CMB Physics
What is the cosmic microwave background (CMB)?
The cosmic microwave background (CMB) is a faint glow of microwave radiation that bathes the entire universe. It is the "fossil light" of the Big Bang: when the universe was very young and very hot, it glowed intensely. As it cooled, this light was "stretched" by the expansion of space and became invisible to the naked eye, but detectable by radio telescopes. Think of it as the residual glow of an oven that has just been turned off, except this glow fills the entire universe.
The cosmic microwave background (CMB) is the oldest light in the universe, a relic radiation field released when the cosmos was just 380,000 years old. It is the most precisely measured cosmological observable, and its study has transformed cosmology from a data-poor, speculative endeavor into a precision science. Every direction we look, we see this ancient light, carrying information about the composition, geometry, and initial conditions of our universe.
A Cosmic Snapshot
The CMB is a nearly perfect blackbody at $T = 2.7255 \pm 0.0006$ K, with tiny temperature fluctuations of order $\Delta T / T \sim 10^{-5}$. These fluctuations encode the density variations in the primordial plasma at the moment of last scattering, the seeds from which all cosmic structure grew.
The Blackbody Spectrum
The CMB spectrum is the most perfect blackbody ever measured in nature. The COBE satellite's FIRAS instrument measured it in 1990, finding a perfect Planck distribution:
$$B(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu / k_B T} - 1}$$with temperature $T = 2.725$ K. The peak of the spectrum lies at a frequency of about 160 GHz (wavelength $\sim 1.9$ mm), firmly in the microwave band. The deviation from a perfect blackbody is less than 1 part in $10^5$: a stunning confirmation that the early universe was in thermal equilibrium.
This observation alone is powerful evidence for the hot Big Bang. No other known process can produce such a perfect blackbody spectrum over the entire sky. The temperature we observe today is related to the temperature at last scattering by:
$$T_0 = \frac{T_{\text{ls}}}{1 + z_{\text{ls}}} = \frac{3000 \text{ K}}{1100} \approx 2.725 \text{ K}$$Temperature Anisotropies
While the CMB is remarkably uniform, it is not perfectly so. Tiny temperature variations, anisotropies, exist at the level of $\Delta T / T \sim 10^{-5}$. These were first detected by the COBE satellite in 1992 and have been mapped with increasing precision by WMAP and Planck.
The temperature field on the sky is decomposed into spherical harmonics:
$$\frac{\Delta T}{T}(\theta, \phi) = \sum_{\ell=1}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\theta, \phi)$$The angular power spectrum $C_\ell$ captures the statistical information:
$$C_\ell = \frac{1}{2\ell + 1} \sum_{m=-\ell}^{\ell} |a_{\ell m}|^2$$The multipole $\ell$ corresponds to an angular scale $\theta \sim 180°/\ell$. Low $\ell$ probes large angular scales; high $\ell$ probes small angular scales.
Acoustic Oscillations
The acoustic peaks in the CMB power spectrum arise from sound waves in the photon-baryon plasma before recombination. The physics is beautifully simple: gravity pulls matter into overdense regions, while photon pressure pushes back. This competition drives oscillations, sound waves, in the plasma.
The fundamental mode corresponds to a perturbation that has had exactly enough time to compress once before recombination. The sound horizon at recombination is:
$$r_s = \int_0^{t_{\text{rec}}} \frac{c_s \, dt}{a(t)}$$where the sound speed in the photon-baryon fluid is:
$$c_s = \frac{c}{\sqrt{3(1 + R)}}, \qquad R = \frac{3\rho_b}{4\rho_\gamma}$$The first peak corresponds to scales that completed exactly one compression by recombination. The second peak corresponds to one full compression and one rarefaction. The third peak corresponds to 1.5 oscillation cycles, and so on.
What the Peaks Tell Us
First peak position ($\ell \approx 220$): Measures the angular size of the sound horizon. This depends on the geometry of the universe. A flat universe ($\Omega_{\text{total}} = 1$) gives the observed position perfectly.
Odd/even peak heights: Odd peaks (1st, 3rd, ...) are enhanced by baryons, which add to the compression. The ratio of odd to even peak heights measures the baryon density $\Omega_b h^2$.
Overall peak height pattern: The relative heights of the peaks constrain the dark matter density $\Omega_c h^2$, because dark matter enhances gravitational compression without providing pressure support.
Damping tail: At high $\ell$, photon diffusion erases small-scale fluctuations (Silk damping). The damping scale constrains the baryon density and the expansion rate.
Polarization: E-modes and B-modes
The CMB is not only characterized by its temperature but also by its polarization. Thomson scattering of anisotropic radiation produces linear polarization. The polarization pattern on the sky can be decomposed into two components:
E-modes (gradient-like, even parity): Produced by the same density perturbations that create temperature anisotropies. E-mode polarization has been measured with high precision and is consistent with the standard model.
B-modes (curl-like, odd parity): Cannot be produced by scalar (density) perturbations at linear order. There are two sources of B-modes:
- Gravitational lensing: E-modes are converted to B-modes by gravitational lensing from large-scale structure. This has been detected.
- Primordial gravitational waves: Tensor perturbations from inflation would produce B-modes at large angular scales. This has not yet been detected. The amplitude is characterized by the tensor-to-scalar ratio $r$, currently bounded by $r < 0.036$.
The detection of primordial B-modes would be a smoking gun for inflation, directly probing the energy scale of inflation: $V^{1/4} \sim 10^{16} \text{ GeV} \times (r/0.01)^{1/4}$.
CMB Missions: COBE, WMAP, Planck
Three generations of space-based CMB experiments have progressively refined our picture:
COBE (1989-1993): The Cosmic Background Explorer made two landmark measurements: FIRAS confirmed the perfect blackbody spectrum, and DMR detected temperature anisotropies for the first time ($\Delta T / T \sim 10^{-5}$ on large angular scales). George Smoot and John Mather received the 2006 Nobel Prize.
WMAP (2001-2010): The Wilkinson Microwave Anisotropy Probe mapped the CMB with 13-arcminute resolution across five frequency bands. WMAP established the "concordance cosmology" and measured fundamental parameters to percent-level precision.
Planck (2009-2013): The European Space Agency's Planck satellite achieved 5-arcminute resolution and measured the CMB in nine frequency bands from 30 to 857 GHz. Planck measured the power spectrum to $\ell \sim 2500$ and determined cosmological parameters to sub-percent precision.
Precision Cosmology: The Planck Results
The six parameters of the standard cosmological model ($\Lambda$CDM), as measured by Planck:
$\Omega_b h^2 = 0.0224$ (baryon density): only 5% of the universe is ordinary matter.
$\Omega_c h^2 = 0.120$ (dark matter density): 27% is cold dark matter.
$H_0 = 67.4$ km/s/Mpc (Hubble constant).
$n_s = 0.965$ (spectral index): confirms the slight tilt predicted by inflation.
$\tau = 0.054$ (optical depth to reionization).
$A_s = 2.1 \times 10^{-9}$ (amplitude of primordial perturbations).
The Next Generation: 2024-2035
Planck retired in 2013 and remains the most complete space-based CMB map. The next leap forward is split between three fronts: more sensitive ground-based CMB experiments, a dedicated space mission for primordial B-modes, and large galaxy surveys whose cross-correlations with the CMB will sharpen every cosmological parameter.
Atacama Cosmology Telescope (ACT, 2007-2022) and South Pole Telescope (SPT, 2007+) have measured the CMB at arcminute resolution from the ground, complementing Planck on small angular scales and providing the first detailed CMB lensing maps. BICEP/Keck at the South Pole leads the hunt for the inflationary B-mode signal, setting the current upper limit $r < 0.036$.
Simons Observatory (SO, 2024-onwards): six telescopes in Chile with 100000 detectors, expected to improve the CMB temperature and polarization sensitivity by a factor of 3 to 5 over Planck. Will produce the deepest CMB lensing map yet and probe inflation down to $r \sim 0.003$.
LiteBIRD (JAXA, launch around 2032): a space mission entirely dedicated to large-angular-scale polarization, hunting for primordial B-modes from inflation. Targets a tensor-to-scalar ratio sensitivity of $r \sim 0.001$, which would either detect the inflationary gravitational-wave background or rule out most simple inflation models.
CMB-S4 (2030s): the planned next-generation US-led ground-based experiment, combining sites in Chile and at the South Pole with around 500000 detectors. Designed to reach $r \sim 5 \times 10^{-4}$, characterise neutrino masses to $\sigma(\sum m_\nu) \sim 20$ meV, and map dark matter via CMB lensing across most of the sky.
Euclid and the CMB partnership
Euclid (ESA, launched July 2023) is not a CMB satellite, but it is the most important new cosmology mission for combining with CMB data. Its visible and near-infrared cameras will map the shapes and redshifts of around 1.5 billion galaxies across one third of the sky over six years. Three measurements matter for the CMB picture:
- Weak gravitational lensing (cosmic shear): the dark-matter distribution traced by tiny distortions of galaxy shapes. Cross-correlated with Planck and SO lensing maps, this pins down $\sigma_8$ and the growth of structure, helping to resolve the existing 2-3$\sigma$ tension with the CMB.
- Baryon acoustic oscillations and redshift-space distortions: the same sound horizon that sets the CMB acoustic peaks shows up as a feature in the galaxy two-point correlation function. Euclid will measure it across cosmic time and break the geometric degeneracy that limits Planck-only constraints.
- Dark energy equation of state: by tracking the growth of structure between $z = 2$ and today, Euclid measures whether dark energy behaves like a cosmological constant. The first cosmology results (Q1 release in 2024, DR1 in 2026) are already among the most precise low-redshift constraints, and joint analyses with Planck are the new standard.
NASA's Roman Space Telescope (launch 2027) will deliver a deeper, narrower survey complementary to Euclid, and DESI (ground-based) is publishing similar BAO measurements right now. Together with Simons Observatory and CMB-S4, this constellation of experiments will turn the next decade into the best chance we have ever had to detect inflationary gravitational waves, weigh the neutrinos, and test whether dark energy is truly constant.
Secondary Anisotropies
Beyond the primordial signal, CMB photons are modified as they travel to us:
- Integrated Sachs-Wolfe effect: Photons gain or lose energy traversing time-varying gravitational potentials (relevant in the dark-energy dominated era)
- Sunyaev-Zel'dovich (SZ) effect: CMB photons scatter off hot electrons in galaxy clusters, shifting their spectrum. This provides a method for finding distant galaxy clusters
- Gravitational lensing: The large-scale mass distribution deflects CMB photons, slightly smoothing the acoustic peaks and generating B-modes
- Reionization: When the first stars formed ($z \sim 6$-$10$), they reionized the intergalactic medium. Thomson scattering by these free electrons partially erases small-scale CMB anisotropies and generates large-scale polarization
Key Insights
- The CMB is a nearly perfect blackbody at $T = 2.725$ K, the most precise blackbody in nature
- Temperature anisotropies at $\Delta T/T \sim 10^{-5}$ encode the physics of the primordial plasma
- Acoustic peaks arise from sound waves in the photon-baryon fluid before recombination
- The first peak position constrains geometry ($\Omega_{\text{total}} = 1$); peak height ratios constrain baryon and dark matter densities
- E-mode polarization confirms the acoustic oscillation picture; primordial B-modes remain the holy grail
- COBE, WMAP, and Planck have made cosmology a precision science, determining fundamental parameters to sub-percent accuracy
- Simons Observatory, LiteBIRD, and CMB-S4 will push polarization sensitivity by another factor of 10, while Euclid and Roman provide the low-redshift cross-checks that the CMB alone cannot supply
- The CMB provides independent confirmation of the baryon density measured by Big Bang nucleosynthesis
Looking Ahead
The CMB tells us that the universe contains far more than ordinary matter. The acoustic peaks demand a large component of non-baryonic dark matter, and the overall expansion history requires dark energy. In the next lesson, we confront one of the deepest mysteries in modern physics: dark matter: its evidence, its properties, and the worldwide hunt to identify it.
- The CMB is the most perfect blackbody ever measured in nature at $T = 2.725$ K, providing powerful evidence that the early universe was in thermal equilibrium.
- Tiny temperature anisotropies at the level of one part in 100,000 encode the physics of the primordial plasma, with acoustic peaks arising from sound waves in the photon-baryon fluid before recombination.
- The position and heights of the acoustic peaks constrain the geometry of the universe ($\Omega_{\text{total}} = 1$), the baryon density, and the dark matter density with sub-percent precision.
- Primordial B-mode polarization from inflationary gravitational waves has not yet been detected and remains one of the most important targets in observational cosmology.