Cosmic microwave background acoustic peaks

In a tightly coupled photon fluid with constant sound speed and no gravitational driving, a Fourier temperature perturbation obeys a harmonic oscillator equation in conformal time. Initial displacement with negligible velocity produces a cosine phase. Modes whose sound-horizon phase reaches an integer multiple of pi are at extrema at last scattering. Their squared amplitudes therefore form a harmonic sequence in this idealization. A random unrelated phase for every mode would erase this simple coherent pattern.

Real baryons add inertia and shift the equilibrium in gravitational potentials, affecting compressions differently from rarefactions. Evolving potentials drive oscillations, radiation and neutrinos affect phases, and photon diffusion damps small scales. A finite visibility width and Doppler terms further modify the observed spectrum. These effects are computed through coupled perturbation and radiative-transfer equations. The elementary oscillator identifies a mechanism but cannot determine baryon density or the complete spectrum from its period alone.

An angular multipole roughly corresponds to a transverse wave number times the distance to last scattering. This gives an acoustic spacing near pi times distance divided by sound horizon. Phase shifts and projection mean that the exact first temperature peak is not simply ell A. Different background and early-time parameter combinations can preserve the acoustic angular scale. Peak heights, damping, polarization, lensing, and external data help distinguish combinations that one angular ruler cannot separate.

Temperature maps also contain foreground emission, instrumental beams, noise, and calibration uncertainty. A likelihood must predict the observed spectra through those effects, with appropriate nuisance parameters and masks. Angular spectra describe ensemble variances, whereas the measured sky supplies only one realization of each multipole’s finite number of modes. More sensitive detectors can reduce instrumental noise, but cannot produce extra independent full-sky realizations of the same primordial temperature field.

For a Gaussian statistically isotropic field, the full-sky multipole coefficients have variance C ell, and the estimated power has a chi-squared sampling distribution. At low multipoles that distribution is appreciably asymmetric, so a Gaussian error bar is only a summary. Cosmic variance describes uncertainty in recovering an ensemble power from one sky; it is not a claim that the measured map changes randomly when a telescope points at it again.

A mask removes information and mixes angular modes. Dividing a naive mode count by the sky fraction offers intuition for error inflation but does not reproduce every cut-sky covariance or foreground-induced bias. Survey geometry, multipole binning, anisotropic noise, and cross-spectra all matter. Inference of a late-time parameter such as H zero from CMB spectra additionally assumes a cosmological model connecting the measured angular and physical scales. That dependence must accompany any quoted numerical result.