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arXiv:hep-th/0411011v5 29 Jan 2005
Landscape, the Scale of SUSY Breaking, and Inflation
Renata Kallosh and Andrei Linde
Department of Physics, Stanford University, Stanford, CA 94305-4060, USA
We argue that in the simplest version of the KKLT model, the maximal value of the Hubble
constant during inflation cannot exceed the present value of the gravitino mass, H<
may have important implications for string cosmology and for the scale of the SUSY breaking in
this model. If one wants to have inflation on high energy scale, one must develop phenomenological
models with an extremely large gravitino mass. On the other hand, if one insists that the gravitino
mass should be O(1 TeV), one will need to develop models with a very low scale of inflation. We
show, however, that one can avoid these restrictions in a more general class of KKLT models based
on the racetrack superpotential with more than one exponent. In this case one can combine a small
gravitino mass and low scale of SUSY breaking with the high energy scale of inflation.
∼m3/2. This
PACS numbers: 11.25.-w, 98.80.-k hep-th/0411011
I. INTRODUCTION
Soon after the invention of inflationary cosmology, it
became clear that our universe may consist of many expo-
nentially large locally homogeneous regions correspond-
ing to different stable or metastable vacuum states, which
could be used for justification of anthropic principle [1].
It was argued that the total number of such states ob-
tained as a result of compactification of 10D or 11D uni-
verse can be exponentially large [2]. Early estimates of
the total number of different vacua in heterotic string
theory gave astonishingly large numbers such as 101500
[3]. A more recent investigation, based on the idea of flux
compactification, gave similarly large number of possible
vacua [4].
However, none of the stable or metastable 4D vacua
which were known in string theory at that time described
dS space or accelerating/inflationary universe.
progress in finding metastable dS vacua with a positive
cosmological constant was achieved only recently [5] (see
also [6], where this problem was addressed for noncriti-
cal string theory). The main idea of KKLT was to find a
supersymmetric AdS minimum taking into account non-
perturbative effects, and then uplift this minimum to dS
state by adding the positive energy density contribution
of D3 branes or D7 branes. The position of the dS mini-
mum and the value of the cosmological constant there
depend on the quantized values of fluxes in the bulk
and on the branes. This provided a starting point for
a systematic investigation of the landscape of all possible
metastable dS string theory vacua [7, 8].
Some
The idea of string theory landscape may be useful, in
particular, for understanding the scale of supersymme-
try breaking. Different points of view on this issue have
been expressed in the literature, with the emphasis on
statistics of string theory flux vacua [9, 10, 11, 12, 13].
Depending on various assumptions, one can either con-
clude that most of the vacua in the string theory land-
scape correspond to strongly broken supersymmetry, or
find a large set of vacua with a small-scale SUSY break-
ing. The methods used in [9, 10, 11, 12, 13] study the
distribution of flux vacua and do not involve the study of
the shape of the effective potential after the uplifting to
dS state. The relation between the parameters of SUSY
breaking, the height of the barrier stabilizing dS vacua
and the Hubble constant during inflation was not studied
so far. This will be the main subject of our investigation.
First of all, we will show that the gravitino mass in
the KKLT scenario with the superpotential used in [5] is
extremely large, m3/2∼ 6 × 1010GeV. This means that
in the context of this model one should either consider
particle phenomenology with superheavy gravitino, see
e.g. [14, 15], or modify the scenario in such a way as to
make it possible to reduce the gravitino mass by many
orders of magnitude.
We will show also that the Hubble parameter during
inflation in the simplest models based on the KKLT sce-
nario cannot exceed the present value of the gravitino
mass, H<
∼m3/2. This means that even if we succeed
to find the models with small gravitino mass, following
[11, 12, 13], we may face an additional problem of finding
successful inflationary models with extremely small H.
We will then suggest a possible resolution of these
problems in the context of a volume modulus stabiliza-
tion model where the gravitino mass is not related to the
scale of inflation and can be made arbitrarily small.
II.GRAVITINO-HUBBLE RELATION IN THE
SIMPLEST KKLT MODEL
Recent ideas on string cosmology rely on a possi-
bility to stabilize string theory moduli, in particular
the dilaton and the total volume modulus.The sim-
Page 2
2
plest KKLT models [5] with the superpotential of the
form W = W0+ Ae−aρand with the K¨ ahler potential
K = −3ln[ρ+ρ] provide the AdS minima for the volume
modulus ρ = σ + iα at finite, moderately large values
of volume. When this potential is supplemented by a
D-type contribution
σ2 from D3 brane [5] or D7 branes
[16], one finds a de Sitter minimum. This simplest KKLT
model has a minimum at some real value of the field ρ:
ρ = σ, α = 0. This minimum is separated from the
Minkowski vacuum of Dine-Seiberg type at infinite vol-
ume of the internal space by a barrier, which makes the de
Sitter minimum metastable with the lifetime t ∼ 1010120
years.
C
100150200 250
Σ
-2
-1
1
2V
FIG. 1: Thin green line corresponds to AdS stabilized poten-
tial for W0 = −10−4, A = 1, a = 0.1. Dashed line shows
the additional term
σ2, which appears either due to the con-
tribution of a D3 brane or of a D7 brane. Thick black line
shows the resulting potential including the
C = 2.6×10−11, which uplifts the AdS minimum to a dS min-
imum. All potentials are shown multiplied by 1015.
C
C
σ2correction with
Since DiW = 0 in the AdS minimum, its depth is given
by
VAdS = − 3eK|W|2. (1)
Here all functions are calculated at σ = σcr, where σcris
the position of the minimum of the potential prior to the
uplifting. We use the units where M2
P= (8πGN)−1= 1.
Before the uplifting, the potential has only one ex-
tremum, at σ = σcr, and its absolute value exponentially
decreases at σ ≫ σcr. When we add the term
minimum shifts upward in such a way that the new dS
minimum is positioned at σ0≈ σcr. This means that the
values of the function eK(σ)|W(σ)|2in the minimum of
the effective potential remain almost unchanged during
the uplifting. Meanwhile, the value of DiW(σ) in the
minimum after the uplifting is no longer equal to zero,
but it still remains relatively small, DiW(σ0) ≪ W(σ0).
At the dS minimum, the total effective potential must
vanish, with the accuracy of 10−120. Therefore one has
C
σ2
C
σ2, the
0≈ −VAdS = 3eK|W|2.
The gravitino mass in the uplifted dS minimum is given
by
m2
3/2(σ0) = eK(σ0)|W(σ0)|2≈ eK(σcr)|W(σcr)|2=VAdS
3
.
(2)
The gravitino mass can be associated with the strength
of supersymmetry breaking at the minimum where the
total potential is approximately vanishing. Indeed,
VKKLT(σ0) = VF+VD= |F|2−3m2
3/2+1
2D2≈ 0 . (3)
This yields
3m2
3/2≈1
2D2+ |F|2. (4)
Now let us discuss the height of the barrier VBwhich
stabilizes dS state after the uplifting. Since the uplifting
is achieved by adding a slowly decreasing function C/σ2
to a potential which rapidly approaches zero at large σ,
the height of the barrier VB is approximately equal (up
to a factor O(1)) to the depth of the AdS minimum VAdS,
see Fig. 1:
VB∼ |VAdS| ∼ m2
3/2. (5)
To complete the list of important features of this
model, let us remember what should be done to use it
for the description of inflation.
The simplest possibility would be to use the extremum
of the potential of the height VB as an initial point for
inflation. A particular realization of this scenario was
proposed in [17]. (In order to do it, it was necessary to
consider a racetrack superpotential with two exponents).
In this case one has an interesting relation between var-
ious parameters of our model and the Hubble constant
during inflation:
H2≈ VB/3 ∼ |VAdS|/3 ∼ m2
3/2. (6)
One may also achieve inflation by considering dynam-
ics of branes in the compactified space. This involves a
second uplifting, which corresponds to a nearly dS (infla-
tionary) potential added to the KKLT potential VKKLT,
for example in D3/D7 case [18]. The added potential
should be flat in the inflaton direction, and, according to
[18], it has a σ−3dependence on the volume modulus:
Vinfl
tot≈ VKKLT(σ) +V (φ)
σ3
. (7)
Here φ is an inflaton field. The resulting potential as a
function of σ is schematically shown in Fig. 2 for different
values of the function V (φ).
figure that the vacuum stabilization is possible in this
model only for sufficiently small values of the inflaton
potential,
It is apparent from this
Vinfl
tot<
∼c VB∼ c |VAdS| ∼ c m2
3/2, (8)
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3
where c ≈ 3 for the original version of the KKLT model.1
4V
100150200 250 Σ
1
2
3
FIG. 2: The lowest curve with dS minimum is the one from
the KKLT model. The second one describes, e.g., the D3/D7
inflationary potential with the term Vinfl=
KKLT potential; it originates from fluxes on D7 brane. The
top curve shows that when the inflationary potential becomes
too large, the barrier disappears, and the internal space de-
compactifies. This explains the constraint H<
V (φ)
σ3
added to the
∼m3/2.
The key reason for the vacuum destabilization is the
σ−ndependence of the inflaton potential, with n > 0 .
As explained in [24, 25], the runaway σ−ndependence of
the energy density in string theory is quite generic. The
σ−3dependence appears explicitly in the D-term contri-
bution to the vacuum energy, which is the source of the
inflationary potential V (φ)/σ3in D3/D7 inflation [18].
In principle, it might be possible to design inflationary
models where the inflaton potential depends on σ and φ
in a more complicated way due to some nonperturbative
effects involving both fields. This could prevent vacuum
destabilization at large energy density. However, no ex-
amples of such models are known.
Equations (6), (8) provide a strong constraint on the
Hubble constant during inflation in a broad class of
KKLT-based inflationary models:
H<
∼m3/2. (9)
One should note that there could be many stages of
inflation in the early universe, some of which could hap-
pen in a vicinity of a different minimum of the effective
potential in stringy landscape, with much higher barri-
ers surrounding it. Thus it is quite possible that at some
stage of the evolution of the universe the Hubble constant
was much greater than m3/2. However, this could not be
the last stage of inflation. We cannot simply jump to the
1This effect is similar to decompactification of space at large H
studied in [19], and to the dilaton destabilization at high temper-
ature discussed in [20] in a different context. A related effect was
also found in [21, 22] and [23] in models of radion stabilization.
KKLT minimum after the tunneling with bubble forma-
tion following some previous stage of inflation, because
such tunneling would create an open universe. After such
tunneling, we will still need to have a long stage of infla-
tion, which should make the universe flat, form the large
scale structure of the observable part of the universe, and
end by a slow roll to the KKLT minimum. Our results
imply that the Hubble constant H at this last and most
important stage of inflation should be smaller than the
present value of the gravitino mass.
III.
INFLATION IN THE SIMPLEST KKLT MODEL
PROBLEMS WITH SUSY BREAKING AND
Now we are ready to formulate a list of unusual features
of this scenario.
1) If one takes the simplest superpotential of the KKLT
model according to [5], one finds, following (1), (2), that
the gravitino mass in this scenario is extremely large,
m3/2∼√VAdS∼ 2.5 × 10−8Mp∼ 6 × 1010GeV. Other
parameters characterizingthe strength of supersymmetry
breaking have similar magnitude.
many orders of magnitude higher than the gravitino mass
O(1 TeV) often discussed in the literature.
These numbers are
In this situation there are two basic choices. The first
idea that comes to mind is to change the parameters of
the KKLT model in such a way as to reduce the scale
of SUSY breaking and the gravitino mass down to the
TeV scale. This is a rather nontrivial task, which is the
subject of many recent investigations [9, 10, 11, 13]. Our
results add two new problems to the list of the problems
studied in these papers.
First of all, if our observations based on the simplest
KKLT model are generic (this question is not addressed
by the methods of Refs. [9, 10, 11, 12, 13]), then the min-
imum of the KKLT potential is extremely shallow, with
the low barrier height VB ∼ m2
density units.This implies that one should be espe-
cially careful when analyzing the possibility that the field
σ during its cosmological evolution may overshoot the
KKLT minimum and roll over the barrier, which will lead
to decompactification of the 4D space [27]; see [17, 28]
for a list of proposed solutions of this problem.
3/2<
∼10−30in Planck
Another problem is that we will need to find inflation-
ary models with H<
∼1 TeV, i.e. with H<
Planck units. Whereas such models can be quite satis-
factory from the cosmological point of view, no explicit
examples of models of such type have been constructed
so far in string theory with stable internal dimensions.
∼10−15in
Another option is to develop particle phenomenology
based on the models with extremely large scale of SUSY
breaking. This is a very interesting possibility, which was
recently discussed, e.g., in [14, 15].
In this paper we are going to suggest a different route
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4
which may help us to solve the problems discussed above.
IV.NEW FEATURES IN THE LANDSCAPE:
SUPERSYMMETRIC MINKOWSKI VACUA AND
LIGHT GRAVITINO
The problems discussed above are related to the fact
that the simplest KKLT potential has only one minimum,
and this minimum occurs at large negative values of the
effective potential. Therefore we will look for a possibil-
ity to stabilize the volume modulus in a supersymmet-
ric Minkowski minimum. We perform an analysis of the
vacuum structure2keeping the tree-level K¨ ahler poten-
tial K = −3ln[(ρ + ρ)] and a racetrack superpotential
similar to the one recently used in the racetrack inflation
scenario [17]
W = W0+ Ae−aρ+ Be−bρ. (10)
Here W0is a tree level contribution which arises from the
fluxes. The exponential terms arise either from Euclidean
D3 branes of from gaugino condensation on D7 branes,
as explained in [5, 17].
At a supersymmetric vacuum DρW = 0. The super-
symmetric Minkowski minimum then lies at
W(σcr) = 0 , DW(σcr) = 0 . (11)
As in KKLT, we simplify things by setting the imaginary
part of the ρ modulus (the axion field α) to zero, and
letting ρ = ρ = σ. (Even though in some models the
condition α = 0 is not satisfied at the minimum of V (ρ)
[17], we have verified that it is satisfied in the model
which we are going to propose; see Fig. 4.) In addition
we take A,a, B,b and W0to be all real and the sign of
A and B opposite.
We find a simple relation between the critical value of
the volume modulus and parameters of the superpoten-
tial
σcr=
1
a − bln
????
aA
bB
????. (12)
Equations (11) require also a particular relation between
the parameters of the superpotential:
− W0= A
????
aA
bB
????
a
b−a
+ B
????
aA
bB
????
b
b−a
(13)
Note that only solutions with non-vanishing value of W0
are possible in this model; these solutions disappear if we
put A or B equal to zero, as in the original version of the
KKLT model.
2We performed the calculations and we plot the corresponding
potentials using the “SuperCosmology” code [29].
The potential, V = eK?GρρDρWDρW − 3|W|2?, as
the function of the real field ρ = ρ = σ is given by
V =e−2(a+b)σ
6σ2
(bBeaσ+ aAebσ)
×
?Beaσ(3 + bσ) + ebσ(A(3 + aσ) + 3eaσW0)?(14)
It vanishes at the minimum which corresponds to
Minkowski space:
VMink(σcr) = 0 ,
∂V
∂σ(σcr) = 0 . (15)
Thus it is possible to stabilize the volume modulus while
preserving Minkowski supersymmetry.
mass in this minimum vanishes.
The gravitino
An example of the model where the vacuum stabiliza-
tion occurs in the supersymmetric Minkowski vacuum is
given by the theory with the superpotential (10) with
A = 1, B = −1.03, a = 2π/100, b = 2π/99, W0 =
−2 × 10−4.
large or small; they are of the same order as the pa-
rameters used in [5]. The resulting potential is shown
in Figs. 3 and 4. The vacuum stabilization occurs at
σ ≈ 62 ≫ 1, which suggests that the effective 4D su-
pergravity approach used in our calculations should be
valid.
None of these parameters is anomalously
6080 100120 140160Σ
-1
1
2
3
4V
FIG. 3: The F-term potential (14), multiplied by 1014, for the
values of the parameters A = 1, B = −1.03, a = 2π/100, b =
2π/99, W0 = −2 × 10−4. A Minkowski minimum at V = 0
stabilizes the volume at σcr ≈ 62. AdS vacuum at V < 0
stabilizes the volume at σcr ≈ 106. There is a barrier pro-
tecting the Minkowski minimum. The height of the barrier is
not correlated with the gravitino mass, which vanishes if the
system is trapped in Minkowski vacuum.
We have found the supersymmetric Minkowski vacuum
prior to adding any nonperturbative terms ∼ C/σ2re-
lated to D3 brane or D7 branes. We assume, as usual,
that by changing the parameters and by adding the term
C/σ2one can fine-tune the value of the potential in its
minimum to be equal to the observed small constant
Λ ∼ 10−120. What is important for us is that in the first
approximation one can make the gravitino mass vanish
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5
6060
7070
8080
90 90
100
ΣΣ
0
2
4
6
8
10
Α
0
1
2
3
4
V
FIG. 4: The potential as a function of the complex field ρ.
The Minkowski minimum occurs at α = Im ρ = 0, as we have
assumed in the analytic investigation.
as compared to all other parameters of the superpoten-
tial. As a result, the value of m3/2in our model does
not have any relation to the height of the potential, and,
correspondingly, to the Hubble constant during inflation.
An important property of our Minkowski (or near-
Minkowski) vacuum, as well as the dS vacuum obtained
by its uplifting, is that the gravitino mass vanishes (or
nearly vanishes) only in its vicinity. Similarly, restora-
tion of supersymmetry in this minimum implies that all
particles whose mass is protected by supersymmetry are
expected to be light in the vicinity of the minimum. How-
ever, supersymmetry breaks down and all of these parti-
cles become heavy once one moves away from the mini-
mum of the effective potential. This is exactly the situa-
tion required for the moduli trapping near the enhanced
symmetry points according to [30] (see also [31, 32]).
This suggests that the moduli trapping may help us to
solve the overshooting problem in our scenario. The fact
that the minimum of the effective potential is simultane-
ously a trapping point is a distinguishing feature of our
model.
The same model may also have AdS vacua defined by
W(σ) ?= 0 , DW(σ) = 0 . (16)
At the AdS minimum one has
− W0= Ae−aσ(1 +2
3aσ) + Be−bσ(1 +2
3bσ) .(17)
The vacuum energy in this minimum is negative,
V (σ) = −3eK|W|2= −(aAe−aσ+ bBe−bσ)2
6σ
.(18)
In our particular example shown in Figs. 3 and 4 the
AdS minimum occurs at σ ∼ 106.
The supersymmetric Minkowski vacuum is absolutely
stable with respect to the tunneling to the vacuum with
a negative cosmological constant. Indeed, tunneling from
a supersymmetric Minkowski vacuum would require cre-
ation of bubbles of a new phase with vanishing total en-
ergy, which is impossible because of the positive energy
theorems [33].
This state may become metastable after the uplifting of
the Minkowski minimum (or of a shallow AdS minimum)
to the dS minimum with Λ ∼ 10−120. Since the tunneling
will occur through the barrier with mostly positive V (σ),
one would expect that the lifetime of the dS space will
be about 1010120years, as in the simplest KKLT model
[5]. However, this question requires a more detailed in-
vestigation. An important distinction of the tunneling in
the simplest model of Ref. [5] and in the model discussed
above is that in [5] the decay leads to spontaneous de-
compactification of internal space in each bubble of the
new phase, whereas in the model proposed in this pa-
per the tunneling to the space with negative cosmologi-
cal constant leads to the development of a cosmological
singularity inside each of the bubbles [34].
Note that dS space never decays completely.
like in old inflation [35] and in eternal inflation scenario
[36, 37], the volume of its non-decayed parts will con-
tinue growing exponentially. In some of its parts, the
scalar field may jump upward to different minima of the
effective potential [38], supporting an eternal process of
self-reproduction of all possible vacuum states in stringy
landscape.
Just
In conclusion, in this paper we have found that the
height of the barrier and the upper bound on the scale of
inflation in the simplest versions of the KKLT model are
directly related to the present value of the gravitino mass.
These observations could require development of particle
phenomenology with large scale of SUSY breaking, or
inflationary models with a very low scale of inflation,
H<
∼m3/2. We suggested a modification of the original
KKLT scenario where the volume stabilization does not
require an uplifting of a deep AdS minimum, and where
the large scale of inflation is compatible with the small
gravitino mass.
It is a pleasure to thank C. Burgess, M. Dine, S.
Kachru, P. Nilles, F. Quevedo, H. Stoica, S. Trivedi, and
S. Watson for useful discussions. This work was sup-
ported in part by NSF grant PHY-0244728.
Page 6
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