**C. The Flatness Problem**

The flatness problem is another example of a fine-tuning problem.
The contribution to the critical density by the baryon density,
based on calculations from nucleosynthesis and the observed
abundance of light elements, are in good agreement with
observations and give _{B}
< 0.1. The radiation density
is negligible and it is believed that non-baryonic dark matter, or
quintessence/dark energy (non zero cosmological constant), will
contribute the remainder of the critical density, yielding
= 1. Although, an
anywhere within the range of 1
causes a problem.

The Friedmann equation (28) can be used to take into
account how changes with
time. Noting that *H* =
/*a*
and =
/
_{c}, one can
divide (28) by
*H*^{2} to obtain,

Using the relationships between the scale factor and time,

and using the definition of *H* yields,

From these relations one can see that
must be very
fine-tuned at early times. For example, requiring
to be
one today, corresponds to a value of
| (1) - 1 | ~
10^{-16} at the time of decoupling and a value of
| (10^{-43}) - 1 | ~
10^{-60} at the Planck epoch. This
value seems unnecessarily contrived and indicates that we live at
a very special time in the universe. That is to say, when the
universe happens to be flat. An alternative is that the universe
has been, is, and always will be flat. However, this is a very
special case and it would be nice to have a mechanism that
explains why the universe is flat. The Big Bang offers no such explanation.