Sean M. Carroll, hep th 0011110 TASI Lectures Cosmology for String Theorists

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feature a precise cancellation between H and Z. In the real world (in contrast to the
world ofhep-th) these are a negligible fraction of all possible states. It is not clear how
SUSY breaking affects the Pre-BB idea.
Perhaps more profoundly, it seems perfectly likely that the appropriate description of
the high-curvature stringy phase will be nothing like a smooth classical spacetime. Evidence
for this comes from matrix theory, not to mention attempts to canonically quantize general
relativity.
There are other, non-stringy, approaches to the very beginning of the universe, and it
would be interesting to know what light can be shed on them by string theory. One is
quantum cosmology , which by some definitions is just the study of the wave function
of the universe, although in practice it has the connotation of minisuperspace techniques
(drastically truncating the gravitational degrees of freedom and quantizing what is left)
[130, 121, 131, 7]. There is also the related idea of creation of baby universes from our own
[132, 133]. This is in principle a conceivable scheme, as closed universes have zero total
energy in general relativity. There is also the hope that string theory will offer some unique
resolution to the question of cosmological (and other) singularities; studies to date have had
some interesting results, but we don t know enough to understand the Big Bang singularity
of the real world [134, 135, 136].
5.2 Extra dimensions and compactification
Of all the features of string theory, the one with the most obvious relevance to cosmology is
the existence of (6? 7?) extra spatial (temporal?) dimensions. The success of our traditional
description of the world as a (3+1)-dimensional spacetime implies that the extra dimensions
must be somehow inaccessible, and the simplest method for hiding them is compactification
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the idea that the extra dimensions describe a compact space of sufficiently small size that
they can only be probed by very high energies.
Of course in general relativity (and even in string theory) spacetime is dynamical, and
it would be natural to expect the compact dimensions to evolve. However, the parameters
describing the size and shape of the compact dimensions show up in our low-energy world as
moduli fields whose values affect the Standard Model parameters. As discussed earlier, we
have good limits on any variation of these parameters in spacetime, and typically appeal to
SUSY breaking to fix their expectation values. This raises all sorts of questions. Why are
three dimensions allowed to be large and expanding while the others are small and essentially
frozen? What is the precise origin of the moduli potentials? What was the behavior of the
extra dimensions in the early universe?
For the most part these are baffling questions, although there have been some provocative
suggestions. One is by Brandenberger and Vafa, who attempted to understand the existence
of three macroscopic spatial dimensions in terms of string dynamics [137]. Consider an n-
torus populated by both momentum modes and winding modes of strings. The momentum
and winding modes are dual to each other under T -duality (R �! 1/R), and have opposite
effects on the dynamics of the torus: the momentum modes tend to make it expand, and
the winding modes to make it contract. (It s counterintuitive, but true.) We can therefore
have a static universe at the self-dual radius where the two effects are balanced. However,
when wound strings intersect they tend to intercommute and therefore unwind. Through
this process, the balance holding the torus at the self-dual radius can be upset, and the
universe will begin to expand, hopefully evolving into a conventional Friedmann cosmology.
But notice that in a sufficiently large number of spatial dimensions, one-dimensional
strings will generically never intersect. (Just as zero-dimensional points will generically
intersect in one dimension but not in two or more dimensions.) The largest number in which
they tend to intersect is three. So we can imagine a universe that begins as a tiny torus in
thermal equilibrium at the self-dual point, until some winding modes happen to annihilate
in some three-dimensional subspace which then begins to expand, forming our universe. Of
course a scenario such as this loses some of its charm in a theory which has not only strings
but also higher-dimensional branes. (Not to mention that toroidal compactifications are not
pheonomenologically favored.)
An alternate route is to take advantage of the existence of these branes, by imagining
that we are living on one. That is to say, that the reason why the extra dimensions are
invisible to us is not simply because they are so very small that low-energy excitations
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cannot probe them, but because we are confined to a three-dimensional brane embedded in
a higher-dimensional space. We know that we can easily construct field theories confined
to branes, for example a U(N) gauge theory by stacking N coincident branes; it is not an
incredible stretch to imagine that the entire Standard Model can be constructed in such a
way (although it hasn t been done yet). Unfortunately, it seems impossible to entirely do
away with the necessity of compactification, since there is one force which we don t know
how to confine to a brane, namely gravity (although see below). [ Pobierz całość w formacie PDF ]

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