Cosmic background fluctuations 97There were thus strong grounds to expect that large-scale fluctuations would be
present at about the 10−^5 level, and it was a significant boost to the credibility of
the gravitational-instability model that such fluctuations were eventually seen.
In more detail, it is possible to relate the COBE anisotropy to the large-scale
portion of the power spectrum. G ́orskiet al(1995), Bunnet al(1995), and White
and Bunn (1995) discuss the large-scale normalization from the two-year COBE
data in the context of CDM-like models. The final four-year COBE data favour
very slightly lower results, and we scale to these in what follows. For scale-
invariant spectra and=1, the best normalization is
COBE⇒^2 (k)=(
k
0. 0737 hMpc−^1) 4
Translated into other common notation for the normalization, this is equivalent to
Qrms−ps= 18. 0 μK, orδH= 2. 05 × 10 −^5 (see e.g. Peacock and Dodds 1994).
For low-density models, the earlier discussion suggests that the power
spectrum should depend onand the growth factorgasP∝g^2 /^2. Because of
the time dependence of the gravitational potential (integrated Sachs–Wolfe effect)
and because of spatial curvature, this expression is not exact, although it captures
the main effect. From the data of White and Bunn (1995), a better approximation
is
^2 (k)∝g^2
^2g^0.^7.This applies for low-models both with and without vacuum energy, with a
maximum error of 2% in density fluctuation provided> 0 .2. Since the
rough power-law dependence ofgisg() ^0.^65 and^0.^23 for open and
flat models respectively, we see that the implied density fluctuation amplitude
scales approximately as−^0.^12 and−^0.^69 respectively for these two cases. The
dependence is weak for open models, but vacuum energy implies much larger
fluctuations for low.
Within the CDM model, it is always possible to satisfy both the large-scale
COBE normalization and the small-scaleσ 8 constraint, by appropriate choice of
andn. This is illustrated in figure 2.20. Note that the vacuum energy affects the
answer; for reasonable values ofhand reasonable baryon content, flat models
requirem 0 .3, whereas open models requirem 0 .5inordertobe
consistent with scale-invariant primordial fluctuations.
2.8.6 Geometrical degeneracy
The statistics of CMB fluctuations depend on a large number of parameters, and
it can be difficult to understand what the effect of changing each one will be.
Furthermore, the effects of some parameters tend to change things in opposite
directions, so that there are degenerate directions in the parameter space, along
which changes leave the CMB unaffected. These were analysed comprehensively
by Efstathiou and Bond (1999), and we now summarize the main results.