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Where does string theory fit in?

   A big complicating factor in understanding string cosmology is understanding string theories. String theories and M theory appear to be limiting cases of some bigger, more fundamental theory. Until that's sorted out, anything we think we know today is potentially up for grabs.
   That being said, there are some basic issues in string theory cosmology:

1. Can string theory make any cosmological predictions relevant to Big Bang physics?
2. What happens to the extra dimensions?
3. Is there Inflation in string theory?

Low energy string cosmology

   The baryonic matter that makes up the nuclei of atoms seems to provide only a small fraction of the total mass in the Universe.

Measured values of Omega

Most of the mass in our Universe appears to occur in the form of dark matter, which is most likely made up of some exotic particle or particles that interact very weakly and have a very large mass.
   String theories require supersymmetry for quantum consistency, and supersymmetric theories require bosons and fermions to come in pairs, because the supercharge operator turns bosons into fermions and vice versa.

Supersymmetry charge

   So supersymmetric theories are good places to look for exotic matter in the form of fermionic superpartners of bosonic particles that carry forces.
   In the Standard Model of particle physics, recall there is a spontaneously broken symmetry that gives mass to the weak interaction gauge bosons through the Higgs potential. The Standard model contains three massive gauge bosons, two charged and one neutral, and a massive neutral Higgs field.
   The Minimal Supersymmetric Standard Model (MSSM) is a supersymmetric version of the Standard Model. The weak interaction gauge bosons and Higgs fields in the MSSM have fermionic superpartners, and the neutral superpartners are called neutralinos. A neutralino would make a good candidate for for dark matter, because it couples with weak interaction strength but should have a high mass.
    But this is true only as long as it is stable. A neutralino would be stable if there were nothing of lower mass that it could decay into, i.e. it is the Lightest Supersymmetric Particle (LSP), and if something called R-parity is conserved.
   The experimental limits on supersymmetric particle masses say that any neutralino LSP out there must have a mass greater than 40 GeV. A neutralino of that mass could give

Omega_LSP

and that's already in the right ballpark for the observed amount of dark matter out there.
   But the success of such a model depends on whether supersymmetry can be broken at the right scale. Supersymmetry breaking has other cosmological implications, such as a cosmological constant with a value that can run away from the very small, but nonzero, value that has recently been observed in the redshifts of supernovae. So this is far from a settled problem.

Cosmology and string duality

   The standard Big Bang cosmology assumes that the Universe began expanding from a state that was very hot, very small, and very highly curved. The Big Bang model agrees so well with observation that it is therefore commonly assumed that any cosmological era that preceded the Big Bang must have involved a Universe that was even hotter and even smaller and more highly curved, until we reach the Planck scale and the Planck temperature, where our ability to describe geometry runs into fundamental quantum limits where gravity is strongly coupled and can no longer be treated as a fixed classical substrate in which particles or strings interact.
   But string theory complicates such a naive monotonic extrapolation backwards through time, temperature and curvature, because in string theory there are symmetries that can obscure the difference between large and small distance, large and small curvature, and large and small coupling strength.
    One such symmetry is T-duality. Recall that with strings quantized in a flat spacetime background, if one dimension is wrapped into a circle of radius R, by identifying xi with xi + 2pR, there are two new kinds of modes added to the spectrum: modes with quantized momentum going around the circle with quantum number n, and modes that wrap around the circle with winding number w. The total mass squared of the string then depends on these two numbers

   This formula has a symmetry under the exchange

T duality

This is T-duality. The self dual point is where

Self dual point

At the self-dual point, extra massless fields enter the dynamics that reflect an enhanced group of symmetries.
   T-duality has been applied to pre-Big Bang cosmology to build a model that is probably wrong, but interesting to study nonetheless.
    A cosmological solution to the vacuum Einstein equations that is homogeneous but not isotropic is the Kasner metric, which can be written as

A homogeneous, nonisotropic cosmology

The set of exponents {pi} as constrained above have the properties that they are all smaller than one, and they can't all have the same sign. If n of the exponents are positive so that the Universe expands as time increases in those n directions, then the remaining D-n exponents are negative, and the Universe shrinks in those directions as time increases.
   String theory has a scalar field called the dilaton, and the Kasner metric in this case extends to

Homogeneous nonisotropic metric with dilation

Again, directions with pi positive expand as time increases, and those with pi negative contract as time increases. Notice that in this case, isotropic solutions are allowed where pi = ± D-1/2.
   For every solution with some set of exponents and dilaton {pi, f(t)}, there is a dual solution with {pi',f'(t)} given by

Pre-Big Bang  duality

So expanding solutions and contracting solutions are dual to one another.
   This duality symmetry has led to an interesting proposal for pre-Big Bang cosmology where the stringy Universe starts out flat, cold and very large instead of curved, hot and very small. This early Universe is unstable and starts to collapse and contract until it reaches the self dual point, where it heats up and starts to expand to give the expanding Universe we observe today.
   One advantage to this model is that it incorporates the very stringy behavior of T duality and the self dual point, so it is a very inherently stringy cosmology. Unfortunately, the model has failings in both the technical and observational categories, so it's no longer considered a viable model for string cosmology.

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