The cosmological parameters affect the anisotropies through the well
understood physics of the evolution of linear perturbations within a
background FRW cosmology.
There are very effective, fast, and publicly-available software codes
for computing the CMB anisotropy, polarization, and matter power spectra,
*e.g.*, `CMBFAST`
[14]
and `CAMB`
[15].
`CMBFAST` is the most extensively used code;
it has been tested over a wide range of
cosmological parameters and is considered to be accurate to better than
the 1% level
[16].

A description of the physics underlying the
*C*_{}s
can be separated into 3 main regions, as shown in
Fig. 1.

**4.1. The Sachs-Wolfe plateau:
100**

The horizon scale (or more precisely, the angle subtended by the Hubble
radius) at last scattering corresponds to
100.
Anisotropies at larger scales have not evolved significantly, and hence
directly reflect the `initial conditions.' The combination of
gravitational redshift and intrinsic temperature fluctuations leads to
*T* / *T*
(1/3)
/
*c*^{2}, where
is the
perturbation to the gravitational potential. This is usually referred to as
the `Sachs-Wolfe' effect
[17].

Assuming that a nearly scale-invariant spectrum of density perturbations
was laid down at early times (*i.e.*, *n*
1, meaning equal power
per decade in *k*), then
( + 1)
*C*_{}
constant
at low s. This effect is
hard to see unless the multipole axis is plotted logarithmically (as in
Fig. 1, but not
Fig. 2).

Time variation in the potentials (*i.e.*, time-dependent metric
perturbations) leads to an upturn in the
*C*_{}s
in the lowest several multipoles; any deviation from a total equation of
state *w* = 0 has such an effect.
So the dominance of the Dark Energy at low redshift makes the lowest
s
rise above the plateau. This is sometimes called the `integrated
Sachs-Wolfe effect' (or ISW Rise),
since it comes from the line integral of
.
It has been confirmed through correlations between the large-angle
anisotropies and large-scale structure
[18].
Specific models can also give additional contributions at low
(*e.g.*, perturbations in the Dark Energy component itself
[19])
but typically these are buried in the cosmic variance.

In principle, the mechanism that produces primordial perturbations would
generate scalar, vector, and tensor modes. However, the vector (vorticity)
modes decay with the expansion of the Universe. Tensors also decay when
they enter the horizon, and so they contribute only to angular scales above
about 1° (see Fig. 1).
Hence some fraction of the low
signal could be
due to a gravity wave contribution, although small amounts of tensors are
essentially
impossible to discriminate from other effects that might raise the level
of the plateau. However the tensors *can* be distinguished
using polarization information (section 6).

**4.2. The acoustic peaks: 100
1000**

On sub-degree scales, the rich structure in the anisotropy spectrum is the consequence of gravity-driven acoustic oscillations occurring before the atoms in the universe became neutral. Perturbations inside the horizon at last scattering have been able to evolve causally and produce anisotropy at the last scattering epoch which reflects that evolution. The frozen-in phases of these sound waves imprint a dependence on the cosmological parameters, which gives CMB anisotropies their great constraining power.

The underlying physics can be understood as follows.
When the proton-electron plasma was tightly coupled to the photons, these
components behaved as a single `photon-baryon fluid', with the photons
providing most of the pressure and the baryons the inertia. Perturbations
in the gravitational potential, dominated by the dark matter component,
are steadily evolving. They drive oscillations in the photon-baryon fluid,
with photon pressure providing the restoring force. The
perturbations are quite small,
O(10^{-5}), and so evolve linearly. That means each
Fourier mode evolves independently and is described by a driven
harmonic oscillator, with frequency
determined by the sound speed in the fluid. Thus, there is an oscillation
of the fluid density, with velocity
/ 2 out of phase and having
amplitude reduced by the sound speed.

After the Universe recombined the baryons and radiation decoupled, and the
radiation could travel freely towards us. At that point the phases of the
oscillations were frozen-in, and projected on the sky as a harmonic
series of peaks. The main peak is the mode that went through 1/4 of
a period, reaching maximal compression. The even peaks are maximal
*under*-densities, which are generally of smaller amplitude because the
rebound has to fight against the baryon inertia. The troughs, which do
not extend to zero power, are partially filled because they are
at the velocity maxima.

An additional effect comes from geometrical projection.
The scale associated with the peaks is the sound horizon at last
scattering, which can be confidently calculated as a physical length scale.
This scale is projected onto the sky, leading to an angular
scale that depends on the background cosmology. Hence the angular
position of the peaks is a sensitive probe of the spatial curvature of the
Universe (*i.e.*,
_{tot}),
with the peaks lying at higher
in open universes and
lower in closed geometry.

One last effect arises from reionization at redshift *z*_{i}.
A fraction of photons will be isotropically scattered at
*z* < *z*_{i},
partially erasing the anisotropies at angular scales smaller than
those subtended by the Hubble radius at *z*_{i}. This
corresponds typically to s
above about a few 10s, depending on the specific
reionization model. The acoustic peaks are
therefore reduced by a factor
*e*^{-2}
relative to the plateau.

These acoustic peaks were a clear theoretical prediction going back to about 1970 [20]. Their empirical existence started to become clear around 1994 [21], and the emergence, over the following decade, of a coherent series of acoustic peaks and troughs is a triumph of modern cosmology. One can think of these peaks as a snapshot of stochastic standing waves. And, since the physics governing them is simple, then one can see how they encode information about the cosmological parameters.

The recombination process is not instantaneous, giving a thickness to the
last scattering surface. This leads to a damping of the anisotropies at the
highest s, corresponding to
scales smaller than that subtended
by this thickness. One can also think of the photon-baryon fluid as having
imperfect coupling, so that there is diffusion between the two components,
and the oscillations have amplitudes that decrease with time.
These effects lead to a damping of the
*C*_{}s, sometimes called Silk damping
[22],
which cuts off the anisotropies at multipoles above about 2000.

An extra effect at high s
comes from gravitational lensing, caused
mainly by non-linear structures at low redshift. The
*C*_{}s are
convolved with a smoothing function in a calculable way, partially
flattening the peaks, generating a power-law tail at the highest
multipoles, and complicating the polarization signal
[23].
This is an example of a `secondary effect', *i.e.*, the processing
of anisotropies
due to relatively nearby structures. Galaxies and clusters of galaxies
give several such effects, but all are expected to be of low amplitude
and are typically only important for the highest
s.