386 Cosmic microwave background anisotropies
the plateau can be used to determine the primordial spectral amplitude and spectral
index. The accuracy is limited mainly by the cosmic variance and by the fact thatCl
for smalllis a weighted integral over modes that also include wavelengths smaller
than the Hubble scale, the contribution of which depends on other parameters, such
as (^) b,
m,etc. This prevents a determination of the spectral slope to better than
10% accuracy. To improve the accuracy and fix other cosmological parameters, one
must go to smaller angular scales and get information about the acoustic peaks and
other features of the spectrum.
The location of the peaks and the spatial curvature of the universeThe acous-
tic peaks arise when the oscillating termO, given by (9.102), is superimposed
on the “hill” given by the nonoscillating contributionN(l)=N 1 +N 2 +N 3 (see
Figure 9.2). The peak locations and the heights depend on both contributions. The
oscillation peaks inOalone come from the superposition of two cosine terms
in (9.102), whose periods differ by a factor of 2. If|A 1 |A 2 ,the peaks are
located at
ln=π!−^1
(
n−1
8
)
, (9.110)
wheren= 1 , 2 , 3 ...and!is given by (9.94). The first term on the right hand side in
(9.102) has a period twice as large as that of the second term and its amplitudeA 1 is
negative. Therefore, it interferes constructively for the odd peaks (n= 1 , 3 ,...) and
destructively for the even peaks (n= 2 , 4 ,...). Moreover, because of the relative
phase shift of the two cosine terms, their maxima do not coincide and the construc-
tive maxima of their sum lies between the closest maxima of the two individual
terms; that is,
l 1 (
6 ÷ 7
8
)
π!−^1 ,l 3 (
2 +
6 ÷ 7
8
)
π!−^1. (9.111)Here the notation 6÷ 7 ,for instance, denotes a number between 6 and 7.If|A 1 |
A 2 , the peaks move closer to the lower bounds of the intervals in (9.111).
In the concordance model, whereξ 0 .6 and (^) mh^275 0 .26 from (9.94) and
(9.111), we find thatl 1 225 ÷265 andl 3 825 ÷ 865 .The situation is made
more complicated by the nonoscillating contributionN. As is clear from Figure 9.2,
the hill causes the first peak to move towards the right and the third peak to move
towards the left.
The even peaks correspond to the multipoles where two terms in (9.102) de-
structively interfere. The second peak should be located at
l 2
(
1 +
6 ÷ 7
8
)
π!−^1 525 ÷ 565 (9.112)