Temperature fluctuations in the CMB arise due to the variations
in the matter density. After last-scattering CMB photons
stream freely to us and the temperature fluctuations are
seen as CMB temperature differences (anisotropy)
across the sky. Anisotropy on a given angular scale is related
to density perturbations with wavelengths corresponding to
the length projected by that angle on the last-scattering surface:
. Until the ions and
electrons ``recombined'' at last-scattering, the photons and ions and
electrons (the baryons) were tightly coupled by Thomson scattering,
and behaved as a single fluid. The gravity-driven collapse of a
perturbation is resisted by the pressure restoring
force of the photons. Fourier mode *k* of the temperature
fluctuation is governed by a harmonic-oscillator like equation,

where *F* is the gravitational forcing term due to the dark matter,
describes the inertia of the fluid,
and primes denote derivatives with respect to (conformal) time
( ). The solutions are acoustic waves.

The large-angular scale (Sachs-Wolfe) plateau ( ) in the angular power spectrum arises from perturbations with periods longer than the age of the Universe at last scattering. CMB photons lose energy climbing out of the potential wells associated with these long-wavelength density perturbations, and the temperature differences seen on the sky reflect the gravitational potential differences on the last-scattering surface. If the density fluctuations are approximately scale-invariant, as inflation and defect theories predict, the plateau in the angular power spectrum is flat.

The baryon - photon fluctuations that produce anisotropy on sub-degree angular scales ( ) have sufficient time to undergo oscillation. At maximum compression (rarefaction) the CMB temperature is higher (lower) than average; neutral compression corresponds to velocity maximum of the fluid, which leads to a Doppler-shifted CMB temperature. Since last-scattering is nearly instantaneous the CMB provides a snapshot of these acoustic oscillations, with different wavelength modes being caught in different phases of oscillation. Because a given multipole is dominated by the effects of a narrow band of Fourier modes ( ), this leads to peaks and valleys in the angular power spectrum. The peaks are modes which were maximally under- or overdense at last-scattering, and the troughs are velocity maxima, which are out of phase with the density maxima.

On the smallest scales ( ) the spectrum is exponentially damped, due to the finite thickness of the last-scattering surface. Features on these angular scales are washed out because last scattering is not a single snapshot, but a montage of snapshots which blurs the fine details.

The precise shape of the power spectrum depends upon cosmological parameters as well as the underlying density perturbations and thereby encodes a wealth of information; see Fig.6. The position of the first peak is sensitive to the total energy density and can be used to determine the geometry of the Universe: . It moves to smaller angles as decreases because the distance to the last-scattering surface increases (the expansion slows less in a low-density universe) and geodesics diverge in negatively curve space (fixed distance on the last-scattering surface subtends a smaller angle).

Other features encode other information. For example, the height of the first peak depends upon the matter and baryon density (which both depend on the Hubble constant) and the presence of a cosmological constant. If the spectrum of density perturbations is not scale invariant, but for example has more power on small scales, the angular power spectrum rises with increasing .

Sun Nov 2 13:44:30 CST 1997