Page 1

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)

Page 3

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

Page 4

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

Page 5

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.