**4.1. What is the Flatness Problem?**

First I will describe the flatness problem and then the inflationary
solution to it. Recent measurements of the total density of the Universe
find 0.95 <
_{o} <
1.05 (e.g. Table 1).
This near flatness is a problem because the Friedmann Equation tells us that
~ 1 is
a very unstable condition - like a pencil balancing on its point. It is
a very special condition that
won't stay there long. Here is an example of how special it is.
Equation 16 shows us that
(^{-1} -1)
*R*^{2} = *constant*. Therefore, we can write,

(27) |

where the right hand side is today and the left hand side is at any arbitrary time. We then have,

(28) |

Redshift is related to the scale factor by
*R* = *R*_{o} / (1 + *z*).
Consider the evolution during matter-domination where
=
_{o}(1
+ *z*)^{3}. Inserting these we get,

(29) |

Inserting the current limits on the density of the Universe,
0.95 < _{o}
< 1.05 (for which -0.05 <
(_{o}^{-1} -1) < 0.05), we get a
constraint on the possible values that
could have
had at redshift *z*,

(30) |

At recombination (when the first hydrogen atoms were formed)
*z*
10^{3} and the constraint on
yields,

(31) |

So the observation that
0.95 < _{o}
< 1.05 today, means that at a redshift of *z* ~ 10^{3}
we must have had 0.99995 <
< 1.000005.
This range is small...special.
However, had to be
even more special earlier on.
We know that the standard big bang successfully predicts the relative
abundances of the
light nuclei during nucleosynthesis between ~ 1 minute and ~ 3 minutes
after the big bang, so let's consider the slightly earlier time, 1
second after the big bang which is about the beginning of the epoch in
which we are confident that the Friedmann Equation holds. The redshift was
*z* ~ 10^{11} and the resulting constraint on the density
at that time was,

(32) |

This range is even smaller and more special,
(although I have assumed matter domination for this calculation, at
redshifts higher than *z*_{eq} ~ 3000, we have
radiation domination and
=
_{o}(1
+ *z*)^{4}. This makes the 1 + *z* in Eq. 30 a (1 +
*z*)^{2} and requires that early values of
be even closer to 1
than calculated here).

To summarize:

(33) (34) (35) |

If the Friedmann Equation is valid at even higher redshifts,
must have been
even closer to one.
These limits are the mathematical quantification behind our previous
statement that:
`If the Universe had started out with a tiny deviation from flatness,
the standard big bang
model would have quickly generated a measurable degree of non-flatness.'
*If* we assume that
could have started
out with any value, then we have a compelling question:
Why should have
been so fine-tuned to 1?

Observing _{o}
1 today can be
compared to a pencil standing on its point. If you walk into a room and
find a pencil standing on its point you think: pencils don't usually
stand on their points. If a pencil is that way
then some mechanism must have recently set it up because pencils won't
stay that way long. Similarly, if you wake up in a universe
that you know would quickly evolve away from
= 1 and yet you
find that _{o}
= 1 then some mechanism must have balanced it very exactly at
= 1.

Another way to state this flatness problem is as an oldness problem. If
_{o}
1 today, then the
Universe cannot have gone through many e-folds of expansion which would
have driven it away from
_{o} = 1.
It cannot be very old. If the pencil is standing on its end, then
the mechanism to push it up must have just finished.
But we see that the Universe *is* old in the sense that it
*has* gone through many e-foldings of expansion (even without
inflation).

If early values of had exceeded 1 by a tiny amount then this closed Universe would have recollapsed on itself almost immediately. How did the Universe get to be so old? If early values of were less than 1 by a tiny amount then this open Universe would have expanded so quickly that no stars or galaxies would have formed. How did our galaxy get to be so old? The tiniest deviation from = 1 grows quickly into a collapsing universe or one that expands so quickly that clumps have no time to form.