Models of supergravity inflation
227
temperatures too low for the necessary abundances to be recreated. It can also
lead to too Iowa baryon number density.
8.2 Models of supergravity inflation
The key thing we require to study inflation is the effective potential for the inflaton
field, which we assume to be a gauge-singlet scalar field~. If we also assume
minimal kinetic terms for ~ arising from (2.151), then the effective potential is of
the form
(8.1)
v=e~·~(I~; +~.wr -3 IW I2 )
as in (2.152), in units where Mp of (2.145) is one. In general [I], we may consider
a superpotential which is a power series in ~,
00
W(~) = JL2 ~::>n~n
(8.2)
n=O
where, as usual, we are not distinguishing notationally between the chiral
superfield and the scalar field in that supermultiplet. When the expectation value
of ~ is real, the explicit effective potential corresponding to the superpotential
(8.2) is (exercise I)
v = JL4e~2[A~ - 3AO + 4AI (A2 - Ao)~
+ (A~ - A~ + 4A~ - 2A2Ao + 6AIA3)~2 + 2(AlAo + 6A2A3 + 4AIA4)~3
+ (l.~ + A~ + 9A~ + 2A21.o + 2AIA3 + 2AoA4 + 16A2A4 + IOAIAS)~4
+ ... ].
(8.3)
A particularly simple case [2] is to take W(~) quadratic in~:
W(~) = JL2(1.o + AI~ + A2~2).
(8.4)
The form of W(~) is further restricted by the requirement of the existence of
a supersymmetry-preserving minimum of V(~) with V = 0 for the following
reasons. We have seen in section 7.7 that the energy scale of the inflationary
potential is expected to be of order 10 16 _10 17 GeV. After inflation has occurred,
~ rolls to a minimum of this potential. This minimum should have V = 0
because otherwise there would be a vacuum energy on the 10 16 _10 17 GeV scale
which could not be cancelled by later supersymmetry breaking on the electroweak
scale. There would then be a large cosmological constant. This minimum
should be a supersymmetry-preserving minimum because otherwise there would
be supersymmetry breaking on a scale too large for the hierarchy problem to
227
temperatures too low for the necessary abundances to be recreated. It can also
lead to too Iowa baryon number density.
8.2 Models of supergravity inflation
The key thing we require to study inflation is the effective potential for the inflaton
field, which we assume to be a gauge-singlet scalar field~. If we also assume
minimal kinetic terms for ~ arising from (2.151), then the effective potential is of
the form
(8.1)
v=e~·~(I~; +~.wr -3 IW I2 )
as in (2.152), in units where Mp of (2.145) is one. In general [I], we may consider
a superpotential which is a power series in ~,
00
W(~) = JL2 ~::>n~n
(8.2)
n=O
where, as usual, we are not distinguishing notationally between the chiral
superfield and the scalar field in that supermultiplet. When the expectation value
of ~ is real, the explicit effective potential corresponding to the superpotential
(8.2) is (exercise I)
v = JL4e~2[A~ - 3AO + 4AI (A2 - Ao)~
+ (A~ - A~ + 4A~ - 2A2Ao + 6AIA3)~2 + 2(AlAo + 6A2A3 + 4AIA4)~3
+ (l.~ + A~ + 9A~ + 2A21.o + 2AIA3 + 2AoA4 + 16A2A4 + IOAIAS)~4
+ ... ].
(8.3)
A particularly simple case [2] is to take W(~) quadratic in~:
W(~) = JL2(1.o + AI~ + A2~2).
(8.4)
The form of W(~) is further restricted by the requirement of the existence of
a supersymmetry-preserving minimum of V(~) with V = 0 for the following
reasons. We have seen in section 7.7 that the energy scale of the inflationary
potential is expected to be of order 10 16 _10 17 GeV. After inflation has occurred,
~ rolls to a minimum of this potential. This minimum should have V = 0
because otherwise there would be a vacuum energy on the 10 16 _10 17 GeV scale
which could not be cancelled by later supersymmetry breaking on the electroweak
scale. There would then be a large cosmological constant. This minimum
should be a supersymmetry-preserving minimum because otherwise there would
be supersymmetry breaking on a scale too large for the hierarchy problem to