414
R. Brandenberger
and compress the radius, then the temperature of the gas will initially increase since
the energy of the momentum modes (which are the light modes for large values of
R) increases. Eventually it becomes thermodynamically preferable to excite higher
and higher energy oscillatory modes. The increase in temperature will level off:
there is a maximal temperature of a gas of strings, the Hagedorn temperature T H
[28]. When R decreases below the string scale, the energy will flow into the winding
modes (which are now the light modes), and the temperature will decrease. Hence
[15], thermodynamic reasoning indicates that there is no temperature singularity in
a stringy early universe cosmology.
String theory also features a new symmetry, T-duality symmetry. For a toroidal
space, this implies that there is a symmetry between a space of radius R and a
dual space of radius 1/R (in string units) obtained by interchanging the momentum
and winding quantum numbers. As already argued in [15], the number of position
operators in a quantum theory of strings must be doubled compared to a theory of
point particles: there is one position operator which is the Fourier transform of the
momentum
|x > =
n
|p > n ,
(3)
where |p > n is the eigenstate of momentum which quantum number n (n ranging
over the integers), and a dual operator | ˜
x > which is dual to the winding number
eigenstates. Physical length l p is measured in terms of |x > if R is large, but in
terms of | ˜
x > if R is small. Hence, as R decreases from some large value towards
zero, l p remains finite (it is an even function of ln(R)). This is another way to see
the non-singularity of a stringy early universe cosmology.
The challenge for string cosmology remains to find consistent equations for the
time-dependent cosmological background. Einstein gravity is not applicable since
it is not consistent with the T-duality symmetry of string theory. In String Gas
Cosmology [15] it was postulated that the universe emerges from a quasi-static
initial Hagedorn phase. Such a phase could emerge from a better understanding of
non-perturbative string theory. If we want to model the dynamics using an effective
field theory, this effective field theory must live in double the number of spatial
dimensions as the topological background contains in order to take into account both
the |x > and | ˜
x > coordinates. A candidate for such a theory is Double Field Theory
[29], a theory which is given by the action for a generalized metric in doubled space.
The cosmology which results if we couple the Double Field Theory action for the
background to “string gas matter” (matter which has an equation of state of radiative
modes for large volumes of the |x > space, and that of winding modes for a small
volume) was recently analyzed in [30]. In this context it can be shown that the
solutions in the string frame are non-singular.
R. Brandenberger
and compress the radius, then the temperature of the gas will initially increase since
the energy of the momentum modes (which are the light modes for large values of
R) increases. Eventually it becomes thermodynamically preferable to excite higher
and higher energy oscillatory modes. The increase in temperature will level off:
there is a maximal temperature of a gas of strings, the Hagedorn temperature T H
[28]. When R decreases below the string scale, the energy will flow into the winding
modes (which are now the light modes), and the temperature will decrease. Hence
[15], thermodynamic reasoning indicates that there is no temperature singularity in
a stringy early universe cosmology.
String theory also features a new symmetry, T-duality symmetry. For a toroidal
space, this implies that there is a symmetry between a space of radius R and a
dual space of radius 1/R (in string units) obtained by interchanging the momentum
and winding quantum numbers. As already argued in [15], the number of position
operators in a quantum theory of strings must be doubled compared to a theory of
point particles: there is one position operator which is the Fourier transform of the
momentum
|x > =
n
|p > n ,
(3)
where |p > n is the eigenstate of momentum which quantum number n (n ranging
over the integers), and a dual operator | ˜
x > which is dual to the winding number
eigenstates. Physical length l p is measured in terms of |x > if R is large, but in
terms of | ˜
x > if R is small. Hence, as R decreases from some large value towards
zero, l p remains finite (it is an even function of ln(R)). This is another way to see
the non-singularity of a stringy early universe cosmology.
The challenge for string cosmology remains to find consistent equations for the
time-dependent cosmological background. Einstein gravity is not applicable since
it is not consistent with the T-duality symmetry of string theory. In String Gas
Cosmology [15] it was postulated that the universe emerges from a quasi-static
initial Hagedorn phase. Such a phase could emerge from a better understanding of
non-perturbative string theory. If we want to model the dynamics using an effective
field theory, this effective field theory must live in double the number of spatial
dimensions as the topological background contains in order to take into account both
the |x > and | ˜
x > coordinates. A candidate for such a theory is Double Field Theory
[29], a theory which is given by the action for a generalized metric in doubled space.
The cosmology which results if we couple the Double Field Theory action for the
background to “string gas matter” (matter which has an equation of state of radiative
modes for large volumes of the |x > space, and that of winding modes for a small
volume) was recently analyzed in [30]. In this context it can be shown that the
solutions in the string frame are non-singular.