D-brane inflation
265
potential with increasing A. as the universe expands can be offset by the behaviour
of e 4l • A more careful treatment shows that this does not affect the outcome.
Once the universe starts to contract, the momentum modes, which have
behaviour dual to the winding modes, play a crucial role. They make a
contribution to E(A.) that increases as A. decreases and oppose the contraction.
In this way, the universe is caused to oscillate between some minimum radius and
some maximum radius within a few orders of magnitude of the Planck scale.
The question then is how the universe can ever expand to a large scale.
The answer is that string winding modes can annihilate totally or partially into
momentum states (with complete annihilation of winding number occurring
between winding modes with equal and opposite winding number). In this way
they are able to reach thermal equilibrium with other string states. In thermal
equilibrium the number of winding modes becomes small as the torus radius
increases and winding-mode masses increase. However, it is difficult to reach
equilibrium if the winding modes find it difficult to collide to annihilate. A
collision corresponds to the two-dimensional world surfaces of the two states
intersecting. Generically, this does not occur when the dimensionality N + I of the
extended spacetime in which the winding modes move is greater than 2 + 2 = 4.
Thus, for N + I > 4, thermal eqUilibrium is not achieved and the winding modes
stop the the universe expanding much beyond the Planck scale. However, for
N + I :c; 4, the winding modes annihilate readily and thermal equilibrium is
reached resulting in a low density of winding modes. The universe is then able to
expand to a large scale.
This argument provides a partial explanation of the three-dimensional nature
of our observed universe. If the universe starts to expand in some number N > 3
of (spatial) dimensions, then the expansion is stopped by the winding modes.
It then oscillates for a while before expanding again in some number N of
dimensions that may differ from the first time. This may happen many times
until finally the universe starts to expand in some number N of dimensions with
N :c; 3. Then, the expansion continues. Of course, this only explains why N :c; 3
and not why N = 3. Therafter, the discussion of the earlier part of this section,
which neglected winding modes, applies and the universe evolves to a standard
radiation-dominated universe.
9.6 D-brane inflation
The discussion so far in this chapter has been in the context of weakly coupled
heterotic string theory. Alternative models of particle theory can be obtained from
type II superstring theories because of the existence of extended so-called 'Dpbrane' solutions which occupy p + I dimensions of spacetime. (See, for example.
[12] and references therein.) As well as closed strings, the theory contains open
strings which are constrained to have their endpoints on D p-branes. Chiral matter
can be obtained from open strings whose endpoints are on D p-branes located at an
265
potential with increasing A. as the universe expands can be offset by the behaviour
of e 4l • A more careful treatment shows that this does not affect the outcome.
Once the universe starts to contract, the momentum modes, which have
behaviour dual to the winding modes, play a crucial role. They make a
contribution to E(A.) that increases as A. decreases and oppose the contraction.
In this way, the universe is caused to oscillate between some minimum radius and
some maximum radius within a few orders of magnitude of the Planck scale.
The question then is how the universe can ever expand to a large scale.
The answer is that string winding modes can annihilate totally or partially into
momentum states (with complete annihilation of winding number occurring
between winding modes with equal and opposite winding number). In this way
they are able to reach thermal equilibrium with other string states. In thermal
equilibrium the number of winding modes becomes small as the torus radius
increases and winding-mode masses increase. However, it is difficult to reach
equilibrium if the winding modes find it difficult to collide to annihilate. A
collision corresponds to the two-dimensional world surfaces of the two states
intersecting. Generically, this does not occur when the dimensionality N + I of the
extended spacetime in which the winding modes move is greater than 2 + 2 = 4.
Thus, for N + I > 4, thermal eqUilibrium is not achieved and the winding modes
stop the the universe expanding much beyond the Planck scale. However, for
N + I :c; 4, the winding modes annihilate readily and thermal equilibrium is
reached resulting in a low density of winding modes. The universe is then able to
expand to a large scale.
This argument provides a partial explanation of the three-dimensional nature
of our observed universe. If the universe starts to expand in some number N > 3
of (spatial) dimensions, then the expansion is stopped by the winding modes.
It then oscillates for a while before expanding again in some number N of
dimensions that may differ from the first time. This may happen many times
until finally the universe starts to expand in some number N of dimensions with
N :c; 3. Then, the expansion continues. Of course, this only explains why N :c; 3
and not why N = 3. Therafter, the discussion of the earlier part of this section,
which neglected winding modes, applies and the universe evolves to a standard
radiation-dominated universe.
9.6 D-brane inflation
The discussion so far in this chapter has been in the context of weakly coupled
heterotic string theory. Alternative models of particle theory can be obtained from
type II superstring theories because of the existence of extended so-called 'Dpbrane' solutions which occupy p + I dimensions of spacetime. (See, for example.
[12] and references therein.) As well as closed strings, the theory contains open
strings which are constrained to have their endpoints on D p-branes. Chiral matter
can be obtained from open strings whose endpoints are on D p-branes located at an