260
Superstring cosmology
as in (7.115). This contrasts with a scale of
(m3/2M p)I/2,.., 101°_1011 GeV
(9.61)
for an effective potential due to the superpotential responsible for low-energy
supersymmetry breaking with soft supersymmetry breaking masses on the scale
of 102-HP GeV. This is the reason for assuming that the superpotential is the
sum of two components. (It should be noted that the discussion of the previous
section employed gaugino condensate superpotentials that had been designed to
be responsible for low-energy supersymmetry breaking.) All of this discussion
applies equally to the use of a modulus field as the inflaton.
Though it is a very attractive idea, it has proved difficult to find a realization
of it in practice. Multiple gaugino condensate potentials for the dilaton, such
as discussed in the previous section, tend to be too steep in the Re S direction
for inflation to occur. If, instead, the overall modulus field T is employed as
the inflaton, with a superpotential consistent with the modular invariance of an
orbifold compactification, up to 20 e-folds of inflation can be obtained. However,
this does not appear to be possible when we demand that the effective potential has
a minimum with unbroken supersymmetry and zero cosmological constant [9].
9.S Ten-dimensional string cosmology
Heterotic string theory begins as a ten-dimensional theory with the need for six
dimensions to be compactified to provide us with the observed four-dimensional
world (except in the case of direct constructions in four dimensions, such as the
free-fermion construction). An attractive possibility is that the compactification
of the six extra dimensions has a cosmological origin. In what follows we shall
treat all spatial dimensions as being wrapped on a torus with all dimensions
initially on the Planck scale. We shall study a mechanism, due to Brandenberger
and Vafa [10], that naturally results in three of the spatial dimensions becoming
very large, corresponding to a flat space, and the rest of the spatial dimensions
remaining on the Planck scale. The dilaton plays a crucial part.
As discussed in section 9.3, we expect the dilaton to acquire a mass of the
order of the electroweak scale, or one or two orders of magnitude larger, when
supersymmetry breaking occurs. For consistent cosmology, it is crucial that the
dilaton does acquire a mass, because it is a scalar field with only gravitational
strength interactions, and a massless field of this kind is inconsistent with solar
system observations. However, there is no a priori objection to the dilaton having
been massless in the early stages of the universe before supersymmetry breaking
at a temperature of around 100 Ge V.
It will be assumed that the gravitational (metric) field and the dilaton field
are slowly varying (adiabatic approximation) so that it is a good approximation to
keep only the leading derivatives in the effective action. It will also be assumed
that N spatial dimensions are large dimensions with time dependence, while the
Superstring cosmology
as in (7.115). This contrasts with a scale of
(m3/2M p)I/2,.., 101°_1011 GeV
(9.61)
for an effective potential due to the superpotential responsible for low-energy
supersymmetry breaking with soft supersymmetry breaking masses on the scale
of 102-HP GeV. This is the reason for assuming that the superpotential is the
sum of two components. (It should be noted that the discussion of the previous
section employed gaugino condensate superpotentials that had been designed to
be responsible for low-energy supersymmetry breaking.) All of this discussion
applies equally to the use of a modulus field as the inflaton.
Though it is a very attractive idea, it has proved difficult to find a realization
of it in practice. Multiple gaugino condensate potentials for the dilaton, such
as discussed in the previous section, tend to be too steep in the Re S direction
for inflation to occur. If, instead, the overall modulus field T is employed as
the inflaton, with a superpotential consistent with the modular invariance of an
orbifold compactification, up to 20 e-folds of inflation can be obtained. However,
this does not appear to be possible when we demand that the effective potential has
a minimum with unbroken supersymmetry and zero cosmological constant [9].
9.S Ten-dimensional string cosmology
Heterotic string theory begins as a ten-dimensional theory with the need for six
dimensions to be compactified to provide us with the observed four-dimensional
world (except in the case of direct constructions in four dimensions, such as the
free-fermion construction). An attractive possibility is that the compactification
of the six extra dimensions has a cosmological origin. In what follows we shall
treat all spatial dimensions as being wrapped on a torus with all dimensions
initially on the Planck scale. We shall study a mechanism, due to Brandenberger
and Vafa [10], that naturally results in three of the spatial dimensions becoming
very large, corresponding to a flat space, and the rest of the spatial dimensions
remaining on the Planck scale. The dilaton plays a crucial part.
As discussed in section 9.3, we expect the dilaton to acquire a mass of the
order of the electroweak scale, or one or two orders of magnitude larger, when
supersymmetry breaking occurs. For consistent cosmology, it is crucial that the
dilaton does acquire a mass, because it is a scalar field with only gravitational
strength interactions, and a massless field of this kind is inconsistent with solar
system observations. However, there is no a priori objection to the dilaton having
been massless in the early stages of the universe before supersymmetry breaking
at a temperature of around 100 Ge V.
It will be assumed that the gravitational (metric) field and the dilaton field
are slowly varying (adiabatic approximation) so that it is a good approximation to
keep only the leading derivatives in the effective action. It will also be assumed
that N spatial dimensions are large dimensions with time dependence, while the