Detailed analysis of the microwave background radiation
gives practically our only direct evidence of conditions before
galaxy formation. The spectral shape and
departures from perfect
isotropies give (temporal and angular) evidence on the existence of
structures at the epoch of recombination, which is the last
view we currently have until galaxies and AGN begin to take shape
at some epoch like *z*=10 (or whatever it is...). Two very useful
references for much of the material in these final lectures are
Peacock's *Cosmological Physics* (Cambridge 1999) and *Galaxy
Formation* by Longair (Springer 1998). The existence of
a recombination epoch happens because the densities of matter
and radiation have different dependences on the cosmic scale factor
(1/*R*³ versus 1/*R*^{4}).

The best-known maps of fluctuations in the background are from the COBE Differential Microwave Radiometer (DMR). These images illustrate the four-year data products as measured at 53 GHz, in galactic coordinates with the galactic center in the middle. At the top is the overall background temperature (blank to an excellent approximation), then in the middle we see the result at a magnified stretch with the mean value removed. This shows the dipole interpreted as a Doppler shift due to motion of the Milky Way with respect to the sphere of matter now seen radiating in the background, and some of the residual contamination from the Galactic foreground. Finally at the bottom we see the all-sky fluctuations with the dipole removed (note, by the way, that we cannot separate any intrinsic dipole structure in the CMBR from effects of our relative motion). The individual fluctuations at high galactic latitudes are now significant (which was an improvement over the initial announcements based on 2-year maps from the data). The Tenerife and South Pole experiments also confirmed individual structures. A further advance and unification of structures on various scales came with the first maps from the Wilkinson Microwave Anisotropy Probe (WMAP), shown below the COBE results.

There are several distinct origins for fluctuations in the CMBR,
telling us about various properties of the baryonic Universe and
important on various scales. Straight density fluctuations will
control just when a given volume element recombines through the
*n*² dependence of recombination rate. Density inhomogeneities
will translate to
temperature fluctuations, since denser regions have shorter
recombination times and thus decouple earlier (at higher redshift
and thus appearing dimmer = colder). In these so-called adiabatic
fluctuations, the temperature amplitude we see is 1/3 the density
amplitude (which traces back to photon density varying as *T*³).
A competing effect, which turns out to be negligible on small scales
compared to density fluctuations but probably dominant on larger
scales, is the *Sachs-Wolfe effect*, basically the gravitational
redshift on photons escaping from denser regions. The fluctuation
amplitude depends on the potential
Φ: &delta T/T = δ Φ
>/3c².
In addition, velocity structure can introduce Doppler fluctuations
in observed temperature, of magnitude
δT/T =
δ
*v* . *r* / c.
The fluctuation amplitudes are always very small, so that linear
perturbation theory can be used.

The horizon size at the last-scattering surface, important not only
for big questions of causal connectivity but in the growth of
perturbations, is 184
(Ω h²)
^{-1/2} Mpc. The spectrum of fluctuations (usually considered
as a power spectrum in wavenember *l* of, for example, Bessel functions
on the sphere) is highly diagnostic of processes in the early
Universe, which has driven much of the work on improving maps
of structure in the CMBR. We expect a Sachs-Wolfe "plateau" for small
values of *l*, leading to an acoustic (or Doppler) peak. The
location of this peak is sensitive to both
Ω
and Λ,
while its amplitude is sensitive almost purely to
Ω.
Models also predict that harmonics of this peak should be seen to
higher values of *l*. These may be explored with the publicly
released **CMBFAST** code (see Seljak and Zaldarriaga 1996 ApJ 469, 437,
more recently updated to include both closed and open geometries,
with the interesting difference that only spatially wrapped spherical
wavenumbers are allowed in the closed case). This typical prediction
was taken from the
MAP WWW site,
which includes further versions showing the effects of varying
cosmological parameters:

The really new aspect of the WMAP results is in covering a wide range of
angualar scales at high sgnal-to-noise ratio in a single experiment,
dramatically reducing the potential role of systematic errors.
(In fact, for low values of *l*, one becvomes limited now by cosmic
variance - the fact that we have only one slice of the early
Universer to observe, and we must work
with whatever it happens to present). In deriving
cosmological quantities, one usually niormalizes the power to what we see
locally (say on 100-Mpc scales). The total density parameter
Ω
_{tot} shifts the location of the first (strongest)
peak to smaller *l* for higher values; this is a curvature measurement
giving the relative scales between then and now. The baryon density
by itself changes the peak ratios of odd and even peaks (through dissipation
in compressive phases of the acoustic motions). Large values of the
cosmological constant
Λ
move the first peak to higher *l* and lower amplitude, while
both baryon density and H_{0} affect the height of the peaks.
A helpful way to see why such effects happen is to note that the Universe
at recombination was pervaded by a field of velocity perturbations
transmitted acoustically (now *that* was different). We see this
field slices with the redshift width of the time it took to complete
recombination, so smaller-scale fluctuations will be damped out. The
peaks and valleys tell what scales had the right timing to constructively
or destructively interfere, since for a given wavnumber *k*
the phase follows *e*^{i s k t} where *s*
is the sound speed.

While brief, the epoch of recombination did have a nonzero
duration, which translates into a finite thickness in redshift.
As set out by Jones and Wyse (1985 A&A 149, 144), the cosmological
dependences cancel almost perfectly giving an optical-depth
function
τ(*z*) =
0.37 (*z*/1000)^{14.25}. Thus the distribution of
last-scattering redshift
(specifically
*e*^{-τ
} *dτ/dz*)
is strongly peaked, almost Gaussian with
mean of *z*=1065 and standard deviation
σ_{z}
= 80. (WMAP data give *z*_{rec} = 1089
± 97).
This will modify the spectrum slightly with respect to a perfect
single-temperature blackbody, and will be manifested by damping
out irregularities due to structure which is smaller than the
comoving depth of this apparent shell. Details of the recombination
process have been recently re-examined by Seager, Sasselov, and Scott (2000,
ApJS 128, 431) using full radiative
transfer codes and sophisticated models for all the relevant ions, who found
that Peebles (1968 ApJ 153, 1) and Zel'dovich et al. (1968 JETP Lett. 28, 146)
did remarkably
well in reducing the problem to approximate differental equations.
Further details exist, but at about the level of uncertainty in some of the
relevant reaction rates. In evaluating the spectrum of the background
radiation, it is noteworthy (see the treatment by Peacock, for example) that
the thermal form of the radiation spectrum was established when
brehmsstrahlung was active at redshift of order 10^{6} and
no major additional input between then and recombination is allowed
by the existing limits on departures from a Planck form.

Further developments should come fast, with NASA's Wilkinson Microwave Anisotropy Probe (WMAP) still operating in the L2 region and and ESA's Planck in the pipeline. These should yield significant measurements of high-order harmonics in the fluctuation spectrum sufficient to measure the basic cosmological parameters by themselves.

Last changes: 11/2009 © 2000-9