Annu. Rev. Astron. Astrophys. 1994. 32: 319-70 Copyright © 1994 by Annual Reviews. All rights reserved

7.2. Open Universes

Until now, we have implicitly assumed that ₀ = 1 as favored by inflationary models. However astronomical evidence generally favors ₀ ~ 0.2( ± 0.1), based on large-scale structure (< 10 Mpc) studies. The negatively-curved spatial hypersurfaces of a low density ( = 0) Friedmann model greatly complicate the analysis of large angular scale anisotropy in the CMB. One expects that the curvature radius will introduce a feature into the low-order multipoles, due to gravitational focusing of geodesics which shifts the power from lower to higher orders. The characteristic angular scale corresponds to the curvature radius of ~ 1/2₀ radians. This effect is analogous to the "ring-of-fire" effect in weakly anisotropic cosmologies (Wilson & Silk 1981, Fabbri et al 1983). Early calculations include those of Kaiser (1982) and Peebles (1982b). This redistribution of the low-order multipole moments was realized in the elegant numerical computations of Wilson (Silk & Wilson 1981, Wilson & Silk 1981, Wilson 1983), who expanded the radiation power spectrum in terms of generalized wave-number (k² + K)^1/2, where K is the spatial curvature, defined as eigenvalues of the Laplace operator for density perturbations in a curved background.

Results similar to those of Wilson were subsequently obtained by Tomita & Tanabe (1983) and Abbott & Schaefer (1986). These latter authors used a gauge-invariant approach, expanded the power spectrum in wave-number k, and included cosmic variance in estimating the multipole moments. Górski & Silk (1989) computed the low-order multipoles for the case of primordial isocurvature (or entropy) fluctuations. Their results were generalized by Gouda et al (1991a, b), who included both primordial adiabatic and entropy fluctuations, and computed the contribution from the integrated Sachs-Wolfe term. This line-of-sight term vanishes in flat ₀ = 1 models but gives an important contribution to all scales in an open model (Hu & Sugiyama 1994). Note that the location of the Doppler peak, 200 / ₀^1/2, also depends on how open the Universe may be (Kamionkowski et al 1994).

Any simple interpretation of the results, however, is complicated by the fact that the concepts of wave number and power-law spectrum must be generalized to curved hyperspaces. They can no longer simply be interpreted in terms of constant curvature fluctuations associated with a spectrum of comoving scales if e.g. n = 1. Kamionkowski & Spergel (1993) have studied power spectra for primordial adiabatic fluctuations that are power laws in either volume (least large-scale power), distance, or eigenvalues of the Laplace operator (most large- scale power), and find that in all cases there is some suppression of the multipoles on scales larger than the curvature scale. In a flat universe, these three definitions of power spectrum would all coincide. There is large uncertainty in defining the spectrum because of the lack of any unique prescription for the initial conditions in a low- universe. Indeed, much of the motivation for a Harrison-Zel'dovich spectrum is lost if ₀ < 1, because one can no longer appeal to simple models of inflation. More complex models that predict fluctuation spectra in open models, whether for primeval curvature perturbations (Lyth & Stewart 1990) or for primordial entropy perturbations (Yokoyama & Suto 1991, Dolgov & Silk 1993), are not compelling.

Given the above, it appears that in low-₀ universes, adiabatic CMB fluctuations are larger than their ₀ = 1 counterparts for the same value of H₀ (of course when ₀ < 1 the requirement of small H₀ is relaxed and the anisotropy can be somewhat reduced). The most detailed model of fluctuations in an open universe to date is the BDM mode) discussed in Section 2.2. Analysis of the spectrum produced by this model is complicated, as radiation drag (from the nonzero x_e) can alter the shape, and the integrated Sachs-Wolfe effect is important on large scales in addition to the secondary fluctuations induced on small scales (Gouda et al 1991a, Hu & Sugiyama 1994c). However, on very large scales ( 10), there is an almost model-independent signature, n_eff 2, to the slope of the radiation power spectrum (Sugiyama & Silk 1994). This model appears to be tightly constrained by current observations (Chiba et al 1993, Hu & Sugiyama 1994b).