and Cosmology

8.6 Angular Fluctuations of the Cosmic Microwave Background
sity to climb out of a potential well. As a result of
this, they loose energy and are redshifted (gravitational
redshift). This effect is partly compensated for
by the fact that, besides the gravitational redshift,
a gravitational time delay also occurs: a photon that
originates in an overdense region will be scattered at
a slightly earlier time, and thus at a slightly higher
temperature of the Universe, compared to a photon
from a region of average density. Both effects always
occur side by side. They are combined under the term
Sachs–Wolfe effect. Its separation into two processes
is necessary only in a simplified description; a general
relativistic treatment of the Sachs–Wolfe effect
jointly yields both processes.
• We have seen that density fluctuations are always
related to peculiar velocities of matter. Hence, the
electrons that scatter the CMB photons for the
last time do not follow the pure Hubble expansion
but have an additional velocity that is closely
linked to the density fluctuations (compare Sect. 7.6).
This results in a Doppler effect: if photons are
scattered by gas receding from us with a speed
larger than that corresponding to the Hubble expansion,
these photons experience an additional redshift
which reduces the temperature measured in that
direction.
• In regions of a higher dark matter density, the baryon
density is also enhanced. On scales larger than
the horizon scale at recombination (see Sect. 4.5.2),
the distribution of baryons follows that of the
dark matter. On smaller scales, the pressure of the
baryon–photon fluid is effective because, prior to recombination,
these two components had been closely
coupled by Thomson scattering. Baryons are adiabatically
compressed and thus get hotter in regions of
higher baryon density, hence their temperature – and
with it the temperature of the photons coupled to
them – is also larger.
• The coupling of baryons and photons is not perfect
since, owing to the finite mean free path of photons,
the two components are decoupled on small spatial
scales. This implies that on small length-scales, the
temperature fluctuations can be smeared out by the
diffusion of photons. This process is known as Silk
damping, and it implies that on angular scales below
about ∼ 5 ′ , only very small primary fluctuations
exist.
Obviously, the first three of these effects are closely coupled
to each other. In particular, on scales > r H,com (z rec )
the first two effects can partially compensate each other.
Although the energy density of matter is, at recombination,
higher than that of the radiation (see Eq. 4.54), the
energy density in the baryon–photon fluid is dominated
by radiation, so that it is considered a relativistic fluid.
Its speed of sound is thus c s ≈ √ P/ρ ≈ c/ √ 3. The high
pressure of this fluid causes oscillations to occur. The
gravitational potential of the dark matter is the driving
force, and pressure the restoring force. These oscillations,
which can only occur on scales below the sound
horizon at recombination, then lead to adiabatic compression
and peculiar velocities of the baryons, hence
to anisotropies in the background radiation.
Secondary anisotropies result, among other things,
from the following effects:
• Thomson scattering of CMB photons. Since the Universe
is currently transparent for optical photons
(since we are able to observe objects at z > 6), it must
have been reionized between z ∼ 1000 and z ∼ 6,
presumably by radiation from the very first generation
of stars and/or by the first QSOs. After this
reionization, free electrons are available again, which
may then scatter the CMB photons. Since Thomson
scattering is essentially isotropic, the direction
of a photon after scattering is nearly independent
of its incoming direction. This means that scattered
photons no longer carry information about the CMB
temperature fluctuations. Hence, the scattered photons
form an isotropic radiation component whose
temperature is the average CMB temperature. The
main effect resulting from this scattering is a reduction
of the measured temperature anisotropies, by the
fraction of photons which experience such scattering.
• Photons propagating towards us are traversing a Universe
in which structure formation takes place. Due to
this evolution of the large-scale structure, the gravitational
potential is changing over time. If it was
time-independent, photons would enter and leave
a potential well with their frequency being unaffected,
compared to photons that are propagating in
a homogeneous Universe: the blueshift they experience
when falling into a potential well is exactly
balanced by the redshift they attain when climbing
out. However, this “conservation” of photon energy
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