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tion [24]. As we will see later on, to satisfy the slow roll
conditions, a necessary condition is that the -parameter,
de ned by:
=M2
plV00
V; (6)
be much smaller than one. If we choose a K ahler potential
K(X;X) with R-symmetry, for instance the canonical one
K(X;X) =XX+:::, where the :::represents a function
ofXX, it is easy to see that from the exponent of (4) we
always get a contribution to equal to 1: = 1 +:::, no
matter which component of Xis taken as the in
aton eld.
This of course violates the slow roll conditions. Since we
are considering a situation with supersymmetry breaking
and gravity (early universe), we cannot exclude supergrav-
ity from the picture, and this leads to the -problem in
these theories.
The simplest way out of this problem without unreason-
able ne tuning, is to have explicit R-symmetry breaking
3in the K ahler potential4. If we have explicit R-breaking,
the expansion of Vfor small elds takes the form:
X=M( +i ) (7)
V=f2(1 +A1( 2+ 2) +B1( 2 2) +:::)(8)
fis the supersymmetry breaking parameter representing
the expectation value of an F-term, and hence with square
mass dimensions. We assume that Vis locally stable at
least during in
ation. Hence A1B1>0. We express the
potential in terms of the dimensionless elds ; . Their
masses can be read o from (8):
m2
=2f2
M2(A1+B1); m2
=2f2
M2(A1B1):(9)
The numbers A1;B1are taken to be O(1).
One could be more explicit, and choose some super-
symmetry breaking superpotential, like W=fX, and
K ahler potential explicitly breaking R-symmetry, like:
K=XX+ (c=M2)(X3X+XX3) +:::as in [27] lead-
ing to an e ective action description of Xfor scales well
belowM. At this stage, we prefer not to consider explicit
examples of UV-completions of the theory.
We consider the beginning of in
ation well below M,
hence the initial conditions are such that ; << 1. In
fact, since is the lighter eld, we take this one to be the
in
aton, and consider that initially ; pf=M . For us
the in
ationary period goes from this scale until the value
of the eld is close to the typical soft breaking scale of the
problemmsoft, where the eld X!XNL(2), at this scale
XNLbehaves like a spurion [27] and as shown in Ref. [27],
the leading couplings to low-energy supersymmetric mat-
ter can be computed as spurion couplings, for instance5,
ifQ;V represent respectively low energy chiral and vector
4R-symmetry is a well-known problem in phenological applica-
tions of supersymmetry. R-symmetry does not allow soft breaking
masses for the gauginos; and spontaneous breaking of the symmetry
may lead to axions with unacceptable couplings. Often one wants to
preserve R-parity to avoid other possible phenomenological disasters.
5The details can be found in[27] section 4, in particular around
equations (4.3,4).super elds, we can have the couplings:
L=Z
d4 XNL
f 2
m2QeVQ (10)
+Z
d2XNL
f1
2BijQiQj+:::+h:c:
plus gauge couplings.
Once we reach the end of in
ation, the eld Xbecomes
nonlinear, its scalar component is a goldstino bilinear and
the period of reheating begins. The details of reheating de-
pend very much on the microscopic model. At this stage
one should provide details of the \waterfall" that turns the
huge amount of energy f2into low energy particles. Part of
this energy will be depleted and converted into low energy
particles through the soft couplings in (10), and hence we
can in principle compute a lower bound on the reheating
temperature. Before making some comments on the re-
heating period, we analyze the cosmological consequences
of a potential as simple as (8), as well as the assumptions
we have made earlier about the in
aton and its range as
in
ation takes place.
3. The In
aton Potential and Slow Roll Conditions
To study the conditions under which our potential pro-
vides in
ation consistent with the latest cosmological con-
straints, we examine the slow-roll parameters, de ned as
[13]:
=M2
pl