and Cosmology

8.6 Angular Fluctuations of the Cosmic Microwave Background
considering the fact that two directions which are separated
by more than ∼ 1 ◦ have a spatial separation at
recombination which is larger than the horizon size
at that time – so spacetime curvature explicitly plays
a role. Fortunately, the physical phenomena that need
to be accounted for are (nearly) all of a linear nature.
This means that, although the corresponding system
of coupled equations is complicated, it can nevertheless
easily be solved, since the solution of a system of
linear equations is not a difficult mathematical problem.
Generally accessible software packages exist (i.e.,
CMBFAST), which compute the power spectrum C l for
any combination of cosmological parameters.
8.6.3 The Fluctuation Spectrum
Horizon Scale. To explain the basic features of CMB
fluctuations, we first point out that a characteristic
length-scale exists at z rec , namely the horizon length.
It is specified by (4.71). For cosmological models with
Ω Λ = 0, the horizon spans an angle of – see (4.72) –
θ H,rec ≈ 1.8 ◦ √ Ω m .
This angle is modified for models with a cosmological
constant; if the Universe is flat (Ω m + Ω Λ = 1), one
finds
θ H,rec ≈ 1.8 ◦ , (8.24)
with a very weak dependence on the matter density,
about ∝ Ωm
−0.1 . As we will demonstrate in the following,
this angular scale of the horizon is directly observable.
Fluctuations on Large Scales. On scales ≫ θ H,rec the
Sachs–Wolfe effect dominates, since oscillations in the
baryon–photon fluid can occur only on scales below
the horizon length. For this reason, the CMB angular
spectrum directly reflects the fluctuation spectrum P(k)
of matter. In particular, for a Harrison–Zeldovich
spectrum, P(k) ∝ k one expects that
l(l + 1)C l ≈ const for l ≪ 180◦ ≃ 100 ,
θ H,rec
and the amplitude of the fluctuations immediately yields
the amplitude of P(k). This flat behavior of the fluctuation
spectrum for n s = 1 is modified by the integrated
Sachs–Wolfe effect.
Sound Horizon and Acoustic Peaks. On angular scales