and Cosmology

8. Cosmology III: The Cosmological Parameters
340
Fig. 8.24. Dependence of the CMB
fluctuation spectrum on cosmological parameters.
Plotted is the square root of
the power per logarithmic interval in l,
Δ T = √ l(l + 1)C l /(2π) T 0 . These power
spectra were obtained from an accurate
calculation, taking into account all
the processes previously discussed in
the framework of perturbation theory in
General Relativity. In all cases, the reference
model is defined by Ω m + Ω Λ = 1,
Ω Λ = 0.65, Ω b h 2 = 0.02, Ω m h 2 = 0.147,
and a slope in the primordial density fluctuation
spectrum of n s = 1, corresponding to
the Harrison–Zeldovich spectrum. In each
of the four panels, one of these parameters
is varied, and the other three remain fixed.
The various dependences are discussed in
detail in the main text
tuations where, starting from some reference model,
individual cosmological parameters are varied. First we
note that the spectrum is basically characterized by
three distinct regions in l (or in the angular scale). For
l 100, l(l + 1)C l is a relatively flat function if – as
in the figure – a Harrison–Zeldovich spectrum is assumed.
In the range l 100, local maxima and minima
can be seen that originate from the acoustic oscillations.
For l 2000, the amplitude of the power spectrum is
strongly decreasing due to Silk damping.
Figure 8.24(a) shows the dependence of the power
spectrum on the curvature of the Universe, thus on
Ω tot = Ω m + Ω Λ . We see that the curvature has two
fundamental effects on the spectrum: first, the locations
of the minima and maxima of the Doppler peaks are
shifted, and second, the spectral shape at l 100 depends
strongly on Ω tot . The latter is a consequence of
the integrated Sachs–Wolfe effect because the more the
world model is curved, the stronger the time variations
of the gravitational potential φ. The shift in the acoustic
peaks is essentially a consequence of the change in
the geometry of the Universe: the size of the sound
horizon depends only weakly on the curvature, but
the angular diameter distance D A (z rec ) is a very sensitive
function of this curvature, so that the angular
scale that corresponds to the sound horizon changes
accordingly.
The dependence on the cosmological constant for
flat models is displayed in Fig. 8.24(b). Here one can
see that the effect of Ω Λ on the locations of the acoustic
peaks is comparatively small, so that these basically
depend on the curvature of the Universe. The most important
influence of Ω Λ is seen for small l.ForΩ Λ = 0,
the integrated Sachs–Wolfe effect vanishes and the
power spectrum is flat (for n s = 1), whereas larger Ω Λ
always produce a strong integrated Sachs–Wolfe effect.
The influence of the baryon density is presented in
Fig. 8.24(c). An increase in the baryon density causes
the amplitude of the first Doppler peak to rise, whereas
that of the second peak decreases. In general, the amplitudes
of the odd-numbered Doppler peaks increase,
and those of the even-numbered peaks decrease with
increasing Ω b h 2 . Furthermore, the damping of fluctuations
sets in at smaller l (hence, larger angular scales)