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
~ 1/2_{0}
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
(*k*^{2} + *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 *H*_{0} (of course when
_{0} < 1
the requirement of small *H*_{0} 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).