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
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
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
The equation of state for the "stuff" in the Universe, combined
with the Einstein equation, tells us that
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
This tells us that an accelerating Universe will have a future event
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.