The minute temperature fluctuations present in the microwave background contain a wealth of information about the fundamental properties of the universe. In order to understand the reasons for this and the kinds of information available, an appreciation of the underlying physical processes generating temperature and polarization fluctuations is required. This section and the following one give a general description of all basic physics processes involved in producing microwave background fluctuations.

First, one practical matter. Throughout these lectures, common
cosmological units will be employed in which
= *c* =
*k*_{b} = 1. All dimensionful quantities can then
be expressed as powers of an energy scale, commonly taken as GeV. In
particular, length and time both have units of [GeV]^{-1}, while
Newton's constant *G* has units of [GeV]^{-2} since it is
defined as
equal to the square of the inverse Planck mass. These units are very
convenient for cosmology, because many problems deal with widely
varying scales simultaneously. For example, any computation of relic
particle abundances (e.g. primordial nucleosynthesis) involves both a
quantum mechanical scale (the interaction cross-section) and a
cosmological scale (the time scale for the expansion of the
universe). Conversion between these cosmological units and physical
(cgs) units can be achieved by inserting needed factors of
, *c*,
and *k*_{b}. The standard textbook by
Kolb and Turner
(1990)
contains an extremely useful appendix on units.

**3.1. Causes of temperature fluctuations**

Blackbody radiation in a perfectly homogeneous and isotropic universe, which is always adopted as a zeroth-order approximation, must be at a uniform temperature, by assumption. When perturbations are introduced, three elementary physical processes can produce a shift in the apparent blackbody temperature of the radiation emitted from a particular point in space. All temperature fluctuations in the microwave background are due to one of the following three effects.

The first is simply a change in the intrinsic temperature of the radiation at a given point in space. This will occur if the radiation density increases via adiabatic compression, just as with the behavior of an ideal gas. The fractional temperature perturbation in the radiation just equals the fractional density perturbation.

The second is equally simple: a doppler shift if the radiation at
a particular point is moving with respect to the
observer. Any density
perturbations within the horizon scale will necessarily be accompanied
by velocity perturbations. The induced temperature perturbation in
the radiation equals the peculiar velocity (in units of *c*, of
course), with motion towards the observer corresponding to a positive
temperature perturbation.

The third is a bit more subtle: a difference in gravitational potential between a particular point in space and an observer will result in a temperature shift of the radiation propagating between the point and the observer due to gravitational redshifting. This is known as the Sachs-Wolfe effect, after the original paper describing it (Sachs and Wolfe, 1967). This paper contains a completely straightforward general relativistic calculation of the effect, but the details are lengthy and complicated. A far simpler and more intuitive derivation has been given by Hu and White (1997) making use of gauge transformations. The Sachs-Wolfe effect is often broken into two pieces, the usual effect and the so-called Integrated Sachs-Wolfe effect. The latter arises when gravitational potentials are evolving with time: radiation propagates into a potential well, gaining energy and blueshifting in the process. As it climbs out, it loses energy and redshifts, but if the depth of the potential well has increased during the time the radiation propagates through it, the redshift on exiting will be larger than the blueshift on entering, and the radiation will gain a net redshift, appearing cooler than it started out. Gravitational potentials remain constant in time in a matter-dominated universe, so to the extent the universe is matter dominated during the time the microwave background radiation freely propagates, the Integrated Sachs-Wolfe effect is zero. In models with significantly less than critical density in matter (i.e. the currently popular CDM models), the redshift of matter-radiation equality occurs late enough that the gravitational potentials are still evolving significantly when the microwave background radiation decouples, leading to a non-negligible Integrated Sachs-Wolfe effect. The same situation also occurs at late times in these models; gravitational potentials begin to evolve again as the universe makes a transition from matter domination to either vacuum energy domination or a significantly curved background spatial metric, giving an additional Integrated Sachs-Wolfe contribution.

The early universe at the epoch when the microwave background
radiation begins propagating freely, around a redshift of *z* = 1100, is
a conceptually simple place. Its constituents are ``baryons'' (including
protons, helium nuclei, and electrons, even though electrons are not
baryons), neutrinos, photons, and dark
matter particles. The neutrinos
and dark matter can be treated as interacting only gravitationally
since their weak interaction cross-sections are too small at this
energy scale to be dynamically or thermodynamically relevant. The
photons and baryons interact electromagnetically, primarily via
Compton scattering of the radiation from the electrons.
The typical interaction energies are low enough for the
scattering to be well-approximated by the simple Thomson cross
section. All other scattering processes (e.g. Thomson scattering from
protons, Rayleigh scattering of radiation from neutral hydrogen) have
small enough cross-sections to be insignificant, so we have four
species of matter with only one relevant (and simple) interaction
process among them. The universe is also very close to being
homogeneous and isotropic, with small perturbations in density and
velocity on the order of a part in 10^{5}. The tiny size of the
perturbations guarantees that linear perturbation theory
around a homogeneous and isotropic
background universe will be an excellent approximation.

Conceptually, the formal description of the universe at this epoch is
quite simple. The unperturbed background cosmology is described by the
Friedmann-Robertson-Walker (FRW) metric, and the evolution of the
cosmological scale factor *a*(*t*) in this metric is given by the
Friedmann equation (see the lectures by Peacock in this volume). The
evolution of the free electron density *n*_{e} is
determined by the
detailed atomic physics describing the recombination of neutral
hydrogen and helium; see
Seager *et
al.* (2000)
for a detailed
discussion. At a temperature of around 0.5 eV, the electrons combine
with the protons and helium nuclei to make neutral atoms. As a result,
the photons cease Thomson scattering and propagate freely to us. The
microwave background is essentially an image of the ``surface of last
scattering''. Recombination must be calculated quite precisely because
the temperature and thickness of this surface depend sensitively on
the ionization history through the recombination process.

The evolution of first-order perturbations in the various energy density components and the metric are described with the following sets of equations:

- The photons and neutrinos are described by distribution
functions
*f*(**x**,**p**,*t*). A fundamental simplifying assumption is that the energy dependence of both is given by the blackbody distribution. The space dependence is generally Fourier transformed, so the distribution functions can be written as (**k**, ,*t*), where the function has been normalized to the temperature of the blackbody distribution and represents the direction in which the radiation propagates. The time evolution of each is given by the Boltzmann equation. For neutrinos, collisions are unimportant so the Boltzmann collision term on the right side is zero; for photons, Thomson scattering off electrons must be included. - The dark matter and baryons are in principle described by
Boltzmann equations as well, but a fluid description incorporating
only the lowest two velocity moments of the distribution functions is
adequate. Thus each is described by the Euler and continuity equations
for their densities and velocities. The baryon Euler equation must
include the coupling to photons via Thomson scattering.
- Metric perturbation evolution and the connection of the metric
perturbations to the matter perturbations are both contained in the
Einstein equations. This is where the subtleties arise. A general
metric perturbation has 10 degrees of freedom, but four of these are
unphysical gauge modes. The physical perturbations include two degrees
of freedom constructed from scalar functions, two from a vector, and
two remaining tensor perturbations
(Mukhanov
*et al.*1992). Physically, the scalar perturbations correspond to gravitational potential and anisotropic stress perturbations; the vector perturbations correspond to vorticity and shear perturbations; and the tensor perturbations are two polarizations of gravitational radiation. Tensor and vector perturbations do not couple to matter evolving only under gravitation; in the absence of a ``stiff source'' of stress energy, like cosmic defects or magnetic fields, the tensor and vector perturbations decouple from the linear perturbations in the matter.

A variety of different variable choices and methods for eliminating
the gauge freedom have been developed. The subject can be fairly
complicated. A detailed discussion and comparison between the
Newtonian and synchronous gauges, along with a complete set of
equations, can be found in
Ma and Bertschinger
(1995);
also see
Hu *et al.* (1998).
An elegant and physically appealing formalism based on an
entirely covariant and gauge-invariant description of all physical
quantities has been developed for the microwave background by
Challinor and
Lasenby (1999)
and Gebbie *et
al.* (2000),
based on earlier work by
Ehlers (1993) and
Ellis and Bruni
(1989).
A more conventional
gauge-invariant approach was originated by
Bardeen (1980) and
developed by
Kodama and Sasaki
(1984).

The Boltzmann equations are partial differential equations, which can be converted to hierarchies of ordinary differential equations by expanding their directional dependence in Legendre polynomials. The result is a large set of coupled, first-order linear ordinary differential equations which form a well-posed initial value problem. Initial conditions must be specified. Generally they are taken to be so-called adiabatic perturbations: initial curvature perturbations with equal fractional perturbations in each matter species. Such perturbations arise naturally from the simplest inflationary scenarios. Alternatively, isocurvature perturbations can also be considered: these initial conditions have fractional density perturbations in two or more matter species whose total spatial curvature perturbation cancels. The issue of numerically determining initial conditions is discussed below in Sec. 5.2.

The set of equations are numerically stiff before last scattering,
since they contain the two widely discrepant time scales: the
Thomson scattering time for electrons and photons, and the (much
longer) Hubble time. Initial conditions must be set with high accuracy
and an appropriate stiff integrator must be employed.
A variety of numerical techniques have been developed for evolving the
equations. Particularly important is the line-of-sight algorithm first
developed by
Seljak and
Zaldarriaga (1996)
and then implemented by
them in the publicly-available CMBFAST code

(see http://www.sns.ias.edu/~matiasz/CMBFAST/cmbfast.html).

The above discussion is intentionally heuristic and somewhat vague because many of the issues involved are technical and not particularly illuminating. My main point is an appreciation for the detailed and precise physics which goes into computing microwave background fluctuations. However, all of this formalism should not obscure several basic physical processes which determine the ultimate form of the fluctuations. A widespread understanding of most of the physical processes detailed below followed from a seminal paper by Hu and Sugiyama (1996), a classic of the microwave background literature.

Two basic time scales enter into the evolution of the microwave
background. The first is the photon scattering time scale
*t*_{s}, the
mean time between Thomson scatterings. The other is the expansion
time scale of the universe, *H*^{-1}, where
*H* = / *a* is the
Hubble parameter. At temperatures significantly greater than 0.5 eV,
hydrogen and helium are completely ionized and
*t*_{s} << *H*^{-1}. The
Thomson scatterings which couple the electrons and photons occur much
more rapidly than the expansion of the universe; as a result, the
baryons and photons behave as a single ``tightly coupled''
fluid. During this period, the fluctuations in the photons mirror the
fluctuations in the baryons. (Note that recombination occurs at around
0.5 eV rather than 13.6 eV because of the huge photon-baryon ratio;
the universe contains somewhere around 10^{9} photons for each baryon,
as we know from primordial nucleosynthesis. It is a useful exercise to
work out the approximate recombination temperature.)

The photon distribution function for scalar perturbations can be
written as
(**k**, *µ*,
*t*) where
*µ* =
**.** and the scalar
character of the fluctuations guarantees
the distribution cannot have any azimuthal directional dependence.
(The azimuthal dependence for vector and tensor perturbations can also
be included in a similar decomposition). The moments of the
distribution are defined as

(1) |

sometimes other normalizations are used. Tight coupling implies that
_{l} = 0 for *l*
> 1. Physically, the *l* = 0 moment corresponds
to the photon energy density perturbation, while *l* = 1
corresponds to the bulk
velocity. During tight coupling, these two moments must match the
baryon density and velocity perturbations. Any higher moments rapidly
decay due to the isotropizing effect of Thomson scattering; this
follows immediately from the photon Boltzmann equation.

In the other regime, for temperatures significantly lower than 0.5 eV,
*t*_{s} >> *H*^{-1} and photons on average
never scatter again until the
present time. This is known as the ``free streaming'' epoch. Since the
radiation is no longer tightly coupled to the electrons, all higher
moments in the radiation field develop as the photons propagate. In a
flat background spacetime, the exact solution is simple to
derive. After scattering ceases, the photons evolve according to the
Liouville equation

(2) |

with the trivial solution

(3) |

where we have converted to conformal time defined by
*d* = *dt*
/ *a*(*t*) and
_{*}
corresponds to the time at which free streaming begins.
Taking moments of both sides results in

(4) |

with *j*_{l} a spherical Bessel function. The process of
free streaming
essentially maps spatial variations in the photon distribution at the
last scattering surface (wavenumber *k*) into angular variations on
the sky today (moment *l*).

In the intermediate regime during recombination,
*t*_{s}
*H*^{-1}. Photons propagate a characteristic distance
*L*_{D} during this
time. Since some scattering is still occurring, baryons experience a
drag from the photons as long as the ionization fraction is
appreciable. A second-order perturbation analysis shows that the
result is damping of baryon fluctuations on scales below
*L*_{D}, known
as Silk damping or diffusion damping. This effect can be modelled
by the replacement

(5) |

although detailed calculations are needed to define *L*_{D}
precisely. As a result of this damping, microwave background
fluctuations are exponentially suppressed on angular scales
significantly smaller than a degree.

**3.6. The resulting power spectrum**

The fluctuations in the universe are assumed to arise from
some random statistical process. We are not interested in the
exact pattern of fluctuations we see from our vantage point,
since this is only a single realization of the process. Rather,
a theory of cosmology predicts an underlying distribution, of
which our visible sky is a single statistical realization.
The most basic statistic describing fluctuations is their
power spectrum. A temperature map on the sky
*T*()
is conventionally expanded in spherical harmonics,

(6) |

where

(7) |

are the temperature multipole coefficients and *T*_{0} is the
mean CMB temperature.
The *l* = 1 term in Eq. (6) is
indistinguishable from the kinematic dipole and is normally ignored.
The temperature angular power spectrum *C*_{l} is then given by

(8) |

where the angled brackets represent an average over statistical
realizations of the underlying distribution. Since we have only a
single sky to observe, an unbiased estimator of *C*_{l} is
constructed as

(9) |

The statistical uncertainty in estimating
*C*^{T}_{l} by a sum of 2*l* + 1
terms is known as ``cosmic variance''.
The constraints *l* = *l'* and *m* = *m'* follow from the
assumption of statistical isotropy:
*C*^{T}_{l} must be independent of the
orientation of the coordinate system used for the harmonic
expansion. These conditions can be verified via an explicit rotation
of the coordinate system.

A given cosmological theory will predict
*C*^{T}_{l} as a function
of *l*, which can be obtained from evolving the temperature
distribution function as described above. This prediction can then be
compared with data from measured temperature differences on the sky.
Figure 1 shows a typical temperature
power spectrum from the inflationary class of
models, described in more detail below. The distinctive sequence of
peaks arise from coherent acoustic oscillations in the fluid during
the tight coupling epoch and are of great importance in precision
tests of cosmological models; these peaks will be discussed in
Sec. 5.
The effect of diffusion damping is clearly visible in the decreasing
power above *l* = 1000. When viewing angular power spectrum plots in
multipole space, keep in mind that *l* = 200 corresponds approximately
to fluctuations on angular scales of a degree, and the angular scale
is inversely proportional to *l*. The vertical axis is conventionally
plotted as
*l* (*l* + 1)*C*^{T}_{l} because the
Sachs-Wolfe temperature
fluctuations from a scale-invariant spectrum of density perturbations
appears as a horizontal line on such a plot.