Models of super gravity inflation
231
rolling across the flat region between q, = 0 and q, = q,~ from a value of q,
sufficiently close to q, = O. However. if initially the universe was in thermal
equilibrium. then thermal effects may put q, well into the region q, > 0 and prevent
enough inflation. This is referred to as the 'thermal constraint' [4]. In general.
for minimal kinetic terms the zero-temperature effective potential V derived from
(2.147) is
V = eG(GiG i - 3)
(8.34)
and from (2.162). the finite-temperature correction V{ to the effective potential
is given by
v{ = constant + ~ T 2 e G (G i G i - 2)
(8.35)
in the limit of a large number N of chiral fields. which is a reasonable
approximation in practice. If the finite-temperature effects are not to destroy the
flatness of the potential. we must require that
av{
av
- = 0 = -
at q, = o.
(8.36)
aq,
aq,
For a single gauge-singlet real scalar field q, with minimal kinetic terms. so that
Gi = G i = G'(q,). these require that
G'(O) = 0
(8.37)
so that
V(O) < O.
(8.38)
It is then impossible for q, to roll to a (supersymmetry-preserving) minimum with
V = O. Thus. the thermal constraint is a very powerful constraint. It may
sometimes be evaded if the inflaton has non-minimal kinetic terms.
However. if a (weakly-coupled) inflaton field q, is out of thennal equilibrium
for temperatures below the Planck scale. then the initial value of q, will. in general.
have a broad distribution [5]. Consequently. it is unlikely that a randomly chosen
horizon volume will possess a (smoothed-out) value of q, close enough to q, = 0
for much inflation to occur. However. any horizon volume which does have a
value of q, close to q, = 0 will undergo inflation and. after inflation has occurred.
most of space will be occupied by such regions. As a result, we are very likely to
find ourselves in a region of space which derived from such a horizon volume at
early times. For this reason. we shall not consider ourselves bound by the thermal
constraint.
We consider next reheating in the context of this simple supergravity model.
For a gauge-singlet inflaton field with only gravitational strength couplings. we
expect a decay rate
m 3
,
r - - 2 '
(8.39)
4J
Mp
231
rolling across the flat region between q, = 0 and q, = q,~ from a value of q,
sufficiently close to q, = O. However. if initially the universe was in thermal
equilibrium. then thermal effects may put q, well into the region q, > 0 and prevent
enough inflation. This is referred to as the 'thermal constraint' [4]. In general.
for minimal kinetic terms the zero-temperature effective potential V derived from
(2.147) is
V = eG(GiG i - 3)
(8.34)
and from (2.162). the finite-temperature correction V{ to the effective potential
is given by
v{ = constant + ~ T 2 e G (G i G i - 2)
(8.35)
in the limit of a large number N of chiral fields. which is a reasonable
approximation in practice. If the finite-temperature effects are not to destroy the
flatness of the potential. we must require that
av{
av
- = 0 = -
at q, = o.
(8.36)
aq,
aq,
For a single gauge-singlet real scalar field q, with minimal kinetic terms. so that
Gi = G i = G'(q,). these require that
G'(O) = 0
(8.37)
so that
V(O) < O.
(8.38)
It is then impossible for q, to roll to a (supersymmetry-preserving) minimum with
V = O. Thus. the thermal constraint is a very powerful constraint. It may
sometimes be evaded if the inflaton has non-minimal kinetic terms.
However. if a (weakly-coupled) inflaton field q, is out of thennal equilibrium
for temperatures below the Planck scale. then the initial value of q, will. in general.
have a broad distribution [5]. Consequently. it is unlikely that a randomly chosen
horizon volume will possess a (smoothed-out) value of q, close enough to q, = 0
for much inflation to occur. However. any horizon volume which does have a
value of q, close to q, = 0 will undergo inflation and. after inflation has occurred.
most of space will be occupied by such regions. As a result, we are very likely to
find ourselves in a region of space which derived from such a horizon volume at
early times. For this reason. we shall not consider ourselves bound by the thermal
constraint.
We consider next reheating in the context of this simple supergravity model.
For a gauge-singlet inflaton field with only gravitational strength couplings. we
expect a decay rate
m 3
,
r - - 2 '
(8.39)
4J
Mp