264
Superstring cosmology
(9.86) can be interpreted as the equation of motion for a particle with position A.
moving in a potential e oll E(A) with a damping term, because ci > < O. Since the
potential decreases as IAI increases (ignoring the A dependence of 4», A. slides
towards increasing values of A.. The radius of the toroidal space is aCt) = el(r)
and, hence, as A increases a increases, when A > O. (There is a duality between
large and small values of aCt) and, hence, between positive and negative values
of A, as noted in (9.84». Thus, we need only discuss A > 0.) Since the spectrum
of a string theory is known, the entropy can be calculated at a given temperature
T in terms of the radius aCt). We know that entropy is constant. Therefore, a
relationship between aCt) and T can be calculated numerically that ensures this.
As IAI decreases, it is found that T increases towards a limit referred to as the
'Hagedorn temperature'. As IAI increases, T falls until the massive string modes
go out of equilibrium and we enter a radiation-dominated era controlled by the
massless string modes, as in the standard model of the universe.
We now ask the question: 'Why is the number N of large spatial dimensions
equal to 31' A possible explanation turns on the presence of winding modes in
string theories compactified on a torus. (Remember that we have taken all spatial
dimensions to be toroidal.) Because of the periodic nature of a torus, the closedstring boundary conditions for the spatial bosonic degrees of freedom Xl ( r , eT)
can be satisfied when
Xk(r, eT + rr) = Xl(r, eT) + 2rr Lk
(9.87)
where centre-of-mass coordinates xl on the torus have the identification
Xl =xl +2rrLk
(9.88)
where Ll are referred to as 'winding numbers' and are proportional to the torus
radius [11]. String modes with non-zero values of Lk are referred to as winding
modes. If pi is the centre-of-mass momentum of the string degree of freedom
Xl, the mass-squared of a string state includes (pi + 2L k)2 and (pk - 2L 1)2. As
the radius of the torus increases, the squared mass of a winding mode increases.
The idea is that string winding modes, unlike other matter densities, will
oppose expansion of the dimensions of the universe. The reason for this is that,
in the presence of winding modes, the behaviour of E(A) is very different from
that discussed earlier. As just discussed, the mass squared of any winding mode
increases as the square of the winding number, for large values of the torus radius,
and so as the square of the torus radius. This effect results in E(A) increasing as
eA. Roughly, the growth of E with A in (9.86) means that>: < 0 (up to a damping
term) so that i eventually becomes negative, A starts to decrease and the radius
of the universe starts to decrease. In this way, the winding modes first stop the
expansion of the universe and then reverse it.
This argument is not quite correct because of the e oll term in (9.86). As
discussed earlier, if ci > starts negative, it remains negative so that 4> decreases
with time. Thus, treating A as the position of a particle, the strengthening of the
Superstring cosmology
(9.86) can be interpreted as the equation of motion for a particle with position A.
moving in a potential e oll E(A) with a damping term, because ci > < O. Since the
potential decreases as IAI increases (ignoring the A dependence of 4», A. slides
towards increasing values of A.. The radius of the toroidal space is aCt) = el(r)
and, hence, as A increases a increases, when A > O. (There is a duality between
large and small values of aCt) and, hence, between positive and negative values
of A, as noted in (9.84». Thus, we need only discuss A > 0.) Since the spectrum
of a string theory is known, the entropy can be calculated at a given temperature
T in terms of the radius aCt). We know that entropy is constant. Therefore, a
relationship between aCt) and T can be calculated numerically that ensures this.
As IAI decreases, it is found that T increases towards a limit referred to as the
'Hagedorn temperature'. As IAI increases, T falls until the massive string modes
go out of equilibrium and we enter a radiation-dominated era controlled by the
massless string modes, as in the standard model of the universe.
We now ask the question: 'Why is the number N of large spatial dimensions
equal to 31' A possible explanation turns on the presence of winding modes in
string theories compactified on a torus. (Remember that we have taken all spatial
dimensions to be toroidal.) Because of the periodic nature of a torus, the closedstring boundary conditions for the spatial bosonic degrees of freedom Xl ( r , eT)
can be satisfied when
Xk(r, eT + rr) = Xl(r, eT) + 2rr Lk
(9.87)
where centre-of-mass coordinates xl on the torus have the identification
Xl =xl +2rrLk
(9.88)
where Ll are referred to as 'winding numbers' and are proportional to the torus
radius [11]. String modes with non-zero values of Lk are referred to as winding
modes. If pi is the centre-of-mass momentum of the string degree of freedom
Xl, the mass-squared of a string state includes (pi + 2L k)2 and (pk - 2L 1)2. As
the radius of the torus increases, the squared mass of a winding mode increases.
The idea is that string winding modes, unlike other matter densities, will
oppose expansion of the dimensions of the universe. The reason for this is that,
in the presence of winding modes, the behaviour of E(A) is very different from
that discussed earlier. As just discussed, the mass squared of any winding mode
increases as the square of the winding number, for large values of the torus radius,
and so as the square of the torus radius. This effect results in E(A) increasing as
eA. Roughly, the growth of E with A in (9.86) means that>: < 0 (up to a damping
term) so that i eventually becomes negative, A starts to decrease and the radius
of the universe starts to decrease. In this way, the winding modes first stop the
expansion of the universe and then reverse it.
This argument is not quite correct because of the e oll term in (9.86). As
discussed earlier, if ci > starts negative, it remains negative so that 4> decreases
with time. Thus, treating A as the position of a particle, the strengthening of the