### 2. THE COSMOLOGICAL NUMBERS

Traditionally, cosmology was the quest for a few numbers. The
first were *H*, *q*, and
. Since 1965 we've had another: the
baryon/photon ratio. This is believed to result from a small
favouritism for matter over antimatter in the early universe -
something that was addressed in the context of ``grand unified
theories'' in the 1970s. (Indeed, baryon non-conservation seems a
prerequisite for any plausible inflationary model. Our entire
observable universe, containing at least 10^{79} baryons, could
not have
inflated from something microscopic if baryon number were strictly
conserved)

In the 1980s non-baryonic matter became almost a natural
expectation, and _{b} /
_{CDM} is another
fundamental number.

Another specially important dimensionless number, Q, tells us how
smooth the universe is. It's measured by

-- The Sachs-Wolfe fluctuations in the microwave background

-- the gravitational binding energy of clusters as a fraction of
their rest mass

-- or by the square of the typical scale of mass- clustering as a
fraction of the Hubble scale.

It's of course oversimplified to represent this by a single number
Q, but insofar as one can, its value is pinned down to be
10^{-5}. (Detailed discussions introduce further numbers: the ratio of
scalar and tensor amplitudes, and quantities such as the ``tilt'', which
measure the deviation from a pure scale-independent Harrison-Zeldovich
spectrum.)

What's crucial is that Q is small. Numbers like and *H*
are only well-defined insofar as the universe possesses ``broad brush''
homogeneity - so that our observational horizon encompasses many
independent patches each big enough to be a fair sample. This wouldn't
be so, and the simple Friedmann models wouldn't be useful
approximations, if Q weren't much less than unity. Q's smallness is
necessary if the universe is to look homogeneous. But it isn't,
strictly speaking, a sufficient condition - a luminous tracer that
didn't weigh much could be correlated on much larger scales without
perturbing the metric. Simple fractal models for the luminous matter
are nonetheless, as Lahav will discuss, strongly constrained by other
observations such as the isotropy of the X-ray background, and of the
radio sources detected in deep surveys.