Cosmic microwave background (CMB)

The cosmic microwave background last-scattering surface is the shell from which the photons observed in the Cosmic microwave background last scattered around photon decoupling.

Cosmic microwave background anisotropies are direction-dependent perturbations of the relic radiation temperature and polarization.

The CMB lensing potential is the line-of-sight projection of gravitational potential that remaps observed CMB directions by the deflection angle $\nabla \psi$.

The CMB angular power spectrum decomposes the variance of temperature or polarization anisotropy by angular multipole. Its large-angle plateau and acoustic peaks encode primordial perturbations and photon-baryon evolution.

Thomson scattering of a local radiation quadrupole generates linear cosmic microwave background polarization. Its acoustic phase follows the photon velocity and is shifted by one quarter-period relative to the monopole temperature oscillation.

For Fourier wavevector $\mathbf{k}$ and photon direction $\hat{\mathbf{p}}$, the temperature perturbation is expanded as

The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination ${\Theta }_0+\Psi$.

In Newtonian gauge, ${\Theta }_0+\Psi$ combines the intrinsic photon temperature perturbation with the gravitational redshift at emission. With constant potentials and negligible baryon loading it oscillates as $\cos (k{r}_s)$ for adiabatic initial conditions.

The Sachs-Wolfe effect is the large-angle CMB temperature anisotropy produced by intrinsic photon-temperature perturbations and gravitational redshift at last scattering. For an adiabatic mode during matter domination, ${\Theta }_0+\Psi =\Psi /3$ on superhorizon scales.

The Doppler CMB anisotropy is the line-of-sight velocity contribution from the last-scattering plasma. With the convention used here it contributes $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_e$ to the observed fractional temperature perturbation.

The integrated Sachs-Wolfe effect is the CMB temperature shift produced when the gravitational potentials traversed by a photon evolve with time. Its line-of-sight source is proportional to ${\Phi }^{\prime }+{\Psi }^{\prime }$.

The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, $v_b=-3{\Theta }_1$ under the convention used here.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.

Before recombination, frequent Thomson scattering couples photons, electrons, and baryons into an acoustic fluid.

Thomson scattering is the low-energy elastic scattering of electromagnetic radiation by a free charged particle. Before cosmological recombination, repeated photon-electron scattering couples photons to the baryon velocity and damps higher photon multipoles.

The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of $k/\Gamma$ and $\mathcal{H}/\Gamma$, where $\Gamma$ is the Thomson scattering rate.

For baryon-loading ratio $R=3{\rho }_b/(4{\rho }_{\gamma })$, the photon-baryon sound speed is $c_s^2=1/[3(1+R)]$.

The comoving sound horizon is the distance traveled by an acoustic wave,

Cosmic microwave background diffusion damping is the suppression of small-angular-scale anisotropy as photons random-walk out of overdense regions during the finite-width recombination epoch. It produces an approximately exponential damping tail in the Cosmic microwave background power spectrum at multipoles $\ell \gtrsim 1000$.

Adiabatic initial conditions perturb every species by the same local time shift, so their relative number-density ratios are initially unperturbed. Photon acoustic modes then begin predominantly as displacement modes with cosine phase.

Equivalently, $\delta {\rho }_i/{\dot{\bar{\rho }}}_i$ is independent of the component $i$. The non-adiabatic pressure perturbation then vanishes, and the comoving curvature perturbation is conserved on superhorizon scales when anisotropic stress and gradient terms can be neglected.

Cosmic microwave background acoustic peaks arise from photon-baryon oscillations caught at successive extrema at recombination. Their approximate phase is set by $k{r}_s=n\pi$ and their angular positions measure the sound horizon relative to the distance to last scattering.

In the absence of collisions, a photon phase-space density is constant along its geodesic. Linearizing this statement gives a first-order transport equation for the direction-dependent temperature perturbation.

Expanding the direction-dependent photon temperature in Legendre polynomials turns its transport equation into an infinite hierarchy. Free streaming couples each multipole ${\Theta }_{\ell }$ only to its neighbours ${\Theta }_{\ell -1}$ and ${\Theta }_{\ell +1}$.

The dipole member of the photon Boltzmann hierarchy is the photon Euler equation. In Newtonian gauge and without collisions,

The line-of-sight solution integrates the gravitational source along the unperturbed photon trajectory. In Fourier space the integrating factor $e^{ik\mu \eta }$ accounts for free streaming between emission and observation.

At linear order in Newtonian gauge, free streaming, gravitational redshift, and Thomson scattering combine into a first-order transport equation for the photon temperature perturbation. The collision term drives the radiation toward its monopole plus the electron bulk-velocity dipole.

The optical depth from conformal time $\eta$ to observation at ${\eta }_0$ isThe factor $e^{-\tau (\eta )}$ is the probability that a photon travels from $\eta$ to the observer without another scattering.

The visibility functionis the probability density for the conformal time of a CMB photon's last scattering.

The CMB line-of-sight solution separates sharply localized last-scattering sources, weighted by the cosmological visibility function, from integrated gravitational sources weighted by $e^{-\tau }$ along the photon trajectory.