**4.4. How big is a causally connected patch of the CMB without and with
inflation?**

From Fig. 4 we can read off
the x axis that the comoving radius of the
base of the small light cone under points **A** or **B** is
*r* = *R*_{o}
~ billion light years.
This is the current size of the patch that was causally connected at
last scattering.
The physical size *D* of the particle horizon today is
*D* 47 billion
light years (Fig. 4).
The fraction *f* of the sky occupied by one causally connected
patch is *f* =
*r*^{2} / 4
*D*^{2}
1/9000. The area of the full sky is about 40, 000 square
degrees (4 steradians). The
area of a causally connected patch is
(*area of the sky*) × *f* = 40,000 / 9,000
4 square degrees.

With inflation, the size of the causally connected patch depends on how
many e-foldings of expansion occurred
during inflation. To solve the horizon problem we need a minimum of ~ 60
e-folds of expansion or an expansion by a factor of
~ 10^{30}. But since this is only a minimum, the full size of a
causally connected patch, although
bigger than the observable universe, will never be known unless it
happens to be between 47 Glyr (our currentm
particle horizon) and 62 Glyr (the comoving size of our particle horizon
at the end of time).

The constraint on the lower limit to the number of e-foldings ~ 60
(or ~ 10^{30} ) comes from the requirement to solve the horizon
problem.
What about the upper limit to the number of e-folds? How big is our
inflationary bubble?
How big the inflationary patch is depends sensitively on
when inflation happened, the height of the inflaton potential and how long
inflation lasted (*t*_{i}, *t*_{e} and
_{inf} at
Eq. 26) -
which in turn depends on the decay rate of the false vacuum.
Without a proper GUT, these numbers cannot be approximated with any
confidence. It is certainly reasonable to expect homogeneity to continue
for some distance beyond our
observable universe but there does not seem to be any reason why it
should go on forever. In eternal inflation models, the homogeneity
definitely does not go on forever
(Liddle & Lyth 2000).

When could inflation have occurred?
The earliest time is the Planck time at 10^{19} GeV or
10^{-43} seconds.
The latest is at the electroweak symmetry breaking at 10^{2} GeV
or 10^{-12} seconds.
The GUT scale is a favorite time at 10^{16} GeV or
10^{-35} seconds.
"Beyond these limits very little can be said for certain about
inflation. So most papers
about inflationary models are more like historical novels than real
history, and they
describe possible interactions that would be interesting instead of
interactions that have
to occur. As a result, inflation is usually described as the
inflationary scenario instead
of a theory or hypothesis. However, it seems quite likely that the
inflation did occur, even
though we don't know when, or what the potential was." - Wright (2003).