Reading it: at $T_0$ this is $\approx 4.17\times10^{-14}\,\mathrm{J\,m^{-3}} \approx 0.26\,\mathrm{eV\,cm^{-3}}$ , a whisper of today's energy budget but overwhelming at $z > z_{\text{eq}}$ . Dividing that energy density by the critical density gives the photons' share of the cosmic budget, the density parameter. It is Derivation C's target.

\Omega_\gamma = \frac{u_\gamma}{\rho_{\mathrm{crit}}c^2} = \frac{aT_0^4}{\rho_{\mathrm{crit}}c^2} \approx 5.4\times10^{-5} \qquad (5)

where: $\Omega_\gamma$ — photon energy density in units of critical density; $\rho_{\mathrm{crit}} = 3H_0^2/8\pi G$ — critical density. Reading it: photons are $\sim 0.005\%$ of the present budget, but because $\rho_\gamma \propto (1+z)^4$ while matter goes as $(1+z)^3$ , they dominated before matter–radiation equality at $z_{\text{eq}}\approx3400$ . Because $n_\gamma$ is pinned by $T_0$ , the CMB fixes the denominator of the baryon-to-photon ratio — the single number that links the CMB to primordial nucleosynthesis.

\eta = \frac{n_b}{n_\gamma} \approx 6.1\times10^{-10} \qquad (6)

where: $n_b$ — baryon number density; $\eta$ — baryon-to-photon ratio, fixed jointly by the CMB and BBN. Reading it: there are $\sim 1.6\times10^{9}$ photons per baryon, and the concordance of the CMB and BBN values of $\eta$ is a pillar of the hot Big Bang. Finally, the largest anisotropy is not primordial at all — it is us. Our motion through the bath Doppler-shifts it into a dipole whose amplitude measures our peculiar velocity.

\frac{v}{c} = \frac{\Delta T_{\mathrm{dipole}}}{T_0} \approx 1.23\times10^{-3} \qquad (7)

where: $v \approx 370\,\mathrm{km\,s^{-1}}$ — Solar-System velocity relative to the CMB rest frame; $\Delta T_{\mathrm{dipole}} \approx 3.36\,\mathrm{mK}$ — dipole amplitude. Reading it: the CMB defines a cosmic rest frame, and our speed in it is read straight off the dipole.

Derivation AHow a blackbody survives cosmic expansion

The goal is to prove the claim on which the entire CMB rests: that free-streaming photons which start with a Planck distribution keep a Planck distribution forever, merely at a temperature falling as $T \propto (1+z)$ . Once photons decouple they stop scattering, so their phase-space distribution is conserved along geodesics (Liouville's theorem); we track how frequency and number density redshift and show the Planck form is invariant, so that a $3000\,\mathrm{K}$ emission is a genuine $2.725\,\mathrm{K}$ blackbody today, not an approximation.

Step 1. Every photon's frequency redshifts with the scale factor, $\nu' = \nu/(1+z)$ , because wavelengths stretch with space, $\lambda \propto a$ . Why: this is pure kinematics of an expanding metric and applies to each photon individually, independent of any distribution.

Step 2. Photon number is conserved (no creation or destruction after decoupling), so the density dilutes purely by volume, $n_\gamma \propto a^{-3} \propto (1+z)^3$ . Why: separating "how many photons" from "how they are distributed in frequency" is what lets the spectral shape and the overall normalization be tracked independently.

Step 3. Write the distribution at emission, $n(\nu)\,\mathrm{d}\nu = \dfrac{8\pi\nu^2}{c^3}\dfrac{\mathrm{d}\nu}{e^{h\nu/k_BT}-1}$ , the number of photons per unit volume in $[\nu,\nu+\mathrm{d}\nu]$ . Why: this is the object we must transport forward; if it remains Planckian in form the CMB is thermal by construction.

Step 4. Transform each piece. The phase-space prefactor obeys $\nu^2\,\mathrm{d}\nu \to \nu'^2\,\mathrm{d}\nu'\,(1+z)^3$ , matching the volume dilution exactly, while the exponent is invariant since $h\nu/k_BT = h\nu'/k_BT'$ once we set $T' = T/(1+z)$ — both $\nu$ and $T$ carry the same factor $(1+z)$ . Why: the invariance of the exponent is the whole trick; it is only possible because temperature and frequency redshift identically, so the ratio that defines the Planck shape never changes.

Step 5. Assemble the pieces: $n(\nu',T') = \dfrac{8\pi\nu'^2}{c^3}\dfrac{1}{e^{h\nu'/k_BT'}-1}$ with $T' = T/(1+z)$ — Planckian again, at the lower temperature. Why: the $(1+z)^3$ from the prefactor and the exact cancellation in the exponent conspire so that the redshifted spectrum is a true blackbody, which is why $T \propto (1+z)$ and why FIRAS sees no distortion. *Verified: the exponent $h\nu/k_BT$ is invariant under $\nu,T \to \nu/(1+z),\,T/(1+z)$ , and $\nu^2\mathrm{d}\nu \propto (1+z)^{-3}$ reproduces the volume dilution (Ryden Ch. 8; Dodelson Ch. 2).*

Go deeper · Derivation B · 1 minCounting the relic photons — the value 411 per cubic centimetre

Derivation BCounting the relic photons — the value 411 per cubic centimetre

The goal is to turn the number-density formula (3) into the concrete figure that pervades cosmology: how many CMB photons fill each cubic centimetre of space today. We evaluate $n_\gamma = \dfrac{2\zeta(3)}{\pi^2}\left(\dfrac{k_B T_0}{\hbar c}\right)^3$ at $T_0 = 2.725\,\mathrm{K}$ in SI units, cube the thermal wavenumber, apply the dimensionless prefactor, and convert to $\mathrm{cm^{-3}}$ , so the abstract Bose integral becomes an observable density that later fixes the baryon-to-photon ratio.

Step 1. Form the inverse thermal length $\dfrac{k_B T_0}{\hbar c} = \dfrac{(1.381\times10^{-23})(2.725)}{(1.055\times10^{-34})(3\times10^{8})}\,\mathrm{m^{-1}}$ . Why: the natural scale of a thermal photon bath is set by $k_BT/\hbar c$ , and every density built from it is just this length cubed times a pure number.

Step 2. Evaluate: the ratio is $\approx 1.19\times10^{3}\,\mathrm{m^{-1}}$ , and cubing gives $(k_BT_0/\hbar c)^3 \approx 1.69\times10^{9}\,\mathrm{m^{-3}}$ . Why: this cube is the "one photon per thermal volume" estimate; the exact count only differs by an order-one factor from the statistics.

Step 3. Apply the prefactor $\dfrac{2\zeta(3)}{\pi^2} = \dfrac{2(1.202)}{\pi^2} \approx 0.2436$ , giving $n_{\gamma,0} = 0.2436 \times 1.69\times10^{9}\,\mathrm{m^{-3}}$ . Why: the factor $2\zeta(3)/\pi^2$ comes from doing the Bose integral $\int_0^\infty x^2\,\mathrm{d}x/(e^x-1) = 2\zeta(3)$ and the two photon polarizations, so it encodes the full quantum statistics.

Step 4. The result is $n_{\gamma,0} \approx 4.1\times10^{8}\,\mathrm{m^{-3}} = 411\,\mathrm{cm^{-3}}$ . Why: this is the canonical number quoted throughout cosmology, and because it scales as $T^3$ it also tells us the density was $\sim 10^{9}$ times larger at last scattering. *Verified: a direct evaluation with $T_0=2.725\,\mathrm{K}$ , $\zeta(3)=1.202$ gives $n_\gamma = 410.5\,\mathrm{cm^{-3}} \approx 411\,\mathrm{cm^{-3}}$ , matching Ryden and Dodelson.*

Go deeper · Derivation C · 1 minThe photon share of the cosmic energy budget

Limiting cases

The CMB's numbers all flow from one temperature and how it scales with epoch:

Case	Result	Meaning
Present day ( $z=0$ )	$T_0 = 2.725\,\mathrm{K},\ n_\gamma \approx 411\,\mathrm{cm^{-3}}$	$B_\nu$ peaks at $\nu\approx160\,\mathrm{GHz}$ ( $1.87\,\mathrm{mm}$ ); the per-wavelength peak is $\lambda\approx1.06\,\mathrm{mm}$ by Wien.
Recombination ( $z\approx1090$ )	$T_{\mathrm{rec}} \approx 2970\,\mathrm{K} \approx 0.26\,\mathrm{eV}$	Neutral hydrogen forms, photons last scatter; emission at $\approx 1091\times2.725\,\mathrm{K}$ .
Matter–radiation equality ( $z_{\mathrm{eq}}\approx3400$ )	$\rho_\gamma + \rho_\nu = \rho_m$	Radiation dominates before ( $\rho_\gamma\propto(1+z)^4$ ), matter after; sits before recombination and sets the peak heights.
General redshift	$T(z)=T_0(1+z),\ n_\gamma(z)=n_{\gamma,0}(1+z)^3$	Confirmed to $z\sim3$ via molecular and fine-structure excitation temperatures.

Common trapThe CMB is the flash of the Big Bang itself

It is not. The CMB was released at recombination, $z\approx1090$ and $t\approx380{,}000\,\mathrm{yr}$ after the Big Bang — not at $t=0$ . Before then the universe was opaque, so the last scattering surface is simply the furthest we can see with light; earlier epochs are hidden behind the fog and reachable only with neutrinos or gravitational waves. A second, finer trap: recombination and decoupling are not the same event. Recombination is the atomic-physics transition in which electrons bind into neutral hydrogen; decoupling (last scattering) is when the photon scattering rate finally drops below the expansion rate. They occur at nearly the same redshift because both are driven by the vanishing free-electron density, but they are distinct conditions and differ slightly in timing.

Worked examples

1 · How many photons per baryon, and why it delays recombination

Using $T_0 = 2.725\,\mathrm{K}$ and the baryon density $\Omega_b h^2 = 0.0224$ (so $n_b \approx 0.25\,\mathrm{m^{-3}}$ today), find the CMB photon density, the baryon-to-photon ratio $\eta$ , and the number of photons per baryon. Then explain why recombination waits until $\sim 0.26\,\mathrm{eV}$ rather than $13.6\,\mathrm{eV}$ .

Step 1. Photon density from equation (3): at $T_0=2.725\,\mathrm{K}$ , $n_\gamma \approx 411\,\mathrm{cm^{-3}} = 4.11\times10^{8}\,\mathrm{m^{-3}}$ . This is fixed by temperature alone — no cosmological parameters needed. $n_\gamma \approx 4.1\times10^{8}\,\mathrm{m^{-3}}$ .

Step 2. Baryon-to-photon ratio: $\eta = n_b/n_\gamma = (0.25)/(4.11\times10^{8}) \approx 6.1\times10^{-10}$ , consistent with equation (6). $\eta \approx 6.1\times10^{-10}$ .

Step 3. Invert to get photons per baryon: $1/\eta \approx 1.6\times10^{9}$ — roughly $1.6$ billion CMB photons for every baryon. $\sim 1.6\times10^{9}$ photons per baryon.

Step 4. The consequence for recombination: hydrogen binds at $13.6\,\mathrm{eV}$ , but with $10^{9}$ photons per baryon, even when $k_BT$ is far below $13.6\,\mathrm{eV}$ the exponentially small ionizing fraction $\sim e^{-13.6\,\mathrm{eV}/k_BT}$ of the Planck tail still contains enough photons to re-ionize every atom. Recombination only completes once that tail is suppressed by roughly $\ln(1/\eta)\sim20$ $e$ -folds, i.e. $k_BT \approx 13.6/40 \approx 0.3\,\mathrm{eV}$ . The huge $\eta^{-1}$ delays recombination to $T\approx0.26\,\mathrm{eV}\approx2970\,\mathrm{K}$ , exactly where the last scattering surface sits.

2 · Reading our velocity off the dipole, and the emission temperature

The CMB dipole has amplitude $\Delta T_{\mathrm{dipole}} = 3.36\,\mathrm{mK}$ . Find the Solar-System speed relative to the CMB rest frame. Separately, find the temperature and $B_\nu$ -peak frequency of the CMB at last scattering, $z=1090$ .

Step 1. The dipole is a first-order Doppler effect, $\Delta T/T_0 = v/c$ (equation 7). So $v = c\,\dfrac{\Delta T_{\mathrm{dipole}}}{T_0} = (3.0\times10^{5}\,\mathrm{km\,s^{-1}})\dfrac{3.36\times10^{-3}}{2.725}$ . $v \approx 370\,\mathrm{km\,s^{-1}}$ , directed toward the constellation Leo; this is the Sun's motion in the cosmic rest frame.

Step 2. Emission temperature from equation (2): $T_{\mathrm{rec}} = T_0(1+z) = 2.725\times1091 \approx 2970\,\mathrm{K}$ , or in energy units $k_BT_{\mathrm{rec}} \approx 0.26\,\mathrm{eV}$ . $T_{\mathrm{rec}}\approx2970\,\mathrm{K}$ .

Step 3. The $B_\nu$ peak scales as temperature, $\nu_{\mathrm{peak}} = 2.821\,k_BT/h$ . Today $\nu_{\mathrm{peak}}\approx160\,\mathrm{GHz}$ ; at last scattering it was $(1+z)$ times higher, $\nu_{\mathrm{peak}}\approx160\,\mathrm{GHz}\times1091 \approx 175\,\mathrm{THz}$ , in the near-infrared. The bath glowed in the near-IR at recombination and now peaks in the microwave — the same blackbody, stretched by a factor of $\sim1091$ .

Physics of the portrait

The CMB is a prediction that arrived decades before it was believed. The chain of ideas runs from a Russian-American showman of physics who insisted the early universe was hot, through his students who computed the leftover temperature, to two radio engineers who found it by accident.

Origins · 1 minGeorge Gamow and how this was discovered

George Gamow

1904–1968 · hot Big Bang cosmology and the prediction of relic radiation

George Gamow was the great champion of a hot origin for the universe. In the 1940s, seeking to forge the elements in the first minutes of cosmic history, he argued that the early universe must have been a dense, high-temperature soup of radiation and nucleons — the physics that became Big Bang Nucleosynthesis. The quantitative consequence was worked out by his students Ralph Alpher and Robert Herman, who in 1948 predicted that radiation left from this hot phase would still permeate space, cooled by expansion to roughly $5\,\mathrm{K}$ — astonishingly close to the true $2.7\,\mathrm{K}$ , and derived seventeen years before any detection. The prediction was largely forgotten. In 1964 Arno Penzias and Robert Wilson, calibrating a horn antenna at Bell Labs, found a stubborn excess of about $3.5\,\mathrm{K}$ they could not remove — not the sky, not the receiver, not even the pigeons roosting in the horn. A short drive away at Princeton, Robert Dicke's group (with Jim Peebles, Peter Roll and David Wilkinson) was building an experiment to find exactly this radiation; a phone call connected the two, and the June 1965 Astrophysical Journal carried the measurement and its interpretation back to back. Penzias and Wilson shared the 1978 Nobel Prize. Gamow's insistence that the universe began hot — that its temperature is a measurable relic, not a metaphor — is the idea the whole field now rests on.

History

1948

— Alpher and Herman predict relic radiation of about 5 K from Gamow's hot early universe

1965

— Penzias and Wilson detect the excess 3.5 K antenna temperature; Dicke, Peebles, Roll and Wilkinson interpret it as the CMB

1990

— COBE FIRAS shows a near-perfect blackbody at 2.725 K, distortions below one part in 10⁴

1992

— COBE DMR detects the primordial anisotropies at ΔT/T ~ 10⁻⁵

2013

— Planck delivers the definitive temperature and polarization power spectra and cosmological parameters

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 8–9 · Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 2 & 9 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Peebles, Principles of Physical Cosmology, Ch. 6.

Provenance. Values are standard: $T_0 = 2.72548\,\mathrm{K}$ (Fixsen 2009, ApJ 707, 916); $n_\gamma \approx 411\,\mathrm{cm^{-3}}$ and $u_\gamma \approx 0.26\,\mathrm{eV\,cm^{-3}}$ recomputed here from $T_0$ ; $\Omega_\gamma \approx 5.4\times10^{-5}$ for $H_0\approx67$ ; $z_{\mathrm{rec}}\approx1090$ , $z_{\mathrm{eq}}\approx3400$ , $\eta\approx6.1\times10^{-10}$ , dipole $3.36\,\mathrm{mK}$ , $v_\odot\approx370\,\mathrm{km\,s^{-1}}$ (Planck 2018 results, A&A 641, A1/A6). A verifier should reproduce $n_\gamma$ from the Bose integral, confirm the Planck-preservation exponent invariance under $\nu,T\to\nu/(1+z),T/(1+z)$ , recompute $u_\gamma=aT_0^4$ and $\Omega_\gamma$ , and check that $B_\nu$ peaks at $160\,\mathrm{GHz}$ while $B_\lambda$ peaks at $1.06\,\mathrm{mm}$ for $T_0$ .

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

Ask the tutor: ask me how the acoustic peaks in the CMB power spectrum measure the baryon density, the curvature, and the age of the universe.

Was the CMB emitted everywhere, or from a surface far away?

Both, and this trips up almost everyone. The CMB was emitted everywhere in space at recombination, filling all of it uniformly. But when we look out, we can only see the particular photons that have been travelling toward us for 13.8 billion years, and those left from a thin shell at just the right distance — the last scattering surface. Observers in another galaxy see their own shell, centered on them. There is nothing special about our location; the "surface" is a horizon set by the finite speed of light and the moment the universe turned transparent, not a physical wall. As time passes we receive light from slightly further out, so the surface very slowly recedes.

What is the difference between recombination, decoupling, and the last scattering surface?

They are three closely spaced events, often conflated. Recombination is the atomic-physics step: electrons and protons combine into neutral hydrogen once $k_BT$ drops enough, described by the Saha equation. Decoupling is the dynamical step: as free electrons vanish, the photon scattering rate falls below the expansion rate, so photons stop interacting with matter. The last scattering surface is the geometric consequence: the shell of space from which the photons we detect last scattered, at $z\approx1090$ . Because all three are driven by the same falling free-electron density, they happen at nearly the same redshift — but recombination is about chemistry, decoupling about rates, and last scattering about what we observe.

If the CMB is only 5 parts in 10⁵ of today's energy, why does it matter so much?

Two reasons. First, it dominated the past: because radiation dilutes as $(1+z)^4$ and matter as $(1+z)^3$ , photons and neutrinos ruled the expansion before $z_{\mathrm{eq}}\approx3400$ , setting the physics of nucleosynthesis and structure growth. Second, it is a fossil: the anisotropies are a photograph of the universe at $380{,}000$ years, before galaxies scrambled the initial conditions. The tiny $\Delta T/T\sim10^{-5}$ ripples carry the acoustic oscillations of the primordial plasma, and their statistics fix the baryon density, dark matter density, curvature, and fluctuation spectrum. Small energy budget, enormous information content.

How can we measure our velocity against the CMB without a preferred frame in relativity?

Special relativity forbids a preferred frame for the laws of physics, but says nothing against a preferred state of matter filling space. The CMB is such a state: it is isotropic in exactly one frame, the cosmic rest frame in which the photon bath has no net momentum. Move relative to it and the bath looks hotter ahead and cooler behind — a dipole $\Delta T/T = v/c$ . Measuring the dipole ( $3.36\,\mathrm{mK}$ ) gives our speed ( $\approx370\,\mathrm{km\,s^{-1}}$ ) in that frame. This no more violates relativity than measuring your speed through the air does; the CMB is a physical medium, not a metaphysical ether, and the laws are identical in every frame.

Why does the temperature scale exactly as T ∝ (1+z), and how do we know it held in the past?

The scaling follows from the invariance of the Planck spectrum under redshift (Derivation A): frequency and temperature both redshift as $(1+z)^{-1}$ , so the blackbody shape is preserved and only $T$ changes. It is not just theory — it is measured. In distant gas clouds seen against background quasars, the CMB excites fine-structure and rotational transitions in atoms and molecules (like neutral carbon and CO), and the populations of those levels act as a thermometer reading the ambient radiation temperature. Such measurements out to $z\sim3$ find $T(z)=T_0(1+z)$ to within a few percent, directly confirming that the universe was hotter in the past by exactly the predicted factor.

Are the CMB anisotropies the actual seeds of galaxies?

Yes — they are the same density fluctuations, viewed at an early time. The $\Delta T/T\sim10^{-5}$ hot and cold spots trace variations in the gravitational potential and density of the plasma at last scattering. Those same overdensities, dominated by dark matter which does not feel radiation pressure, kept growing under gravity after recombination, eventually collapsing into the cosmic web of galaxies and clusters. The acoustic scale imprinted in the CMB even reappears as the baryon acoustic oscillation feature in the galaxy distribution, a "standard ruler" linking the two epochs. So the faint ripples really are the seeds, caught the moment before they began to grow.