Physical Foundations of Cosmology

390 Cosmic microwave background anisotropies

constant. The dependence of the peak location onh 75 is modest, so a very accu-
rate measurement of the microwave background is required to obtain a reasonable
constraint onh 75. For example, if the locations and heights of the peaks are deter-
mined to 1% accuracy, then the expected accuracy for the Hubble parameter will be
about 7%.

Reconsideration of the spectral tiltUntil now we have been assuming that the pri-
mordial spectrum of inhomogeneities is scale-invariant with spectral indexnS= 1.
Inflation predicts that there should be a small deviation from perfect scale invariance,
typicallynS 0 .92–0. 97 .The above derivation for the microwave background fluc-
tuations can easily be modified to account for these deviations.

Problem 9.14Show that the multipolesClfor a primordial spectrum with tilt
nSare modified by a factor proportional tolnS−^1 compared with a scale-invariant
spectrum.

When we include uncertainty in the spectral tilt, then the heights and locations

of the first two peaks are insufficient to determine (^) b, (^) mandh 75 separately. Here
is where the third acoustic peak comes into play. The height of the third peak is
not as sensitive to (^) bh^275 and (^) mh^275 as the first two peaks, but it is sensitive to the
spectral index. Fixing these parameters and the height of the first peak, the ratio of
the third peak height to the first,r≡H 3 /H 1 , changes by a factor
r
r

∼ 1 −

(

l 3

l 1

) 1 −nS

∼(nS− 1 )ln

(

l 3

l 1

)

. (9.113)

For instance, ifnS 0. 95 ,the height of the third peak decreases by about 7%
compared with the case fornS=1.

SummaryThus we have seen what a powerful tool the microwave background
power spectrum can be. The general shape – a plateau at large angular scales and
acoustic peaks at small angular scales – confirms that the spectrum is predomi-
nantly nearly scale-invariant and adiabatic. (The higher-order correlation functions
should be used to show that the spectrum is also Gaussian.) This supports the basic
predictions of the inflationary/big bang paradigm. Then we can proceed to use the
quantitative details of the spectrum−the plateau and the heights and locations of
the acoustic peaks−to determine the primary cosmological parameters.
Our analysis is valid for a limited parameter set; the inclusion of other physical
effects or variants of the best-fit model weakens to some extent the conclusions
that can be drawn from measurements of the anisotropy. For example, secondary
reionization by early star formation at redshiftsz>20 or so reduces the small
angular scale power in a way that is difficult to disentangle from tilting the spectral