Cosmological perturbation theory and Boltzmann codes

Setting up. Start with a perfectly smooth expanding universe (the FLRW model). Add small perturbations to the metric (the geometry) and to the density, velocity and temperature of each ingredient: photons, neutrinos, ordinary (baryonic) matter and cold dark matter. Keep only terms linear in the perturbation. This is accurate while the ripples are small.

Gauges. Splitting the universe into smooth part plus ripples is a choice, called a gauge. Different choices describe the same physics but make different-looking equations. Two common choices are the conformal Newtonian gauge, which looks like Newtonian gravity, and the synchronous gauge. Physical predictions must not depend on the choice.

Fourier modes. Because the linearized equations treat different spatial wave-numbers k separately, each mode can be evolved by itself. The starting values come from inflation, which predicts a nearly scale-invariant power spectrum: the amplitude of ripples is almost the same on all sizes, with slightly more on large scales. Two numbers describe it: the amplitude A_s and the tilt n_s.

The Boltzmann hierarchy. Photons and neutrinos are not simple fluids, so we track how their distribution depends on direction. We expand that dependence into multipoles (monopole, dipole, quadrupole and higher), which gives a long chain of coupled equations. In practice the chain is cut off at a finite length.

Tight coupling and acoustic oscillations. Early on, photons scatter off electrons very frequently (Thomson scattering), so photons and baryons move as one fluid. Gravity compresses it and radiation pressure pushes it back, producing sound waves with speed c_s about c/√3 reduced slightly by the baryon loading. The distance a sound wave can travel before the universe becomes transparent is the sound horizon, and it acts as a standard ruler: the baryon acoustic oscillation (BAO) scale. Photon diffusion damps small scales (Silk damping).

Recombination. About 380,000 years after the Big Bang, electrons and nuclei combine and photons escape. The detailed ionization history must be computed accurately, with codes such as RECFAST or HyRec, because the data are now very precise.

Output. CAMB and CLASS solve these equations on a grid of k and time and use a line-of-sight integral to quickly give the CMB temperature and polarization spectra, and the matter power spectrum.

Limits. At late times and small scales, matter clumps strongly and the linear approximation fails; that needs simulations (see the N-body entry).