The Cosmic Microwave Background

The cosmic microwave background is a layered observation. Primordial curvature perturbations evolve into photon, baryon, and metric sources; Thomson scattering selects a last-scattering distribution and generates polarization; free propagation projects three-dimensional modes onto the sky; lensing remaps that sky; and an instrument records a beam-, mask-, and foreground-contaminated data vector. Only after all of those maps are stated can a likelihood constrain cosmological parameters. This chapter follows that chain without confusing a physical source, an angular observable, and a statistical estimate.

Helpful background. Einstein Equations and relativistic matter introduce the metric and radiation sources. The cosmology overview locates the thermal and perturbation history; Probability and Inference for Gravity prepares covariance and posterior calculations; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics background used for propagation and wave projection.

Enter the CMB transfer chain

The six pages support two reader routes.

1. For the physical transfer route, read Line-of-Sight CMB Anisotropies, Acoustic Oscillations and Peak Structure, CMB Polarization and E/B Modes, and CMB Lensing in order. This route follows a primordial scalar mode through visibility, sound waves, angular projection, spin-2 polarization, and gravitational remapping.
2. For the data-to-posterior route, establish the source and spectrum conventions on the first three pages, then read CMB Lensing, Foregrounds, Beams, and Calibration, and CMB Likelihoods and Parameter Inference. This route separates a late-time four-point reconstruction from primary two-point spectra, then carries named survey products through nuisance modeling and correlated inference.

The order matters. Peak positions cannot be read without the projection kernel; E and B cannot be assigned to individual Stokes pixels; lensing power is not the same observable as peak smoothing; and a parameter table cannot be detached from the likelihood that produced it.

One convention package from transport to spectra

The theory pages use spatially flat Newtonian gauge,

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right], \qquad e^{i\mathbf k\cdot\mathbf x}.$$

The conformal Thomson rate is positive, $\dot\kappa=an_e\sigma_T\gt 0$. Optical depth is measured from emission time to the observer,

$$\kappa(\eta)=\int_\eta^{\eta_0}\dot\kappa\,d\tilde\eta, \qquad g(\eta)=\dot\kappa e^{-\kappa}.$$

Consequently $g$ is positive and $\int g\,d\eta=1-e^{-\kappa(0)}$. This convention differs from sources that call the derivative of optical depth with respect to forward conformal time negative; copying only one sign reverses collision or visibility terms.

Temperature moments use

$$\Theta(\mathbf k,\mu)=\sum_{\ell=0}^{\infty} (-i)^\ell(2\ell+1)\Theta_\ell(\mathbf k)P_\ell(\mu), \qquad \mu=\hat{\mathbf k}\cdot\mathbf n.$$

For the scalar velocity convention $\mathbf n\cdot\mathbf v_b=-i\mu v_b$, the Doppler source projects with $+v_bj_\ell'$. For unlensed linear adiabatic scalar transfer, primordial power and angular power are related by

$$\mathcal P_{\mathcal R}(k)=\frac{k^3}{2\pi^2}P_{\mathcal R}(k), \qquad C_\ell^{XY}=4\pi\int d\ln k\, \mathcal P_{\mathcal R}(k)T_\ell^X(k)T_\ell^{Y*}(k).$$

Independent tensor or isocurvature initial modes require their own spectra and transfers, including cross-spectra when the seeds are correlated. Lensing subsequently remaps the primary prediction; its B power is not a linear scalar $T_\ell^B$ inserted into this formula.

These definitions make it possible to compare the analytic line-of-sight solution, an Einstein–Boltzmann output, and a likelihood spectrum without a hidden factor of $2\pi$, $k^3$, or $2\ell+1$.

Source, propagation, and measurement are different layers

The source layer contains the effective temperature $\Theta_0+\Phi$, baryon velocity, time-varying Weyl potential $\Phi'+\Psi'$, and the Thomson quadrupole. The line-of-sight integral weights local scattering sources by $g$ and freely propagating metric evolution by $e^{-\kappa}$. Spherical Bessel functions then project wavenumber $k$ across comoving distance $D$ into a broad distribution of multipoles. The useful large-$\ell$ guide $\ell\sim kD$ is not an equality. Hu and Dodelson 2002, §§3.1–3.8, preprint printed pp. 9–25 (Open PDF) develops the acoustic source and its angular projection.

Before recombination, the tightly coupled photon–baryon fluid has

$$R=\frac{3\rho_b}{4\rho_\gamma}, \qquad c_s^2=\frac{1}{3(1+R)}, \qquad r_s(\eta_*)=\int_0^{\eta_*}c_s\,d\eta.$$

The leading peak spacing comes from $\ell_A=\pi D_M(\eta_*)/r_s(\eta_*)$, while baryon loading, radiation driving, diffusion, visibility width, neutrino stress, and projection change phases and heights. A local radiation quadrupole generates the spin-2 field $Q\pm iU$; parity-even E and parity-odd B are nonlocal full-sky combinations. Linear scalar perturbations generate primordial E but no primordial B. Tensors, vectors, lensing, rotation, foregrounds, and leakage are distinct routes to B.

Lensing belongs to the propagation layer. With the convention

$$\widetilde X(\mathbf n)=X(\mathbf n+\nabla\phi), \qquad \kappa_{\rm lens}=-\frac12\nabla^2\phi,$$

a constant deflection changes Fourier phases but not total power; a varying deflection couples modes and supports quadratic reconstruction. The symbol $\kappa_{\rm lens}$ is convergence and must not be confused with the optical depth $\kappa(\eta)$ above.

The measurement layer begins only after maps are calibrated, beam- and transfer-corrected, masked, split, and converted into spectra or higher-point estimators. Foreground frequency laws, bandpass averages, leakage, noise, and mode-coupling covariance are part of this layer. For a declared bandpower normalization,

$$D_\ell^{XY}=\frac{\ell(\ell+1)}{2\pi}C_\ell^{XY}, \qquad m_b=\sum_\ell W_{b\ell}D_\ell+F_b,$$

where the window $W_{b\ell}$ and foreground term $F_b$ belong to a specific release. The theory spectrum is not itself the measured data vector.

What each page contributes

Page	Central object	Decisive check
Line of sight	visibility-weighted transfer integral	recover the instantaneous Sachs–Wolfe result
Acoustic peaks	driven photon–baryon oscillator	recover $c_s=1/\sqrt3$ and spacing $\pi/r_s$ when $R=0$
Polarization	spin-2 E/B decomposition	show scalar linear B vanishes and parity forbids TB/EB conditionally
Lensing	remapping and off-diagonal covariance	preserve power and phase under an exact constant shift; converge for a varying shift
Foregrounds	multifrequency instrument model	vary frequency, mask, or component assumptions with calibration propagated
Likelihood	versioned data, covariance, priors, and sampler	recover a released marginal and residual, then change a declared cut

The first three checks are analytic. The lensing page adds a deterministic repository benchmark and an ACT DR6 likelihood-amplitude check. The final two pages use ACT DR6.02 and Planck 2018 as dated examples of how released maps, spectra, windows, covariance, nuisance parameters, and chains fit together. Those examples are not mission catalogs and their numerical values are not evergreen constants. Their likelihood definitions are given in Planck Collaboration 2020 V, §§2–3.4, pp. 4–53 (Open PDF) and Louis et al. 2025, §§6.1–7.3, pp. 22–29 (Open PDF).

Boundaries and onward routes

This chapter assumes metric photon propagation, linear primary transfer, and the usual temperature and linear-polarization observables unless a page states otherwise. Detailed recombination kinetics belong to Recombination and Photon Decoupling. The gauge-invariant field and kinetic equations belong to Cosmological Perturbations and Gauge Invariance. Primordial tensor generation belongs to Inflation and Primordial Fluctuations, and nonlinear matter evolution belongs to Relativistic Structure Formation.

Spectral distortions, circular polarization, detailed mission engineering, nonlinear secondary anisotropies beyond the stated foreground models, and a survey-by-survey parameter compilation are outside this chapter. The evidence cutoff for its ACT and Planck examples is 2026-08-19. A new likelihood, calibration, covariance, map release, or foreground treatment should update the observational pages without changing the analytic definitions merely to match a new parameter central value.

Review the chapter

Assume instantaneous last scattering at $\eta_*$, constant equal potentials, no Doppler or polarization source, and a matter-era adiabatic superhorizon mode. First find the temperature transfer to multipole $\ell$. Next take a one-dimensional sky mode $T(x)=A\cos nx$ and remap it by a constant deflection $\alpha$. Show what changes in Fourier space and what does not. Finally, suppose its measured amplitude is $d=A+f+\epsilon$, with $\epsilon\sim\mathcal N(0,\sigma^2)$ and a foreground prior $f\sim\mathcal N(0,s_f^2)$. Find the likelihood for $A$ after marginalizing $f$.

Solution

In the stated Sachs–Wolfe limit, $\Theta_0+\Phi=\Phi/3$ and the visibility is a delta function. Therefore

$$\Theta_\ell(k,\eta_0)=\frac{\Phi(k,\eta_*)}{3} j_\ell[k(\eta_0-\eta_*)].$$

The constant remapping gives

$$\widetilde T(x)=A\cos[n(x+\alpha)].$$

The $+n$ Fourier coefficient is multiplied by $e^{in\alpha}$ and the $-n$ coefficient by $e^{-in\alpha}$. Their moduli, hence the mode and total power, are unchanged. A spatially varying deflection would mix different modes instead of producing only these phases.

Marginalization convolves the two Gaussian errors:

$$\mathcal L(d\mid A) =\int df\,\mathcal N(d;A+f,\sigma^2)\mathcal N(f;0,s_f^2) =\mathcal N(d;A,\sigma^2+s_f^2).$$

Thus the physical transfer fixes the mean model, constant lensing remapping preserves its ideal power, and foreground uncertainty broadens the inference. Fixing $f=0$ would incorrectly report uncertainty $\sigma$ rather than $\sqrt{\sigma^2+s_f^2}$.

References

• Hu, W., and S. Dodelson, “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216 (2002), DOI, Open PDF.
• Louis, T., et al. (ACT Collaboration), “The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and ΛCDM Parameters,” arXiv:2503.14452v2 (2025), Open PDF, data products.
• Planck Collaboration, “Planck 2018 Results. V. CMB Power Spectra and Likelihoods,” Astronomy & Astrophysics 641 (2020) A5, DOI, Open PDF.