Cosmic Microwave Background

Relic radiation used to measure cosmological temperature, spectrum, and anisotropies.

Conceptual context

The Cosmic Microwave Background is relic light from the epoch when matter stopped scattering photons continuously. It is not a visible photograph of a distant wall, but a thermal map of the whole sky. Each direction brings radiation with a characteristic $\lambda$ and a spectral curve of $B_{\lambda}$ that matches a blackbody with extraordinary precision. Reading that curve lets us infer a temperature without touching the early Universe. The key question is how a spectrum observed today, cooled and stretched by expansion, still carries the thermal fingerprint of a much hotter cosmic plasma.

🟢 Essential level

The Cosmic Microwave Background acts as a cosmic ember: its brightness is not concentrated at one $\lambda$, but spread across a full blackbody curve.

By locating the $\lambda_{max}$ and examining how $B_{\lambda}$ falls on both sides, we infer the $T$ that best fits the curve. A hotter background shifts the maximum toward shorter $\lambda$ and raises the total brightness; a colder one shifts it toward longer microwaves. Because cosmic expansion stretches every photon wavelength, the radiation cools over time. Therefore the measured $T_0$ can be used with $z$ to reconstruct the earlier $T_z$ of the photon bath. The curve acts as a remote thermometer, and its full shape prevents a single noisy point from masquerading as a temperature measurement of the whole sky.

The FIRAS instrument aboard the COBE satellite measured the CMB spectrum in the 1990s with extraordinary precision: the curve matches a perfect blackbody to within one part in 10,000, with no detectable deviations. That result, together with the accidental discovery of the radiation by Penzias and Wilson in 1965, cemented the cosmic background as the strongest observational evidence for the hot Big Bang.

🔵 Formal level

Basic formulas

Formally, the spectrum is modeled as blackbody radiation. The central quantity is $B_{\lambda}$, controlled by $\lambda$ and $T$. The shape of the curve is described by:

Planck law in wavelength

This distribution combines $h$, $c$ and $k_B$ into a single blackbody curve.

This relation is more than a point calculation: it fixes the whole spectral silhouette. For a given temperature, there is a $\lambda_{max}$ where $B_{\lambda}$ reaches its maximum. That peak position gives a direct thermal estimate through:

Wien displacement law

The $b_W$ dimensionally anchors the temperature-peak product.

Real student questions

Does the CMB come from a distant wall? No. It arrives from every direction because the observable Universe was once filled with plasma and photons.

Why use a blackbody? Because the radiation was in thermal equilibrium before decoupling, and its $B_{\lambda}$ retains that shape with very high precision.

Is $T$ the temperature of empty space? Not exactly. It describes the relic photon bath, not an ordinary material temperature.

Do anisotropies contradict uniformity? No. Uniformity is approximate; small $\Delta_{T}$ are the signal that lets us study primordial structure.

Final synthesis

The Cosmic Microwave Background reveals the thermal history of the Universe because its $B_{\lambda}$ preserves a blackbody form. $\lambda_{max}$, $T_0$, $z$, and anisotropies turn a microwave curve into a fossil archive of the early cosmos.