SU-ITP-2007-4
April 4, 2007
Testing String Theory with CMB
Renata Kallosh and Andrei Linde
Department of Physics, Stanford University, Stanford, CA 94305
Abstract
Future detection/non-detection of tensor modes from inflation in CMB observations
presents a unique way to test certain features of string theory. Current limit on the ratio
of tensor to scalar perturbations, r = T/S, is r <∼ 0.3, future detection may take place
for r >∼ 10
−2 − 10−3. At present all known string theory inflation models predict tensor
modes well below the level of detection. Therefore a possible experimental discovery of
tensor modes may present a challenge to string cosmology.
The strongest bound on r in string inflation follows from the observation that in most
of the models based on the KKLT construction, the value of the Hubble constant H during
inflation must be smaller than the gravitino mass. For the gravitino mass in the usual
range, m3/2 <∼ O(1) TeV, this leads to an extremely strong bound r <∼ 10
−24. A discovery
of tensor perturbations with r >∼ 10
−3 would imply that the gravitinos in this class of
models are superheavy, m3/2 >∼ 10
13 GeV. This would have important implications for
particle phenomenology based on string theory.
http://arxiv.org/abs/0704.0647v1
1 Introduction
Cosmological observations may provide several interesting ways of testing string theory, which is
important for its further development. For example, a discovery of the cosmological acceleration
corresponding to the existence of the cosmological constant Λ ∼ 10−120 (in Planck unitsMp = 1,
where Mp = 2.435× 1018 GeV) initially was viewed as a problem for string theory. For a while
is was not known how to describe an accelerating 4D universe in a vacuum state with a positive
energy density. Eventually the problem was resolved by the KKLT construction [1] (developing
on [2]), which allowed to explain acceleration in a metastable vacuum state. Earlier and further
investigation of these issue [3], combined with the ideas of eternal inflation [4, 5], resulted in
the development of the idea of inflationary multiverse [5, 6] and string landscape scenario [7],
which may have important implications for the general methodology of theoretical physics.
There are some other ways in which cosmology can be used for testing string theory. Much
attention of string theory and cosmology communities during the recent few years, starting with
[8], was dedicated to the possible future detection of cosmic strings produced after inflation
[9, 10]. It is viewed as a possible window of a string theory into the real world. If detected,
cosmic strings in the sky may test various ideas in string theory and cosmology.
One may also try to check which versions of string theory lead to the best description of in-
flation, in agreement with the existing measurements of the anisotropy of the cosmic microwave
background radiation produced by scalar perturbations of metric [11]. These measurements
provide an important information about the structure of the inflaton potential [12, 13, 14, 15].
In particular, observational constraints on the amplitude of scalar perturbations, in the slow
roll approximation, imply that
V 3/2
≃ 5× 10−4 , (1.1)
whereas the spectral index of the scalar perturbations is given by
ns = 1− 3
≈ 0.95± 0.02 (1.2)
if the ratio of tensor perturbations to scalar perturbations is sufficiently small, r ≪ 0.1. For
larger values of r, e.g. for r ∼ 0.2, ns = 0.98± 0.02.
However, these data give rather indirect information about V : One can reduce the overall
scale of energy density by many orders of magnitude, change its shape, and still obtain scalar
perturbations with the same properties.
In this sense, a measurement of the tensor perturbations (gravitational waves) [16], or of
the tensor-to scalar ratio r = T/S, would be especially informative, since it is directly related
to the value of the inflationary potential and the Hubble constant during inflation [12],
r = 8
≈ 3× 107 V ∼ 108 H2. (1.3)
The last part of this equation follows from Eg. (1.1) and from the Einstein equation H2 = V/3.
The purpose of this note is to address the issues of string cosmology in view of the possibility
that tensor modes in primordial spectrum may be detected. We will argue here that the possible
detection of tensor modes from inflation may have dramatic consequences for string theory and
for fundamental physics in general. The current limit on the ratio of tensor to scalar fluctuations
is r < 0.3. During the next few years one expects to probe tensor modes with r ∼ 0.1 and
gradually reach the level of r ∼ 0.01. It is believed that probing below r ∼ 10−2 − 10−3 will be
“formidably difficult” [17]. However, the interval between r = 0.3 and r ∼ 10−3 is quite large,
and it can be probed by the cosmological observations.
Expected amplitude of tensor perturbations in stringy inflation appears to be very low,
r ≪ 10−3, see in particular [18, 19]. In Section 2 we will briefly review their results, as well
as some other recent results concerning string theory inflation [20]. In Section 3 we give some
independent arguments using the relation between the maximal value of the Hubble constant
during inflation and the gravitino mass [21], which suggest that in the superstring models based
on generic KKLT construction the amplitude of tensor perturbations in string theory inflation
with m3/2 <∼ 1 TeV should be extremely small, r <∼ 10−24.
One could argue therefore that the experimental detection of tensor modes would be in a
contradiction with the existing models of string cosmology. Let us remember, however, that
many of us did not expect the discovery of the tiny cosmological constant Λ ∼ 10−120, and
that it took some time before we learned how to describe acceleration of the universe in the
context of string theory. Since there exists a class of rather simple non-stringy inflationary
models predicting r in the interval 0.3 <∼ r <∼ 10−3 [22, 23, 24, 28, 25, 26], it makes a lot of
sense to look for tensor perturbations using the CMB experiments. It is important to think,
therefore, what will happen if the cosmological observations will discover tensor perturbations
in the range 10−3 < r < 0.3. As we will see, this result would not necessarily contradict string
theory, but it may have important implications for the models of string theory inflation, as well
as for particle phenomenology based on string theory.
2 Tensor modes in the simplest inflationary models
Before discussing the amplitude of tensor modes in string theory, we will briefly mention what
happens in general non-stringy inflationary models.
The predicted value of r depends on the exact number of e-foldings N which happened after
the time when the structure was formed on the scale of the present horizon. This number,
in turn, depends on the mechanism of reheating and other details of the post-inflationary
evolution.
For N ∼ 60, one should have r ∼ 0.14 for the simplest chaotic inflation model m2φ2/2, and
r ∼ 0.28 for the model λφ4/4. In the slow-roll approximation, one would have r = 8/N for the
model m2φ2/2 and 16/N for the model λφ4/4 [12].
If one considers the standard spontaneous symmetry breaking model with the potential
V = −
(φ2 − v2)2 , (2.1)
with v = m/
λ, it leads to chaotic inflation with the tensor to scalar ratio which can take any
value in the interval 10−2 <∼ r <∼ 0.3, for N ∼ 60. The value of r depends on the scale of the
spontaneous symmetry breaking v [23, 24], see Fig. 1. The situation in the so-called natural
inflation model [25] is very similar [26], except for the upper branch of the curve above the green
star (the first star from below) shown in Fig. 1, which does not appear in natural inflation.
Figure 1: Possible values of r and ns in the theory
(φ2 − v2)2 for different initial conditions
and different v, for N = 60. In the small v limit, the model has the same predictions as the
theory λφ4/4. In the large v limit it has the same predictions as the theory m2φ2. The branch
above the green star (the first star from below) corresponds to inflation which occurs while the
field rolls down from large φ, as in the simplest models of chaotic inflation. The lower branch
corresponds to the motion from φ = 0, as in new inflation.
If one considers chaotic inflation with the potential including terms φ2, φ3 and φ4, one can
considerably alter the properties of inflationary perturbations [27] and cover almost all parts of
the area in the (r, ns) plane allowed by the latest observational data [28].
However, in all of these models the value of r is large because the change of the inflation
field during the last 60 e-folds of inflation is greater than Mp = 1 [29], which is not the case in
many other inflationary models, such as new inflation [30] and hybrid inflation [31], see [29, 32]
for a discussion of this issue. Therefore the bet for the possibility of the observational discovery
of tensor modes in non-stringy inflationary models would be a bet for the triumph of simplicity
over majority.
3 Existing models of string theory inflation do not pre-
dict a detectable level of tensor modes
String theory at present has produced two classes of models of inflation: brane inflation and
modular inflation, see [10, 20, 33] for recent reviews. The possibility of a significant level of
tensor modes in known brane inflation models was carefully investigated by several authors.
The following conclusion has been drawn from our analysis of the work performed by Bean,
Shandera, Tye, and Xu [19]. They compared the brane inflationary model to recent cosmological
data, including WMAP 3-year cosmic microwave background (CMB) results, Sloan Digital Sky
Survey luminous red galaxies (SDSS LRG) power spectrum data and Supernovae Legacy Survey
(SNLS) Type 1a supernovae distance measures. When they used the bound on the distance in
the warped throat geonetry derived by Baumann and McAllister [18], it became clear that in all
currently known models of brane inflation (including DBI models [34]) the resulting primordial
spectrum could not simultaneously have significant deviation from the slow roll behavior and
satisfy the bound [18]. Moreover the slow roll inflation models that satisfy the bound have
very low tensors not measurable by current or even upcoming experiments. The known models
of brane inflation include the motion of a D3 brane down a single throat in the framework of
the KKLMMT scenario [9]. In short, the bound on an inflaton field, which is interpreted as a
distance between branes, does not permit fields with vev’s of Planckian scale or larger, which
would lead to tensor modes. A work on the improved derivation of the bound including the
breathing mode of the internal geometry is in progress [35].
At present, there is still a hope that it may be possible to go beyond the simplest models of
brane inflation and evade the constraint on the field range. However, this still has to be done
before one can claim that string theory has a reliable class of brane inflation models predicting
tensor modes, or, on the contrary, that brane inflation predicts a non-detectable level of tensor
modes.
All known models of modular inflation in string theory (no branes) do not predict a de-
tectable level of gravity waves [33], [20]. The only string theory inspired version of assisted
inflation model [36], N-flation [37], would predict a significant level of tensors, as in chaotic and
natural inflation [22, 25, 26], if some assumptions underlying the derivation of this model would
be realized. The main assumption is that in the effective supergravity model with numerous
complex moduli, tn =
+ iM2R2n, all moduli R
n quickly go to their minima. Then only the
axions φn
remain to drive inflation. The reason for this assumption is that the Kähler potential
depends only on the volume modulus of all two-cycles, R2n = − i2M2 (tn − t̄n), but is does not
depend on the axions φn
(tn + t̄n), so one could expect that the axion directions in the
first approximation remain flat. Recently this issue was re-examined in [20], and it was found
that in all presently available models this assumption is not satisfied. The search for models in
various regions of the string theory landscape which would support assumptions of N-flation is
in progress [38].
Thus at present we are unaware of any string inflation models predicting the detectable
level of gravitational waves. However, a search for such models continues. We should mention
here possible generalizations on N-flation, new types of brane inflation listed in Sec. 5 of [19]
and some work in progress on DBI models in a more general setting [39].
We may also try to find a string theory generalization of a class of inflationary models in
N = 1 d = 4 supergravity, which has shift symmetry and predict large tensor modes. One
model is a supergravity version [40] of chaotic inflation, describing fields Φ and X with
(Φ + Φ̄)2 +XX̄ , W = mΦX . (3.1)
This model effectively reproduces the simplest version of chaotic inflation with V = 1
m2φ2,
where the inflaton field is φ = i(Φ− Φ̄). Here the prediction for r, depending on the number of
e-foldings, is 0.14 <∼ r <∼ 0.20.
Another model is a supergravity version [20] of natural inflation [25].
(Φ + Φ̄)2 , W = w0 +Be
−bΦ . (3.2)
This model has an axion valley potential in which the radial part of the complex field quickly
reaches the minimum. Therefore this model effectively reproduces natural inflation with the
axion playing the role of the inflaton with potential V = V0(1 − cos(bφ)) where φ = i(Φ − Φ̄).
Here the possible range of r, depending on the number of e-foldings and the axion decay constant
2 b)−1, is approximately 5× 10−3 <∼ r <∼ 0.20 [26].
Both models have one feature in common. They require shift symmetry of the canonical
Kähler potential K = 1
(Φ + Φ̄)2,
Φ → Φ + iδ , δ = δ̄ . (3.3)
The inflaton potential appears because this shift symmetry is slightly broken by the superpo-
tential.
If supersymmetry will be discovered in future, one would expect that inflationary potential
should be represented by a supergravity potential, or even better, by the supergravity effective
potential derivable from string theory. It is gratifying that at least some supergravity models
capable of prediction of large amplitude of tensor perturbations from inflation are available.
So far, neither of the supergravity models in (3.1), (3.2) with detectable level of gravity waves
was derived from string theory.1 It would be most important to study all possible corners of the
landscape in a search of models which may eventually predict detectable tensor fluctuations, or
prove that it is not possible. The future data on r will make a final judgment on the theories
discussed above.
If some models in string cosmology with r > 10−3 will be found, one can use the detection of
gravity waves for testing models of moduli stabilization in string theories, and in this way relate
cosmology to particle physics. The main point here is that the value of the Hubble constant
during inflation is directly measurable in case that gravity waves are detected.
4 Scale of SUSY breaking, the gravitino mass, and the
amplitude of the gravitational waves in string theory
inflation
So far, we did not discuss relation of the new class of models with particle phenomenology. This
relation is rather unexpected and may impose strong constraints on particle phenomenology
1There is a difference between arbitrary N = 1, d = 4 supergravity model of the general type and models
derived from string theory where various fields in effective supergravity theory have some higher-dimensional
interpretation, like volumes of cycles, distance between branes etc. However, there are situations in string theory
when the actual value of the Kähler potential is not known and therefore models like (3.1), (3.2) are not a priori
excluded.
and on inflationary models: In simplest models based on the KKLT mechanism the Hubble
constant H should be smaller than the present value of the gravitino mass [21],
H <∼ m3/2 . (4.1)
The reason for this bound is that the mass of gravitino at the supersymmetric KKLT minimum
with DW = 0 before the uplifting is given by 3m23/2 = |VAdS|. Uplifting of the AdS minimum
to the present nearly Minkowski vacuum is achieved by adding to the potential a term of the
type of C/σn, where σ is the volume modulus and n = 3 for generic compactification and n = 2
for the highly warped throat geometry. Since the uplifting is less significant at large σ, the
barrier created by the uplifting generically is a bit smaller than |VAdS|. Adding the energy of
the inflaton field leads to an additional uplifting. Since it is also proportional to an inverse
power of the volume modulus, it is greater at the minimum of the KKLT potential than at the
top of the barrier. Therefore adding a large vacuum energy density to the KKLT potential,
which is required for inflation, may uplift the minimum to the height greater than the height
of the barrier, and destabilize it, see Fig. 2. This leads to the bound (4.1).
100 150 200 250 Σ
Figure 2: The lowest curve with dS minimum is the potential of the KKLT model. The
second one shows what happens to the volume modulus potential when the inflaton potential
Vinfl =
V (φ)
added to the KKLT potential. The top curve shows that when the inflaton potential
becomes too large, the barrier disappears, and the internal space decompactifies. This explains
the constraint H <∼ m3/2.
One should note that an exact form of this bound is a bit more complicated than (4.1),
containing additional factors which depend logarithmically on certain parameters of the KKLT
potential. However, unless these parameters are exponentially large or exponentially small, one
can use the simple form of this bound, H <∼ m3/2.
Therefore if one believes in the standard SUSY phenomenology with m3/2 <∼ O(1) TeV, one
should find a realistic particle physics model where the nonperturbative string theory dynamics
occurs at the LHC scale (the mass of the volume modulus is not much greater than the gravitino
mass), and inflation occurs at a density at least 30 orders of magnitude below the Planck energy
density. Such models are possible, but their parameters should be substantially different from
the parameters used in all presently existing models of string theory inflation.
An interesting observational consequence of this result is that the amplitude of the grav-
itational waves in all string inflation models of this type should be extremely small. Indeed,
according to Eq. (1.3), one has r ≈ 3× 107 V ≈ 108 H2, which implies that
r <∼ 10
8 m23/2 , (4.2)
in Planck units. In particular, for m3/2 <∼ 1 TeV ∼ 4 × 10−16 Mp, which is in the range most
often discussed by SUSY phenomenology, one has
r <∼ 10
−24 . (4.3)
If CMB experiments find that r >∼ 10−2, then this will imply, in the class of theories described
above, that
m3/2 >∼ 10
−5 Mp ∼ 2.4× 1013 GeV , (4.4)
which is 10 orders of magnitude greater than the standard gravitino mass range discussed by
particle phenomenologists.
There are several different ways to address this problem. First of all, one may consider
KKLT models with the racetrack superpotential containing at least two exponents and find
such parameters that the supersymmetric minimum of the potential even before the uplifting
occurs at zero energy density [21], which would mean m3/2 = 0. Then, by a slight change of
parameters one can get the gravitino mass squared much smaller than the height of the barrier,
which removes the constraint H <∼ m3/2.
If we want to increase the upper bound on H from 1 TeV up to 1013 GeV for m3/2 ∼ 1
TeV, we would need to fine-tune the parameters of the model of Ref. [21] with a very high
accuracy. Therefore it does not seem easy to increase the measurable value of r in the model
of [21] from 10−24 up to 10−3. However, this issue requires a more detailed analysis, since this
model is rather special: In its limiting form, it describes a supersymmetric Minkowski vacuum
without any need of uplifting, and it has certain advantages with respect to vacuum stability
being protected by supersymmetry were discussed in [41]. Therefore it might happen that this
model occupies a special place in the landscape which allows a natural way towards large r.
We will discuss now several other models of moduli stabilization in string theory to see
whether one can overcome the bound (4.2).
A new class of moduli stabilization in M-theory was recently developed in [42]. In particular
cases studied numerically, the height of the barrier after the uplifting is about Vbarrier ≈ 50m23/2,
in some other cases, Vbarrier ≤ O(500) m23/2 [43]. It seems plausible that for this class of
models, just as in the simplest KKLT models, the condition that Vbarrier ≥ 3H2 is required for
stabilization of moduli during inflation. Since the gravitino mass in this model is in the range
from 1 TeV to 100 TeV, the amplitude of the tensor modes is expected to be negligibly small.
Another possibility is to consider the large volume compactification models with stringy
α′ corrections taken into account [44]. At first glance, this also does not seem to help. The
AdS minimum at which moduli are stabilized before the uplifting is not supersymmetric, which
means that generically in AdS minimum 3m23/2 = |V |AdS + eK |DW |2 ≥ |V |AdS. Upon uplift-
ing, generically the height of the barrier is not much different from the absolute value of the
potential in the AdS minimum, Vbarrier ∼ |V |AdS. As the result, the situation with the destabi-
lization during inflation may seem even more difficult than in the simplest KKLT models: the
extra term due to broken supersymmetry eK |DW |2 6= 0 tends to increase the gravitino mass
squared as compared to |V |AdS. This decreases the ratio of the height of the barrier after the
uplifting to the gravitino mass squared. However, a more detailed investigation of this model
is required to verify this conjecture. As we already mentioned, an important assumption in
the derivation of the constraint H <∼ m3/2 in the simplest version of the KKLT model is the
absence of exponentially large parameters. Meanwhile the volume of compactification in [44] is
exponentially large. One should check whether this can help to keep the vacuum stabilized for
large H .
But this class of models offers another possible way to address the low-H problem: In
the phenomenological models based on [44] the gravitino mass can be extremely large. Phe-
nomenological models with superheavy gravitinos were also considered in [45, 46]. In particu-
lar, certain versions of the split supersymmetry models allow gravitino masses in the range of
1013 − 1014 GeV [46]. Therefore in such models the constraint H <∼ m3/2 is quite consistent
with the possibility of the discovery of tensor modes with 10−3 <∼ r <∼ 0.3 if the problems with
constructing the corresponding inflationary models discussed in the previous section will be
resolved.
We would like to stress that we presented here only a first scan of possibilities available
in string cosmology with regard to detectability of the tensor modes, and so far the result
is negative. More studies are required to have a better prediction of r in string cosmology.
It would be most important either to construct a reliable inflationary model in string theory
predicting tensors with 10−3 <∼ r <∼ 0.3, or prove a no-go theorem. If tensor modes will not
be detected, this issue will disappear; the attention will move to more precise values of the tilt
of the spectrum ns, non-gaussianity, cosmic strings and other issues which will be clarified by
observations in the next few years.
However, a possible discovery of tensor modes may force us to reconsider several basic
assumptions of string cosmology and particle phenomenology. In particular, it may imply
that the gravitino must be superheavy. Thus, investigation of gravitational waves produced
during inflation may serve as a unique source of information about string theory and about the
fundamental physics in general.
Acknowledgments
We are grateful to D. Baumann, R. Bean, S.E. Church, G. Efstathiou, S. Kachru, L. Kofman,
D. Lyth, L. McAllister, V. Mukhanov, S. Shenker, E. Silverstein and H. Tye for very stimulating
discussions. This work was supported by NSF grant PHY-0244728.
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	Introduction
	Tensor modes in the simplest inflationary models
	Existing models of string theory inflation do not predict a detectable level of tensor modes
	Scale of SUSY breaking, the gravitino mass, and the amplitude of the gravitational waves in string theory inflation

  Future detection/non-detection of tensor modes from inflation in CMB
observations presents a unique way to test certain features of string theory.
Current limit on the ratio of tensor to scalar perturbations, r=T/S, is r <
0.3, future detection may take place for r > 10^{-2}-10^{-3}. At present all
known string theory inflation models predict tensor modes well below the level
of detection. Therefore a possible experimental discovery of tensor modes may
present a challenge to string cosmology.
  The strongest bound on r in string inflation follows from the observation
that in most of the models based on the KKLT construction, the value of the
Hubble constant H during inflation must be smaller than the gravitino mass. For
the gravitino mass in the usual range, m_{3/2} < O(1) TeV, this leads to an
extremely strong bound r < 10^{-24}. A discovery of tensor perturbations with r
> 10^{-3} would imply that the gravitinos in this class of models are
superheavy, m_{3/2} > 10^{13} GeV. This would have important implications for
particle phenomenology based on string theory.

Introduction
Cosmological observations may provide several interesting ways of testing string theory, which is
important for its further development. For example, a discovery of the cosmological acceleration
corresponding to the existence of the cosmological constant Λ ∼ 10−120 (in Planck unitsMp = 1,
where Mp = 2.435× 1018 GeV) initially was viewed as a problem for string theory. For a while
is was not known how to describe an accelerating 4D universe in a vacuum state with a positive
energy density. Eventually the problem was resolved by the KKLT construction [1] (developing
on [2]), which allowed to explain acceleration in a metastable vacuum state. Earlier and further
investigation of these issue [3], combined with the ideas of eternal inflation [4, 5], resulted in
the development of the idea of inflationary multiverse [5, 6] and string landscape scenario [7],
which may have important implications for the general methodology of theoretical physics.
There are some other ways in which cosmology can be used for testing string theory. Much
attention of string theory and cosmology communities during the recent few years, starting with
[8], was dedicated to the possible future detection of cosmic strings produced after inflation
[9, 10]. It is viewed as a possible window of a string theory into the real world. If detected,
cosmic strings in the sky may test various ideas in string theory and cosmology.
One may also try to check which versions of string theory lead to the best description of in-
flation, in agreement with the existing measurements of the anisotropy of the cosmic microwave
background radiation produced by scalar perturbations of metric [11]. These measurements
provide an important information about the structure of the inflaton potential [12, 13, 14, 15].
In particular, observational constraints on the amplitude of scalar perturbations, in the slow
roll approximation, imply that
V 3/2
≃ 5× 10−4 , (1.1)
whereas the spectral index of the scalar perturbations is given by
ns = 1− 3
≈ 0.95± 0.02 (1.2)
if the ratio of tensor perturbations to scalar perturbations is sufficiently small, r ≪ 0.1. For
larger values of r, e.g. for r ∼ 0.2, ns = 0.98± 0.02.
However, these data give rather indirect information about V : One can reduce the overall
scale of energy density by many orders of magnitude, change its shape, and still obtain scalar
perturbations with the same properties.
In this sense, a measurement of the tensor perturbations (gravitational waves) [16], or of
the tensor-to scalar ratio r = T/S, would be especially informative, since it is directly related
to the value of the inflationary potential and the Hubble constant during inflation [12],
r = 8
≈ 3× 107 V ∼ 108 H2. (1.3)
The last part of this equation follows from Eg. (1.1) and from the Einstein equation H2 = V/3.
The purpose of this note is to address the issues of string cosmology in view of the possibility
that tensor modes in primordial spectrum may be detected. We will argue here that the possible
detection of tensor modes from inflation may have dramatic consequences for string theory and
for fundamental physics in general. The current limit on the ratio of tensor to scalar fluctuations
is r < 0.3. During the next few years one expects to probe tensor modes with r ∼ 0.1 and
gradually reach the level of r ∼ 0.01. It is believed that probing below r ∼ 10−2 − 10−3 will be
“formidably difficult” [17]. However, the interval between r = 0.3 and r ∼ 10−3 is quite large,
and it can be probed by the cosmological observations.
Expected amplitude of tensor perturbations in stringy inflation appears to be very low,
r ≪ 10−3, see in particular [18, 19]. In Section 2 we will briefly review their results, as well
as some other recent results concerning string theory inflation [20]. In Section 3 we give some
independent arguments using the relation between the maximal value of the Hubble constant
during inflation and the gravitino mass [21], which suggest that in the superstring models based
on generic KKLT construction the amplitude of tensor perturbations in string theory inflation
with m3/2 <∼ 1 TeV should be extremely small, r <∼ 10−24.
One could argue therefore that the experimental detection of tensor modes would be in a
contradiction with the existing models of string cosmology. Let us remember, however, that
many of us did not expect the discovery of the tiny cosmological constant Λ ∼ 10−120, and
that it took some time before we learned how to describe acceleration of the universe in the
context of string theory. Since there exists a class of rather simple non-stringy inflationary
models predicting r in the interval 0.3 <∼ r <∼ 10−3 [22, 23, 24, 28, 25, 26], it makes a lot of
sense to look for tensor perturbations using the CMB experiments. It is important to think,
therefore, what will happen if the cosmological observations will discover tensor perturbations
in the range 10−3 < r < 0.3. As we will see, this result would not necessarily contradict string
theory, but it may have important implications for the models of string theory inflation, as well
as for particle phenomenology based on string theory.
2 Tensor modes in the simplest inflationary models
Before discussing the amplitude of tensor modes in string theory, we will briefly mention what
happens in general non-stringy inflationary models.
The predicted value of r depends on the exact number of e-foldings N which happened after
the time when the structure was formed on the scale of the present horizon. This number,
in turn, depends on the mechanism of reheating and other details of the post-inflationary
evolution.
For N ∼ 60, one should have r ∼ 0.14 for the simplest chaotic inflation model m2φ2/2, and
r ∼ 0.28 for the model λφ4/4. In the slow-roll approximation, one would have r = 8/N for the
model m2φ2/2 and 16/N for the model λφ4/4 [12].
If one considers the standard spontaneous symmetry breaking model with the potential
V = −
(φ2 − v2)2 , (2.1)
with v = m/
λ, it leads to chaotic inflation with the tensor to scalar ratio which can take any
value in the interval 10−2 <∼ r <∼ 0.3, for N ∼ 60. The value of r depends on the scale of the
spontaneous symmetry breaking v [23, 24], see Fig. 1. The situation in the so-called natural
inflation model [25] is very similar [26], except for the upper branch of the curve above the green
star (the first star from below) shown in Fig. 1, which does not appear in natural inflation.
Figure 1: Possible values of r and ns in the theory
(φ2 − v2)2 for different initial conditions
and different v, for N = 60. In the small v limit, the model has the same predictions as the
theory λφ4/4. In the large v limit it has the same predictions as the theory m2φ2. The branch
above the green star (the first star from below) corresponds to inflation which occurs while the
field rolls down from large φ, as in the simplest models of chaotic inflation. The lower branch
corresponds to the motion from φ = 0, as in new inflation.
If one considers chaotic inflation with the potential including terms φ2, φ3 and φ4, one can
considerably alter the properties of inflationary perturbations [27] and cover almost all parts of
the area in the (r, ns) plane allowed by the latest observational data [28].
However, in all of these models the value of r is large because the change of the inflation
field during the last 60 e-folds of inflation is greater than Mp = 1 [29], which is not the case in
many other inflationary models, such as new inflation [30] and hybrid inflation [31], see [29, 32]
for a discussion of this issue. Therefore the bet for the possibility of the observational discovery
of tensor modes in non-stringy inflationary models would be a bet for the triumph of simplicity
over majority.
3 Existing models of string theory inflation do not pre-
dict a detectable level of tensor modes
String theory at present has produced two classes of models of inflation: brane inflation and
modular inflation, see [10, 20, 33] for recent reviews. The possibility of a significant level of
tensor modes in known brane inflation models was carefully investigated by several authors.
The following conclusion has been drawn from our analysis of the work performed by Bean,
Shandera, Tye, and Xu [19]. They compared the brane inflationary model to recent cosmological
data, including WMAP 3-year cosmic microwave background (CMB) results, Sloan Digital Sky
Survey luminous red galaxies (SDSS LRG) power spectrum data and Supernovae Legacy Survey
(SNLS) Type 1a supernovae distance measures. When they used the bound on the distance in
the warped throat geonetry derived by Baumann and McAllister [18], it became clear that in all
currently known models of brane inflation (including DBI models [34]) the resulting primordial
spectrum could not simultaneously have significant deviation from the slow roll behavior and
satisfy the bound [18]. Moreover the slow roll inflation models that satisfy the bound have
very low tensors not measurable by current or even upcoming experiments. The known models
of brane inflation include the motion of a D3 brane down a single throat in the framework of
the KKLMMT scenario [9]. In short, the bound on an inflaton field, which is interpreted as a
distance between branes, does not permit fields with vev’s of Planckian scale or larger, which
would lead to tensor modes. A work on the improved derivation of the bound including the
breathing mode of the internal geometry is in progress [35].
At present, there is still a hope that it may be possible to go beyond the simplest models of
brane inflation and evade the constraint on the field range. However, this still has to be done
before one can claim that string theory has a reliable class of brane inflation models predicting
tensor modes, or, on the contrary, that brane inflation predicts a non-detectable level of tensor
modes.
All known models of modular inflation in string theory (no branes) do not predict a de-
tectable level of gravity waves [33], [20]. The only string theory inspired version of assisted
inflation model [36], N-flation [37], would predict a significant level of tensors, as in chaotic and
natural inflation [22, 25, 26], if some assumptions underlying the derivation of this model would
be realized. The main assumption is that in the effective supergravity model with numerous
complex moduli, tn =
+ iM2R2n, all moduli R
n quickly go to their minima. Then only the
axions φn
remain to drive inflation. The reason for this assumption is that the Kähler potential
depends only on the volume modulus of all two-cycles, R2n = − i2M2 (tn − t̄n), but is does not
depend on the axions φn
(tn + t̄n), so one could expect that the axion directions in the
first approximation remain flat. Recently this issue was re-examined in [20], and it was found
that in all presently available models this assumption is not satisfied. The search for models in
various regions of the string theory landscape which would support assumptions of N-flation is
in progress [38].
Thus at present we are unaware of any string inflation models predicting the detectable
level of gravitational waves. However, a search for such models continues. We should mention
here possible generalizations on N-flation, new types of brane inflation listed in Sec. 5 of [19]
and some work in progress on DBI models in a more general setting [39].
We may also try to find a string theory generalization of a class of inflationary models in
N = 1 d = 4 supergravity, which has shift symmetry and predict large tensor modes. One
model is a supergravity version [40] of chaotic inflation, describing fields Φ and X with
(Φ + Φ̄)2 +XX̄ , W = mΦX . (3.1)
This model effectively reproduces the simplest version of chaotic inflation with V = 1
m2φ2,
where the inflaton field is φ = i(Φ− Φ̄). Here the prediction for r, depending on the number of
e-foldings, is 0.14 <∼ r <∼ 0.20.
Another model is a supergravity version [20] of natural inflation [25].
(Φ + Φ̄)2 , W = w0 +Be
−bΦ . (3.2)
This model has an axion valley potential in which the radial part of the complex field quickly
reaches the minimum. Therefore this model effectively reproduces natural inflation with the
axion playing the role of the inflaton with potential V = V0(1 − cos(bφ)) where φ = i(Φ − Φ̄).
Here the possible range of r, depending on the number of e-foldings and the axion decay constant
2 b)−1, is approximately 5× 10−3 <∼ r <∼ 0.20 [26].
Both models have one feature in common. They require shift symmetry of the canonical
Kähler potential K = 1
(Φ + Φ̄)2,
Φ → Φ + iδ , δ = δ̄ . (3.3)
The inflaton potential appears because this shift symmetry is slightly broken by the superpo-
tential.
If supersymmetry will be discovered in future, one would expect that inflationary potential
should be represented by a supergravity potential, or even better, by the supergravity effective
potential derivable from string theory. It is gratifying that at least some supergravity models
capable of prediction of large amplitude of tensor perturbations from inflation are available.
So far, neither of the supergravity models in (3.1), (3.2) with detectable level of gravity waves
was derived from string theory.1 It would be most important to study all possible corners of the
landscape in a search of models which may eventually predict detectable tensor fluctuations, or
prove that it is not possible. The future data on r will make a final judgment on the theories
discussed above.
If some models in string cosmology with r > 10−3 will be found, one can use the detection of
gravity waves for testing models of moduli stabilization in string theories, and in this way relate
cosmology to particle physics. The main point here is that the value of the Hubble constant
during inflation is directly measurable in case that gravity waves are detected.
4 Scale of SUSY breaking, the gravitino mass, and the
amplitude of the gravitational waves in string theory
inflation
So far, we did not discuss relation of the new class of models with particle phenomenology. This
relation is rather unexpected and may impose strong constraints on particle phenomenology
1There is a difference between arbitrary N = 1, d = 4 supergravity model of the general type and models
derived from string theory where various fields in effective supergravity theory have some higher-dimensional
interpretation, like volumes of cycles, distance between branes etc. However, there are situations in string theory
when the actual value of the Kähler potential is not known and therefore models like (3.1), (3.2) are not a priori
excluded.
and on inflationary models: In simplest models based on the KKLT mechanism the Hubble
constant H should be smaller than the present value of the gravitino mass [21],
H <∼ m3/2 . (4.1)
The reason for this bound is that the mass of gravitino at the supersymmetric KKLT minimum
with DW = 0 before the uplifting is given by 3m23/2 = |VAdS|. Uplifting of the AdS minimum
to the present nearly Minkowski vacuum is achieved by adding to the potential a term of the
type of C/σn, where σ is the volume modulus and n = 3 for generic compactification and n = 2
for the highly warped throat geometry. Since the uplifting is less significant at large σ, the
barrier created by the uplifting generically is a bit smaller than |VAdS|. Adding the energy of
the inflaton field leads to an additional uplifting. Since it is also proportional to an inverse
power of the volume modulus, it is greater at the minimum of the KKLT potential than at the
top of the barrier. Therefore adding a large vacuum energy density to the KKLT potential,
which is required for inflation, may uplift the minimum to the height greater than the height
of the barrier, and destabilize it, see Fig. 2. This leads to the bound (4.1).
100 150 200 250 Σ
Figure 2: The lowest curve with dS minimum is the potential of the KKLT model. The
second one shows what happens to the volume modulus potential when the inflaton potential
Vinfl =
V (φ)
added to the KKLT potential. The top curve shows that when the inflaton potential
becomes too large, the barrier disappears, and the internal space decompactifies. This explains
the constraint H <∼ m3/2.
One should note that an exact form of this bound is a bit more complicated than (4.1),
containing additional factors which depend logarithmically on certain parameters of the KKLT
potential. However, unless these parameters are exponentially large or exponentially small, one
can use the simple form of this bound, H <∼ m3/2.
Therefore if one believes in the standard SUSY phenomenology with m3/2 <∼ O(1) TeV, one
should find a realistic particle physics model where the nonperturbative string theory dynamics
occurs at the LHC scale (the mass of the volume modulus is not much greater than the gravitino
mass), and inflation occurs at a density at least 30 orders of magnitude below the Planck energy
density. Such models are possible, but their parameters should be substantially different from
the parameters used in all presently existing models of string theory inflation.
An interesting observational consequence of this result is that the amplitude of the grav-
itational waves in all string inflation models of this type should be extremely small. Indeed,
according to Eq. (1.3), one has r ≈ 3× 107 V ≈ 108 H2, which implies that
r <∼ 10
8 m23/2 , (4.2)
in Planck units. In particular, for m3/2 <∼ 1 TeV ∼ 4 × 10−16 Mp, which is in the range most
often discussed by SUSY phenomenology, one has
r <∼ 10
−24 . (4.3)
If CMB experiments find that r >∼ 10−2, then this will imply, in the class of theories described
above, that
m3/2 >∼ 10
−5 Mp ∼ 2.4× 1013 GeV , (4.4)
which is 10 orders of magnitude greater than the standard gravitino mass range discussed by
particle phenomenologists.
There are several different ways to address this problem. First of all, one may consider
KKLT models with the racetrack superpotential containing at least two exponents and find
such parameters that the supersymmetric minimum of the potential even before the uplifting
occurs at zero energy density [21], which would mean m3/2 = 0. Then, by a slight change of
parameters one can get the gravitino mass squared much smaller than the height of the barrier,
which removes the constraint H <∼ m3/2.
If we want to increase the upper bound on H from 1 TeV up to 1013 GeV for m3/2 ∼ 1
TeV, we would need to fine-tune the parameters of the model of Ref. [21] with a very high
accuracy. Therefore it does not seem easy to increase the measurable value of r in the model
of [21] from 10−24 up to 10−3. However, this issue requires a more detailed analysis, since this
model is rather special: In its limiting form, it describes a supersymmetric Minkowski vacuum
without any need of uplifting, and it has certain advantages with respect to vacuum stability
being protected by supersymmetry were discussed in [41]. Therefore it might happen that this
model occupies a special place in the landscape which allows a natural way towards large r.
We will discuss now several other models of moduli stabilization in string theory to see
whether one can overcome the bound (4.2).
A new class of moduli stabilization in M-theory was recently developed in [42]. In particular
cases studied numerically, the height of the barrier after the uplifting is about Vbarrier ≈ 50m23/2,
in some other cases, Vbarrier ≤ O(500) m23/2 [43]. It seems plausible that for this class of
models, just as in the simplest KKLT models, the condition that Vbarrier ≥ 3H2 is required for
stabilization of moduli during inflation. Since the gravitino mass in this model is in the range
from 1 TeV to 100 TeV, the amplitude of the tensor modes is expected to be negligibly small.
Another possibility is to consider the large volume compactification models with stringy
α′ corrections taken into account [44]. At first glance, this also does not seem to help. The
AdS minimum at which moduli are stabilized before the uplifting is not supersymmetric, which
means that generically in AdS minimum 3m23/2 = |V |AdS + eK |DW |2 ≥ |V |AdS. Upon uplift-
ing, generically the height of the barrier is not much different from the absolute value of the
potential in the AdS minimum, Vbarrier ∼ |V |AdS. As the result, the situation with the destabi-
lization during inflation may seem even more difficult than in the simplest KKLT models: the
extra term due to broken supersymmetry eK |DW |2 6= 0 tends to increase the gravitino mass
squared as compared to |V |AdS. This decreases the ratio of the height of the barrier after the
uplifting to the gravitino mass squared. However, a more detailed investigation of this model
is required to verify this conjecture. As we already mentioned, an important assumption in
the derivation of the constraint H <∼ m3/2 in the simplest version of the KKLT model is the
absence of exponentially large parameters. Meanwhile the volume of compactification in [44] is
exponentially large. One should check whether this can help to keep the vacuum stabilized for
large H .
But this class of models offers another possible way to address the low-H problem: In
the phenomenological models based on [44] the gravitino mass can be extremely large. Phe-
nomenological models with superheavy gravitinos were also considered in [45, 46]. In particu-
lar, certain versions of the split supersymmetry models allow gravitino masses in the range of
1013 − 1014 GeV [46]. Therefore in such models the constraint H <∼ m3/2 is quite consistent
with the possibility of the discovery of tensor modes with 10−3 <∼ r <∼ 0.3 if the problems with
constructing the corresponding inflationary models discussed in the previous section will be
resolved.
We would like to stress that we presented here only a first scan of possibilities available
in string cosmology with regard to detectability of the tensor modes, and so far the result
is negative. More studies are required to have a better prediction of r in string cosmology.
It would be most important either to construct a reliable inflationary model in string theory
predicting tensors with 10−3 <∼ r <∼ 0.3, or prove a no-go theorem. If tensor modes will not
be detected, this issue will disappear; the attention will move to more precise values of the tilt
of the spectrum ns, non-gaussianity, cosmic strings and other issues which will be clarified by
observations in the next few years.
However, a possible discovery of tensor modes may force us to reconsider several basic
assumptions of string cosmology and particle phenomenology. In particular, it may imply
that the gravitino must be superheavy. Thus, investigation of gravitational waves produced
during inflation may serve as a unique source of information about string theory and about the
fundamental physics in general.
Acknowledgments
We are grateful to D. Baumann, R. Bean, S.E. Church, G. Efstathiou, S. Kachru, L. Kofman,
D. Lyth, L. McAllister, V. Mukhanov, S. Shenker, E. Silverstein and H. Tye for very stimulating
discussions. This work was supported by NSF grant PHY-0244728.
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	Introduction
	Tensor modes in the simplest inflationary models
	Existing models of string theory inflation do not predict a detectable level of tensor modes
	Scale of SUSY breaking, the gravitino mass, and the amplitude of the gravitational waves in string theory inflation
