It is most useful to describe the CMB anisotropy on the celestial sphere by spherical-harmonic multipole moments,

The multipole moments, which are determined by the underlying
density perturbations, can only be described statistically.
They have zero mean,
i.e., , and if the underlying density fluctuations
are described by a gaussian random process, as inflation
predicts, the angular power spectrum,
,
contains all possible information.
(The angled brackets indicate the average over all observers
in the Universe; the absence of a preferred direction in the Universe
implies that is independent of *m*.)
If the density fluctuations are nongaussian, as other models predict,
then higher-order correlations functions contain additional information.

Temperature differences between points on the sky separated by angle are related to those multipoles with spherical-harmonic indices around . The rms fractional temperature fluctuation for a given angular separation is then

The angle subtends a length on the surface of last scattering That would now, by the Hubble expansion of the universe, be about 200Mpc per degree (1Mpc is approximately 3 million light-years). Therefore, the corresponding th multipole is determined by density fluctuations on that wavelength scale. For example, the density fluctuations of wavelength around 2Mpc, which seed galaxies, subtend an angle of around an arcminute; those of 20Mpc that seed clusters of galaxies subtend about 10 arcminutes; and those of around 200Mpc that seed the largest structures seen today subtend about 1 degree. (All of these distances were a thousand times smaller at the time of last scattering, when the linear size of the universe was a thousand times smaller. But it is conventional to quote ``comoving separations'' as they would be now.)

The two competing models for the origin of the primeval density perturbations involve the physics of the early Universe. The first holds that around after the Big-Bang, a very short burst of tremendous expansion (called inflation) stretched quantum fluctuations on subatomic scales to astrophysical size and that these fluctuations became density perturbations when the vacuum energy that drove inflation decayed into radiation and matter. According to this inflationary scenario, the density perturbations are almost ``scale invariant:'' That is to say, fluctuations in the gravitational potential were the same magnitude ( ) on all scales. Figure 2 shows the angular power spectrum predicted by inflation.

The competing theory holds that the density perturbations were seeded by topological defects formed even earlier ( ) in a cosmological phase transition associated with spontaneous symmetry breaking in the theory that unifies the fundamental forces and particles. Depending upon how the symmetry is broken, these defects might be point-like (global monopoles), one-dimensional (cosmic strings), or three-dimensional (spacetime textures).

It is the gravitational effects of such defects that would induce perturbations thousands of years later in the matter distribution. Although these perturbations would also be approximately scale invariant the power spectrum of CMB anisotropy would be very different from what we expect from inflation, because density perturbations would have originated so much later than in the inflationary scenario. The current anisotropy data appear to be consistent with inflation and inconsistent with the topological defect scenario (see Fig.2).

Both inflation and defect models require nonbaryonic dark matter. So do the dynamical measurements of galaxies and clusters that indicate there is much more gravitating matter than can be accounted for by luminous objects or even by dark baryons. The notably successful theory of big-bang nucleosynthesis constrains the baryon density to be less than 100f the ``critical mass density'' above which the Hubble expansion would eventually become a contraction. But the dynamical observations indicate that dark matter contributes at least 200f the critical density, and inflation favors precisely the critical density. The observed level of CMB anisotropy provides additional circumstantial evidence: If there were only baryons, the level of primeval inhomogeneity required to produce the observed structure would lead to an anisotropy that is about ten times larger than that observed. (See Box 1.)

The non-baryonic matter may be ``cold'' (slow moving) or ``hot'' (fast). If most of the dark matter is cold, then structure forms hierarchically - from galaxies to clusters of galaxies to superclusters; if it is mostly hot, then superclusters form first and then fragment into clusters and galaxies. There is now good evidence that galaxies formed first (the bulk around redshifts of two to three) and that clusters of galaxies and superclusters formed later, which strongly favors the (mostly) cold dark matter picture. This, together with measurements of CMB anisotropy, have made inflation+cold dark matter the working hypothesis for how structure formed in the Universe.

The precise shape of the angular power spectrum depends not only on the underlying inflation model, but also, in a well understood way, on cosmological parameters such as the Hubble constant, the mass density and the composition of the dark matter. (See Box 2.) Therefore, the 2500 or so independent multipoles that can be measured have enormous potential to determine cosmological parameters and to test theories of the early Universe.

Sun Nov 2 13:44:30 CST 1997