There are three key facts we know about the properties of the
Universe: galaxies recede, there's a faint microwave glow coming from
all directions in the sky, and the Universe is mostly hydrogen and
helium.

In 1929 Edwin Hubble published a claim that the radial velocities of
galaxies are proportional to their distance.  His claim was based on
the measurement of the galaxies' redshifts and estimates of their
distances.  The redshift is a measure of how much the wavelength of a
spectral line has been shifted from the value measured in
laboratories; if assumed to occur because of the Doppler effect, the
redshift of a galaxy is then a measure of its radial velocity.  His
estimates of the galaxies' distances was based on the brightness of a
particular kind of star (a pulsating star known as a Cepheid).

The constant of proportionality in Hubble's relationship (v = H * d,
where v is a velocity and d is a distance) is known as Hubble's
parameter or Hubble's constant.  Hubble's initial estimate was that
the Hubble parameter is 464 km/s/Mpc (in other words, a galaxy 1 Mpc =
3 million light years away would have a velocity of 464 km/s).  We
know now that Hubble didn't realize that there are two kinds of
Cepheid stars.  Various estimates of the Hubble parameter today are
between 50--100 km/s/Mpc.

Hubble also measured the number of galaxies in different directions
and at different brightness in the sky.  He found approximately the
same number of faint galaxies in all directions (though there is a
large excess of bright galaxies in the northern sky).  When a
distribution is the same in all directions, it is isotropic.  When
Hubble looked for galaxies four times fainter than a particular
brightness, he found approximately 8 times more galaxies than he found
that were brighter than this cutoff.  A brightness 4 times smaller
implies a doubled distance.  In turn, doubling the distance means one
is looking into a volume that is 8 times larger.  This result
indicates that the Universe is close to homogeneous or it has a
uniform density on large scales.  (Of course, the Universe is not
really homogeneous and isotropic, because it contains dense regions
like the Earth.  However, if you take a large enough box, you will
find about the same number of galaxies in it, no matter where you
place the box.  So, it's a reasonable approximation to take the
Universe to be homogeneous and isotropic.)  Surveys of very large
regions confirm this tendency toward homogeneity and isotropy on the
scales larger than about 300 million light years.

The case for an isotropic and homogeneous Universe became much
stronger after Penzias & Wilson announced the discovery of the Cosmic
Microwave Background in 1965.  They observed an excess brightness at a
wavelength of 7.5 cm, equivalent to the radiation from a blackbody
with a temperature of 3.7+/-1 degrees Kelvin. (The Kelvin temperature
scale has degrees of the same size as the Celsius scale, but it is
referenced at absolute zero, so the freezing point of water is 273.15
K.)  A blackbody radiator is an object that absorbs any radiation that
hits it and has a constant temperature.  Since then, many astronomers
have measured the intensity of the CMB at different wavelengths.
Currently the best information on the spectrum of the CMB comes from
the FIRAS instrument on the COBE satellite.  The COBE data are
consistent with the radiation from a blackbody with T = 2.728 K.  (In
effect, we're sitting in an oven with a temperature of 2.728 K.)  The
temperature of the CMB is almost the same all over the sky.  Over the
distance from which the CMB travels to us, the Universe must be
exceedingly close to homogeneous and isotropic.  These observations
have been combined into the so-called Cosmological Principle: The
Universe is *homogeneous* and *isotropic*.

If the Universe is expanding---as the recession of galaxies
suggests---and it is at some temperature today, then in the past
galaxies would have been closer together and the Universe would have
been hotter.  If one continues to extrapolate backward in time, one
reaches a time when the temperature would be about that of a star's
interior (millions of degrees; galaxies at this time would have been
so close that they would not retain their form as we see them today).
If the temperature was about that of a star's interior, then fusion
should have been occurring.

The majority of the Universe is hydrogen and helium.  Using the
known rate of expansion of the Universe, one can figure out how long
fusion would have been occurred.  From that one predicts that,
starting with pure hydrogen, about 25% of it would have been fused to
form deuterium (heavy hydrogen), helium (both helium-4 and helium-3),
and lithium; the bulk of the fusion products would helium-4.
Observations of very old stars and very distant gas show that the
abundance of hydrogen and helium is about 75% to 25%.