232
Inflation in supergravity
Also, from the double derivative of the effective potential, a mass-squared m~ of
the inflaton of order
2
J.'41~
m. ' " _'_'"2
(8.40)
MS P
is to be expected, allowing for (;) - Mp owing to the effects of quantum gravity
and restoring factors of Mp. Thus,
61~
r. '" ~
(8.41)
MS .
p
The reheating temperature of (7.72) is
,,3~/2
TR - (r ... Mp)1/2 _ _ _
(8.42)
."
M2 p
ignoring the difference between m p and Mp. Using the estimate (8.33), this gives
TR - 1011 GeV.
(8.43)
8.3 D-term supergravity inflation
To arrange for sufficient inflation in the model of the previous section, it was
necessary to take ,,21A21 ..... 10- 3 , which is an unnatural fine-tuning of the
superpotential. This is a generic feature of supergravity models where the positive
value of the effective potential during inflation is due to a non-zero F -term in
the sense that I ~ + t/li. Wl 2 #: 0 in (2.156). (The terminology is because
a w / a;i + t/I'. W is the generalization to the supergravity context of the auxiliary
field usually denoted by Fi in the construction of the globally-supersymmetric
Lagrangian.) When all relevant fields are gauge singlet and assuming minimal
kinetic terms, the effective potential for the inflaton takes the form (8.1). There is
a term quadratic in ;, namely VoI/>*t/I, where Vo == V(O). Keeping only this term
and assuming a real inflaton,
V"(;) - Vo
(8.44)
and then, from (8.25), the number of e-folds of inflation is
Ne _ 3H 2
(8.45)
Wol·
But when ; ::: 1/>0 ::: 0,
H2::: iVo.
(8.46)
Thus,
Ne -I.
(8.47)
Inflation in supergravity
Also, from the double derivative of the effective potential, a mass-squared m~ of
the inflaton of order
2
J.'41~
m. ' " _'_'"2
(8.40)
MS P
is to be expected, allowing for (;) - Mp owing to the effects of quantum gravity
and restoring factors of Mp. Thus,
61~
r. '" ~
(8.41)
MS .
p
The reheating temperature of (7.72) is
,,3~/2
TR - (r ... Mp)1/2 _ _ _
(8.42)
."
M2 p
ignoring the difference between m p and Mp. Using the estimate (8.33), this gives
TR - 1011 GeV.
(8.43)
8.3 D-term supergravity inflation
To arrange for sufficient inflation in the model of the previous section, it was
necessary to take ,,21A21 ..... 10- 3 , which is an unnatural fine-tuning of the
superpotential. This is a generic feature of supergravity models where the positive
value of the effective potential during inflation is due to a non-zero F -term in
the sense that I ~ + t/li. Wl 2 #: 0 in (2.156). (The terminology is because
a w / a;i + t/I'. W is the generalization to the supergravity context of the auxiliary
field usually denoted by Fi in the construction of the globally-supersymmetric
Lagrangian.) When all relevant fields are gauge singlet and assuming minimal
kinetic terms, the effective potential for the inflaton takes the form (8.1). There is
a term quadratic in ;, namely VoI/>*t/I, where Vo == V(O). Keeping only this term
and assuming a real inflaton,
V"(;) - Vo
(8.44)
and then, from (8.25), the number of e-folds of inflation is
Ne _ 3H 2
(8.45)
Wol·
But when ; ::: 1/>0 ::: 0,
H2::: iVo.
(8.46)
Thus,
Ne -I.
(8.47)