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Where does string theory fit in?

Inflation vs. the giant brane collision

   Inflation is still the preferred cosmological model of astrophysicists. But efforts to derive a suitable inflationary potential from the low energy limit of superstring theory have met with many obstacles. The dilaton field would seem to be an obvious candidate for the inflaton, but in perturbative low energy string theory the dilaton has no potential, the field is massless and couples to gravity solely through its kinetic energy, which is positive and would slow down the expansion of the Universe rather than speed it up.
   String theories contain other scalar field called moduli, but the moduli are also massless in perturbative string theory, and their nonperturbative potentials are still unknown. Any nonperturbative physics that fixes stable minima for these fields controls the supersymmetry breaking scale, the sizes of compactified dimensions, the value of the cosmological constant, and the dynamics of the inflaton field, and that's why deriving a string theory inflationary model has been such a challenge.
   But inflationary models suffer from a conceptual inadequacy in that they are constructed using a combination of relativistic quantum field theory and classic general relativity. String theory is a theory of quantum gravity. And so string theory ought to be able to describe cosmology on a more fundamental level than inflationary models are capable of describing.
   The discovery of extended fundamental structures in string theory called D-branes has brought forth some startling new ideas for the structure of spacetime. The first such model by Horava and Witten started with M-theory in eleven spacetime dimensions, compactified on a 6-dimensional Calabi-Yau space, leaving four space dimensions and time. The four space dimensions are bounded by two three-dimensional surfaces, or branes, separated by some distance R between the three-branes in the fourth direction. One of those three-branes, called the visible brane, can be seen as the three-dimensional world on which we live. The other three-dimensional brane is called the hidden brane, and we never see it. The volume V of the Calabi-Yau space varies from the visible brane to the hidden brane, and each brane has a different set of E8 gauge supermultiplets living on it, with the gauge couplings of fields living on the visible and hidden branes related by

Horava-Witten couplings

This model is an effective five-dimensional theory, because the value of R is large compared to the size of the Calabi-Yau space.
   This Horava-Witten world is not a cosmological model, but this picture has been applied to cosmology with interesting and controversial results. The latest version of braneworld cosmology is the giant brane collision model, also known as the Ekpyrotic Universe, or the Big Splat.
   The Ekpyrotic Universe starts out as a cold, flat, static five-dimensional spacetime that is close to being a supersymmetric BPS state, meaning a state invariant under some special subalgebra of the supersymmetry algebra. The four space dimensions of the bulk are bounded by two three-dimensional walls or three-branes, and one of those three-branes makes up the space that we live on. 
   But how does the Universe evolve to give the Big Bang cosmology for which there is so much observational evidence? The Ekpyrotic theory postulates that there is a third three-brane loose between the two bounding branes of the four dimensional bulk world, and when this brane collides with the brane on which we live, the energy from the collision heats up our brane and the Big Bang occurs in our visible Universe as described elsewhere in this site.
   This proposal is quite new, and it remains to be seen whether it will survive careful scrutiny.

The problem with acceleration

   There is a problem with an accelerating Universe that is fundamentally challenging to string theory, and even to traditional particle theory. In eternal inflation models and most quintessence models, the expansion of the Universe accelerates indefinitely. This eternal acceleration leads to some contradictions in the mathematical assumptions made about spacetime in the fundamental formulations of quantum field theories and string theories.
   According to the Einstein equation, for the usual case of a four-dimensional spacetime where space is homogeneous and isotropic, the acceleration of the scale factor depends on the energy density and the pressure of the "stuff" in the Universe as

Acceleration of scale factor

The equation of state for the "stuff" in the Universe, combined with the Einstein equation, tells us that

Equation of state and scale factor

   The boundary of the region beyond which an observer can never see is called that observer's event horizon. In cosmology, the event horizon is like the particle horizon, except that it is in the future and not in the past. In the class of spacetimes we've been looking at, the amount of the future that an observer at some time t0 would be able to see were she or he to live forever is given by

Cosmological future event horizon

This tells us that an accelerating Universe will have a future event horizon, because

Condition for positive acceleration

   From the point of view of human philosophy or the internal consistency of Einstein's theory of relativity, there is no problem with a cosmological event horizon. So what if we can't ever see some parts of the Universe, even if we were to live forever?
   But a cosmological event horizon is a major technical problem in high energy physics, because of the definition of relativistic quantum theory in terms of the collection of scattering amplitudes called the S Matrix. One of the fundamental assumptions of quantum relativistic theories of particles and strings is that when incoming and outgoing states are infinitely separated in time, they behave as free noninteracting states.
   The presence of an event horizon implies a finite Hawking temperature and the conditions for defining the S Matrix cannot be fulfilled. This lack of an S Matrix is a formal mathematical problem not only in string theory but also in particle theories.


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