272
Superstring cosmology
be of order unity in the post-big-bang universe and so the initial value of", must
be tuned to be very small. Moreover, the density perturbations due to the dilaton
(playing the role of the inflaton here) are not consistent with the COBE data.
However, it has been argued that this last problem can be overcome if the density
perturbations required by COBE are due to an axion field in a non-perturbative
potential. (See the third reference in [16].)
9.8 M-theory cosmology-the ekpyrotic universe
Another example of a cosmology in which there is a period of evolution of the
universe prior to the big bang can be obtained by considering the collision of the
so-called 'boundary branes' which occur in M-theory. We first review briefly the
M-theory description of strongly-coupled heterotic string theory [18].
Strongly-coupled heterotic string theory in ten dimensions is known to be
dual to a theory in 1 1 dimensions. If the strongly-coupled heterotic string theory
has six dimensions compacified on a Calabi- Yau manifold or orbifold X, then the
dual theory has seven dimensions compactified on X x SI/Z2. Here SI is in the
X 10 direction with
-1rP ~ X IO ~ 1rP
(9.120)
and the identification
X IO ..... X IO + 21rp.
(9.121)
The action of the Z2 group is
Z2 : X IO .... _XIO.
(9.122)
Consequently, there are two fixed points under the action of Z2, namely
XIO=o
and
X IO = 1Cp.
(9.123)
Because of the Z2 symmetry (9.122), the quotiented circle is equivalent to a line
segment of length 1rp.
The low-energy limit of this 1 I-dimensional theory is II-dimensional
supergravity. The E8 gauge fields of the observable sector (and the observable
chiral matter) live on one end of the line segment (on one four-dimensional
boundary brane) while the E8 gauge fields of the hidden sector live on the other
end of the line segment (on the other boundary brane). The gravitational fields
propagate in the I I-dimensional 'bulk'. The dilaton expectation value e. in the
ten-dimensional theory is reinterpreted, up to a numerical factor, as the length of
the line segment in the 1 1 -dimensional theory in M; I units.
The idea behind M-theory cosmology [19] (the so-called 'ekpyrotic
universe-the universe being consumed by fire and reconstituted out of fire,
as in Stoic philosophy) is that two four-dimensional boundary branes may
move towards each other and collide before separating again. The size of the
fifth dimension (which is the separation of the two four-dimensional boundary
Superstring cosmology
be of order unity in the post-big-bang universe and so the initial value of", must
be tuned to be very small. Moreover, the density perturbations due to the dilaton
(playing the role of the inflaton here) are not consistent with the COBE data.
However, it has been argued that this last problem can be overcome if the density
perturbations required by COBE are due to an axion field in a non-perturbative
potential. (See the third reference in [16].)
9.8 M-theory cosmology-the ekpyrotic universe
Another example of a cosmology in which there is a period of evolution of the
universe prior to the big bang can be obtained by considering the collision of the
so-called 'boundary branes' which occur in M-theory. We first review briefly the
M-theory description of strongly-coupled heterotic string theory [18].
Strongly-coupled heterotic string theory in ten dimensions is known to be
dual to a theory in 1 1 dimensions. If the strongly-coupled heterotic string theory
has six dimensions compacified on a Calabi- Yau manifold or orbifold X, then the
dual theory has seven dimensions compactified on X x SI/Z2. Here SI is in the
X 10 direction with
-1rP ~ X IO ~ 1rP
(9.120)
and the identification
X IO ..... X IO + 21rp.
(9.121)
The action of the Z2 group is
Z2 : X IO .... _XIO.
(9.122)
Consequently, there are two fixed points under the action of Z2, namely
XIO=o
and
X IO = 1Cp.
(9.123)
Because of the Z2 symmetry (9.122), the quotiented circle is equivalent to a line
segment of length 1rp.
The low-energy limit of this 1 I-dimensional theory is II-dimensional
supergravity. The E8 gauge fields of the observable sector (and the observable
chiral matter) live on one end of the line segment (on one four-dimensional
boundary brane) while the E8 gauge fields of the hidden sector live on the other
end of the line segment (on the other boundary brane). The gravitational fields
propagate in the I I-dimensional 'bulk'. The dilaton expectation value e. in the
ten-dimensional theory is reinterpreted, up to a numerical factor, as the length of
the line segment in the 1 1 -dimensional theory in M; I units.
The idea behind M-theory cosmology [19] (the so-called 'ekpyrotic
universe-the universe being consumed by fire and reconstituted out of fire,
as in Stoic philosophy) is that two four-dimensional boundary branes may
move towards each other and collide before separating again. The size of the
fifth dimension (which is the separation of the two four-dimensional boundary