FIG. 1. The bands in comoving wavenumber k probed by CMB primary and secondary anisotropy experiments, in particular by the satellites Cosmic Background Explorer (COBE), Microwave Anisotropy Probe (MAP), and Planck, and by LSS observations are contrasted. Mpc, megaparsec (3.09×1022 m). The width of the CMB photon decoupling region and the sound crossing radius (∆τγdec, csτγdec) define the effective acoustic peak range. Sample (linear) gravitational potential power spectra [actually ] are also plotted. The region at low k gives the 4-yr Differential Microwave Radiometer (DMR) error bar on the Φ amplitude in the COBE regime. The solid data point in the cluster-band denotes the Φ constraint from the abundance of clusters, and the open data point at 10h−1 Mpc denotes a Φ constraint from streaming velocities (for Ωtot =1, ΩΛ=0). The open squares are estimates of the linear Φ power from current galaxy clustering data by ref. 1. A bias is “allowed” to (uniformly) raise the shapes to match the observations. The corresponding linear density power spectra, are also shown rising to high k. Models are the “standard” ns=1 cold dark matter (CDM) model (labeled Γ=0.5), a tilted (ns=0.6, Γ=0.5) CDM (TCDM) model, and a model with the shape modified (Γ=0.25) by changing the matter content of the Universe.

is related to the usual index, ns, by vs= ns−1.] The transport problem (see below) is dependent upon physical processes, and hence on physical parameters. A partial list includes the Hubble parameter h, various mean energy densities [Ωtot, ΩB, ΩΛ, Ωcdm, Ωhdm]h2, and parameters characterizing the ionization history of the Universe—e.g., the Compton optical depth τC from a reheating redshift zreh to the present. Instead of Ωtot, we prefer to use the curvature energy parameter, Ωk≡1−Ωtot, thus zero for the flat case. In this space, the Hubble parameter h=(Σjjh2))1/2, and the age of the Universe, t0, are functions of the Ωjh2. The density in nonrelativistic (clustering) particles is ΩnrBcdmhdm. The density in relativistic particles, Ωer, includes photons, relativistic neutrinos, and decaying particle products, if any, Ωer, the abundance of primordial helium, etc., should also be considered as parameters to be determined. The count is thus at least 17. Estimates of errors on a smaller 9-parameter inflation set for the MAP and Planck satellites are given in the final section.

The arena in which CMB theory battles observation is the anisotropy power spectrum in multipole space. Fig. 2, which shows how primary C values vary with some of these cosmic parameters. Here The

FIG. 2. The anisotropy data for experiments up to March 1997 (top panel) and optimal combined bandpower estimates (lower panels) are compared with secondary values (top panel) and various primary sequences. The kinematic Sunyaev-Zeldovich is off scale and the thermal SZ is low; compare the primary values. Dusty emission from early galaxies may lead to high signals, but the power is concentrated at higher and higher frequency. The next panels are sequences of 13-Gyr models with variations in the following parameters: ns, 0.85 to 1.25 (panel 2 from the top): ΩΛ, 0 to 0.87 (H0 from 50 to 90) (panel 3); Ωk, 0 to 0.84 (H0 from 50 to 65) (panel 4); and ΩBh2, 0.003 to 0.05 (panel 5). Panel 6 shows that sample defect values from Pen, Seljak, and Turok (2) do not fare well compared with the current data; values from ref. 3 are similar. Panels 7 and 8 show forecasts of errors for the satellites MAP and Planck.

values are normalized to the 4-yr DMR(53+90+31) (A+ B) data (47). The arena for LSS theory is the of Fig. 1.

For a given model, the early universe is uniquely related to late-time power spectrum measures of relevance for the CMB, such as the quadrupole or averages over ℓ-bands B, and to LSS measures, such as the rms density fluctuation level on the 8 h−1 Mpc (cluster) scale, σ8, so any of these can be used in

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