**3.1. Spatial flatness**

Of the listed properties, spatial flatness is the only one which
refers to the global properties of the Universe.
^{(1)}
It is particularly
pertinent because of the original strong statements that spatial
flatness was an inevitable prediction of inflation, later retracted
with the discovery of a class of models - the open inflation models
[8,
9]
- which cunningly utilize quantum tunnelling to generate homogeneous
open Universes. In the recent `tunnelling from nothing' instanton models
of Hawking and Turok
[9],
any observed curvature has the interesting
interpretation of being a relic from the initial formation of the
Universe which managed to survive the inflationary epoch.

If we convince ourselves that, to a high degree of accuracy, the
Universe is spatially flat, that will strengthen the likelihood that the
simplest models of inflation are correct. However, an accurate
measurement of the curvature is not a *test* of the full
inflationary paradigm, because whatever the outcome of such a
measurement there do remain inflation models which make that prediction.
This point has recently been stressed by Peebles
[7].
The likelihood will
have shifted to favour some inflation models at the expense of others,
but the total likelihood of inflation will be
unchanged. ^{(2)} Only if a
rival class of theories can be invented, which predict say a
negative-curvature Universe in a way regarded as more compelling than
the open inflation models, will measurements of the curvature acquire
the power to test the inflationary paradigm.

I should also mention that the standard definition of inflation - a
period where the scale factor *a*(*t*) undergoes accelerated
expansion
- is a rather general one, and in particular any classical solution
to the flatness problem using general relativity must involve
inflation. This follows directly from writing the Friedmann equation
as

An example is the pre big bang cosmology [11], which is now viewed as a novel type of inflation model rather than a separate idea. This makes it hard to devise alternative solutions to the flatness problem; open inflation models use quantum tunnelling but in fact still require classical inflation after the tunnelling, and presently the only existing alternative is the variable-speed-of-light theories [10] which violate general relativity.

Before continuing on to the properties of perturbations in the Universe,
there's a final point worth bearing in mind concerning inflation as a
theory of the global Universe. As I've said, there now seems little
prospect that any observations will come along which might rule out the
model. But it is interesting that while that is true now, it was
*not* true when inflation was first devised. An example is the
question
of the topology of the Universe. We now know that if there is any
non-trivial topology to the Universe, the identification scale is at
least of order the Hubble radius, and I expect that that can be
consistent with inflation (though I am unaware of any detailed
investigation of the issue). However, from observations available in
1981 it was perfectly possible that the identification scale could have
been much much smaller. Since inflation will stretch the topological
identification scale, that would have set an upper limit on the amount
of inflation strong enough to prevent it from solving the horizon and
flatness problems. The prediction of no small-scale topological
identification has proven a successful one. Another example of a test
that could have excluded inflation, but didn't, is the now-observed
absence of a global rotation of our observable Universe
[12].

^{1} Inflation is
also responsible for solving the horizon problem, ensuring a Universe
close to homogeneity, but this is no longer a useful test as it is
already observationally verified to high accuracy through the
near isotropy of the cosmic microwave background.
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^{2} Indeed, the only existing alternative
to inflation
in explaining spatial flatness is the variable-speed-of-light (VSL)
theories
[10], which may
be able to solve the problem
without inflation, though at the cost of abandoning Lorentz invariance.
There are no available alternatives at all to inflation in explaining an
open Universe, so one might say that observation of negative curvature
modestly *improves* the likelihood of inflation amongst known
theories, by eliminating the VSL theories from consideration.
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