13.3 Cosmic Microwave Background Radiation

Cosmic Microwave Background (CMB)

The cosmic microwave background is the oldest light in the observable universe, a thermal relic from when the cosmos was roughly 380,000 years old. Studying the CMB gives us direct observational constraints on the Big Bang model, the geometry of the universe, and the initial conditions that seeded all cosmic structure. This section covers the CMB's discovery, its temperature anisotropies, the angular power spectrum, and polarization.

Discovery and Characteristics of CMB

The CMB dates to the epoch of recombination, when the universe cooled enough for electrons and protons to form neutral hydrogen. Before this, photons were tightly coupled to the baryon-electron plasma via Thomson scattering. Once neutral atoms formed, photons decoupled and free-streamed through the universe. That radiation is what we detect today as the CMB.

Arno Penzias and Robert Wilson discovered it accidentally in 1964 while calibrating a radio antenna at Bell Labs. They found a persistent, isotropic excess noise at microwave frequencies that couldn't be attributed to any terrestrial or instrumental source. This turned out to be the thermal radiation predicted by Gamow, Alpher, and Herman in the late 1940s.

Key properties:

• The CMB is an almost perfect blackbody spectrum, following Planck's law: the spectral radiance of an ideal thermal emitter at equilibrium. COBE's FIRAS instrument confirmed this to extraordinary precision, with deviations smaller than 50 parts per million.
• Its present-day temperature is $T_0 \approx 2.725 \text{ K}$.
• The spectrum peaks at roughly 160 GHz (about 1.9 mm wavelength), placing it squarely in the microwave band. You can find the peak frequency using Wien's displacement law: $\nu_{\text{max}} \approx 5.88 \times 10^{10} \, T$ Hz/K.
• The radiation is highly isotropic, appearing nearly uniform in every direction, which is strong evidence for a hot, dense, homogeneous early universe.

Temperature Anisotropies and Their Significance

Despite its near-uniformity, the CMB contains tiny temperature fluctuations across the sky. These anisotropies are the imprint of density perturbations in the primordial plasma at the time of last scattering.

• Fluctuations are on the order of $\Delta T / T \sim 10^{-5}$, corresponding to roughly 30 μK variations around the mean temperature.
• The largest anisotropy is the dipole ($l = 1$), with an amplitude of about 3.4 mK. This is kinematic in origin: it reflects the motion of the Solar System relative to the CMB rest frame at roughly 370 km/s. The dipole is subtracted before analyzing the cosmological signal.
• The remaining fluctuations at $l \geq 2$ are cosmological. They trace back to quantum fluctuations generated during inflation, which were stretched to macroscopic scales and then amplified by gravitational instability.
• These density perturbations are the seeds of all large-scale structure. Regions of slight overdensity in the CMB correspond to locations where matter would later collapse to form galaxies, galaxy clusters, and the cosmic web.

Satellite Missions and CMB Observations

Three flagship satellite missions have progressively sharpened our view of the CMB:

• COBE (launched 1989): The FIRAS instrument confirmed the blackbody spectrum with unprecedented precision. The DMR instrument made the first detection of temperature anisotropies at large angular scales ($l \lesssim 20$), earning Mather and Smoot the 2006 Nobel Prize.
• WMAP (2001–2010): Mapped the full sky in five frequency bands with angular resolution of about 0.2°. WMAP pinned down key cosmological parameters (age of the universe, baryon density, dark matter density) to percent-level precision and provided strong evidence for a flat geometry and a nearly scale-invariant primordial power spectrum.
• Planck (2009–2013): Achieved the highest resolution (~5 arcminutes) and sensitivity to date across nine frequency bands. Planck's measurements refined the cosmological parameters further and tightened constraints on inflation, neutrino masses, and deviations from the standard $\Lambda$CDM model.

Each successive mission improved angular resolution and frequency coverage, which is critical for separating the cosmological signal from astrophysical foregrounds (dust, synchrotron, free-free emission).

CMB Power Spectrum

Understanding the Angular Power Spectrum

The angular power spectrum is the primary statistical tool for extracting physics from the CMB. It quantifies how much temperature variance exists at each angular scale on the sky.

The procedure works like this:

1. Expand the CMB temperature field $\Delta T(\hat{n})/T$ on the sphere in terms of spherical harmonics $Y_{lm}$: $\frac{\Delta T(\hat{n})}{T} = \sum_{l,m} a_{lm} Y_{lm}(\hat{n})$
2. The coefficients $a_{lm}$ encode the amplitude of fluctuations at multipole $l$ and azimuthal mode $m$.
3. The angular power spectrum is defined as the variance of these coefficients, averaged over $m$: $C_l = \frac{1}{2l+1} \sum_{m=-l}^{l} |a_{lm}|^2$
4. By convention, the quantity typically plotted is $\mathcal{D}_l = l(l+1)C_l / 2\pi$, which gives a roughly flat spectrum on large scales and makes the acoustic peaks visually prominent.

The multipole moment $l$ is inversely related to angular scale: $\theta \sim 180°/l$. So $l = 2$ corresponds to the largest cosmological scales (quadrupole), while $l \sim 1000$ probes sub-degree scales.

Acoustic Peaks and Their Implications

The most striking feature of the CMB power spectrum is a series of acoustic peaks, which arise from sound waves (baryon-acoustic oscillations) in the photon-baryon plasma before recombination.

Here's the physical picture: before decoupling, overdense regions gravitationally compress the plasma, raising the temperature. Radiation pressure resists compression and drives the plasma back out. This sets up standing acoustic oscillations in the plasma. At the moment of recombination, these oscillations freeze in, and the pattern of compressions and rarefactions is imprinted on the CMB.

What each peak tells you:

• First peak ($l \approx 220$): Corresponds to the mode that had time to compress exactly once before recombination. Its angular position is sensitive to the spatial curvature of the universe. The observed position at $l \approx 220$ is consistent with a spatially flat universe ($\Omega_k \approx 0$).
• Odd-numbered peaks (1st, 3rd, 5th...): These represent modes caught at maximum compression. They are enhanced by increasing the baryon density $\Omega_b h^2$, because baryons add inertial mass to the oscillation, deepening the compression phase.
• Even-numbered peaks (2nd, 4th...): These represent modes caught at maximum rarefaction. The ratio of odd to even peak heights directly constrains the baryon-to-photon ratio.
• Damping tail ($l \gtrsim 1000$): At small angular scales, photon diffusion during recombination (Silk damping) exponentially suppresses the anisotropies. The damping scale depends on the baryon density and the expansion rate.

Current measurements resolve 5–7 distinct peaks, and fitting the full peak structure simultaneously constrains the Hubble constant $H_0$, the matter density $\Omega_m$, the baryon density $\Omega_b$, and the spectral index $n_s$ of primordial perturbations.

The Sachs-Wolfe Effect and Large-Scale Structure

On large angular scales ($l \lesssim 30$), the dominant contribution to the CMB anisotropy comes from the Sachs-Wolfe effect, which links temperature fluctuations to gravitational potentials.

There are two components:

• Ordinary (SW) Sachs-Wolfe effect: Photons climbing out of gravitational potential wells at the surface of last scattering lose energy and appear cooler. Conversely, photons from underdense regions appear slightly warmer. For adiabatic perturbations on large scales, the relation is $\Delta T / T = \Phi / 3$, where $\Phi$ is the gravitational potential. This produces the roughly flat plateau in $\mathcal{D}_l$ at low $l$.
• Integrated Sachs-Wolfe (ISW) effect: As CMB photons travel from the last scattering surface to us, they pass through time-varying gravitational potentials. In a matter-dominated universe with $\Omega_m = 1$, potentials are constant and the ISW effect vanishes. But during radiation domination (early ISW) and during dark energy domination (late ISW), potentials decay, giving photons a net energy boost or loss. The late ISW effect provides independent evidence for dark energy.

The Sachs-Wolfe contributions are particularly useful for constraining inflationary models, since they probe the largest scales where the primordial power spectrum is least affected by causal physics.

CMB Polarization

Types and Origins of CMB Polarization

The CMB is weakly polarized because Thomson scattering produces linear polarization when the incident radiation field has a local quadrupole anisotropy. At the epoch of recombination, the photon distribution develops a quadrupole from the velocity gradients in the acoustic oscillations, generating a net polarization signal.

The polarization field on the sky is decomposed into two geometrically distinct patterns:

• E-modes (gradient, or curl-free pattern): Generated by scalar (density) perturbations. E-modes have been well measured and are correlated with the temperature anisotropies in a predictable way. Their amplitude is roughly 10% of the temperature signal, peaking at a few μK.
• B-modes (curl, or divergence-free pattern): Two sources produce B-modes. Primordial gravitational waves (tensor perturbations from inflation) generate B-modes directly, with a signal that peaks at large angular scales ($l \lesssim 100$). Gravitational lensing of E-modes by intervening large-scale structure converts some E-mode power into B-modes, producing a signal that peaks at smaller scales ($l \sim 1000$). The lensing B-mode has been detected; the primordial signal has not yet been confirmed.

The ratio of tensor to scalar perturbation amplitudes is parameterized by $r$, the tensor-to-scalar ratio. Current upper bounds from Planck and BICEP/Keck place $r < 0.036$ (95% confidence), which already rules out several classes of inflationary models.

Observational Techniques and Challenges

Measuring CMB polarization is significantly harder than measuring temperature, because the signals are so faint.

• E-mode polarization is about a factor of 10 weaker than the temperature anisotropies. Primordial B-modes, if they exist at detectable levels, could be another factor of 10–100 weaker still.
• Experiments use arrays of highly sensitive transition-edge sensor (TES) bolometers or kinetic inductance detectors cooled to sub-Kelvin temperatures.
• Ground-based experiments at the South Pole and the Atacama Desert (BICEP/Keck Array, POLARBEAR/Simons Array, SPT, ACT) target specific angular scales and frequency bands. Balloon-borne instruments like SPIDER observe from above most of the atmosphere to reduce atmospheric noise.
• Foreground contamination is the biggest systematic challenge. Galactic thermal dust emission (dominant at high frequencies) and synchrotron radiation (dominant at low frequencies) both produce polarized signals that can mimic or obscure the primordial B-mode signal. Multi-frequency observations are essential for component separation.
• The 2014 BICEP2 announcement of a B-mode detection at $r \approx 0.2$ was later shown to be largely attributable to polarized dust foregrounds, underscoring how critical foreground modeling is.

Implications for Cosmology and Fundamental Physics

CMB polarization measurements carry information that temperature data alone cannot provide.

• E-mode measurements have independently confirmed the $\Lambda$CDM model and tightened constraints on the optical depth to reionization $\tau$, which tells us when the first stars and galaxies reionized the intergalactic medium.
• A confirmed detection of primordial B-modes would be direct evidence for a gravitational wave background generated during inflation. This would pin down the energy scale of inflation: $V^{1/4} \sim 10^{16} \text{ GeV} \times (r/0.01)^{1/4}$, connecting CMB observations to physics near the grand unification scale.
• Different inflationary models predict different values of $r$ and different relationships between $r$ and the spectral tilt $n_s$. Precise polarization measurements can discriminate between large-field models (which generically predict larger $r$) and small-field models.
• Combining polarization with temperature data breaks parameter degeneracies, improving constraints on neutrino masses, the number of relativistic species $N_{\text{eff}}$, and possible deviations from general relativity on cosmological scales.
• Next-generation experiments (CMB-S4, LiteBIRD) aim to reach sensitivities of $\sigma(r) \sim 10^{-3}$, which would either detect primordial gravitational waves or rule out all large-field inflation models.