Towards an explicit model of large field inflation
Accepted: May
Towards an explicit model of large eld in ation
Uppsala 0
Sweden 0
0 3001 Leuven , Belgium
1 Institutionen for fysik och astronomi, Uppsala University
2 Institute for Theoretical Physics, KU Leuven
The unwinding in ation mechanism is studied in a type IIB cation where all moduli are stabilized using 0 corrections of the large volume scenario. We consider the backreaction on the geometry due to the presence of antiD3 branes as well as the backreaction of in ation on the Kahler moduli, and compute the resulting corrections to the slowroll potential. By taking large ux numbers, we are able to nd in ationary epochs where backreaction e ects are under control, the in aton traverses a superPlanckian of scalar perturbations is consistent with observation.
Cosmology of Theories beyond the SM; Flux compacti cations; Dbranes

1 Introduction
2
3
4
5
2.1
2.2
3.1
3.2
3.3
4.1
4.2
5.1
5.2
5.3
Stabilizing moduli
Complex structure moduli
Kahler moduli
The in ating sector
The unwinding mechanism
Unwinding in KlebanovStrassler
Unwinding in the Sdual of KlebanovStrassler
Realizing in ation
Backreaction
Constraints on parameter space
Examples
Backreaction of antibranes on net D3 charge Backreaction on Kahler moduli Breaking supersymmetry
6
Conclusions A Examples in KlebanovStrassler 1 4
the Planck collaboration [
1
]. While this may be considered a triumph of the in ationary
paradigm, in order to move forward with our exploration of the early universe and the high
energy physics that was at play, it is necessary to cull in ationary models that cannot be
embedded in a UV complete theory of quantum gravity (read: string theory). Since the
parameters of UV complete models of in ation should be xed by the vacuum expectation
values (vevs) of string moduli, it is reasonable to expect that this set of models will be
far more restricted, and therefore far more predictive, than the full set of EFT models.
A UV complete embedding of in ation is desirable for its increased predictivity and the
glimpse it provides of quantum gravity, and it is moreover essential for large eld theories
of in ation, where the in aton traverses a superPlanckian
eld range and therefore the
expansion in = ( eld over cuto ) implicit in any EFT construction, does not converge.
{ 1 {
There is a growing literature of various \swampland" conjectures [2{5] which aims
at formulating general principles which can be used to distinguish UVcompletable EFT's,
that lie somewhere in the string landscape, from the rest which are relegated to the
swampland. The importance of understanding such underlying principles of quantum gravity is
di cult to overstate, however to date we lack a proof for any of the conjectures. While
there is a growing body of suggestive arguments stemming from AdS/CFT, black hole
physics, and folk theorems, some of the primary evidence for the conjectures lies in a lack
of counterexamples. Thus, we view the best way to further support or sharpen said
conjectures is to search for a counterexample. Speci cally, we aim to nd a large eld model of
in ation that takes into account backreaction on string moduli. This would help to sharpen
and quantify the claim of [
3
]: \We cannot have a slow roll in ation where the distance in
the scalar moduli space is much bigger than Planck length and still use the same e ective
eld theory."
We succeed in
nding a large eld model that takes into account the backreaction
on string moduli by embedding the unwinding in ation mechanism [6, 7] in a
KlebanovStrassler [8] (KS) throat region of a compact manifold.
We use the setting of warped
orientifold compacti cations of type IIB on CalabiYau threefolds [
9
] where all complex
structure moduli are stabilized at tree level. Additionally, we use the large volume scenario
(LVS) [10, 11] to break the noscale structure and stabilize a model with two Kahler moduli.
The unwinding mechanism achieves slow roll in ation by gradually decreasing the positive
vacuum energy sourced by antiD3 branes. This is mediated by a 5brane bubble which
expands, crossing many times over the S3 at the tip of the KS geometry. As the 5brane
bubble moves across the S3, it removes both threeform
ux and antiD3 branes via
braneux annihilation [12]. The repeated motion over a compact cycle, removing ux with each
pass , realizes a
ux cascade [13] which gradually decreases the fourdimensional vacuum
energy. For a more detailed description of this process, see section 3.1.
One of the main purposes of this work is to fortify the original embedding of unwinding
in ation in string theory [7] by taking into account backreaction. The backreaction e ects
we consider are 1) the e ect of antiD3 brane charge on the warping of the KS geometry,
and 2) the evolution of the Kahler moduli's potential due to the depletion of antiD3 charge
during the unwinding process. We compute these e ects and
nd scenarios in which they
do not spoil in ation. While we are able to explicitly compute the contribution to the slow
roll potential due to Kahler moduli backreaction, we only consider the leading UV e ect of
the backreacted antibranes. An explicit calculation of backreaction involving the
supersymmetry breaking e ects of the antibranes remains beyond the scope of this work, however
the backreaction of antiD3 branes in the KS geometry is extremely wellstudied [14{19],
making future e orts to compute these higher order backreaction particularly tractable.
Further, we estimate the e ects of supersymmetry breaking backreaction and argue that it
is tunably small in the UV. Due to this wellstudied setting, this model serves as an ideal
setting to further solidify or rule out the claim of a transPlanckian eld range.
We nd that in order to obtain large eld ranges where backreaction e ects do not spoil
in ation and all moduli are stabilized at high mass, we need large numbers of threeform
ux in the KS throat. Unless there are additional sources of negative D3 charge outside of
{ 2 {
the throat region, which enters with opposite sign in the tadpole condition (3.3), the
threefrom
ux in the KS region implies an Euler characteristic for the associated CalabiYau
fourfold that is larger than known examples. Consequently, we do not have an explicit
global example of a CalabiYau threefold which accommodates our model; hence towards
an explicit model of large eld in ation.
Beyond the consideration of underlying principles in quantum gravity, we are also
interested in whether this model can reproduce the observed cosmic microwave background
(CMB.) After all, it is the aim of every model of in ation to explain the observed features
in the CMB and additionally teach us about the high energy physics at play in the early
universe. In this respect, the current embedding of the model has passed the rst test
for remaining a viable model of in ation. We are able to achieve an in ationary period
of at least 60 efolds that gives rise to the correct amplitude of Gaussian curvature
perturbations. We further expect a rich phenomenology including equilateral and resonant
nonGaussianity and observable tensors. However, the details of these observables will be
sensitive to the explicit details of the global CalabiYau threefold through the highly
nonlinear constraints that CalabiYau with di erent moduli masses will impose on parameter
space. In light of this, we see little point in computing speci c observables in lieu of a
fully explicit embedding. Rather, we take the embedding using a generic Swisscheese style
manifold with two Kahler moduli as a proof of principle that this model is viable and
warrants further study.
Finally, it is necessary to mention some important caveats to this work, and the
construction of positive vacuum energy solutions in string theory in general. First, recent
work [20] has shown that the antibrane uplift in the KKLT [21] construction of de Sitter
vacua with a single Kahler modulus is not su cient to achieve positive vacuum energy.
This proof does not directly apply to time dependent backgrounds, or to the LVS
stabilization mechanism we use, and therefore it presents no immediate obstacle to our model.
However, by showing that, in the case of a single Kahler modulus, the antibrane uplift
does not simply add to the Kahler moduli potential, [20] invites serious reservations as to
the accuracy of all de Sitter constructions using antibranes. A better understanding of the
positive energy added by antibranes in time dependent backgrounds with multiple Kahler
moduli would certainly bene t the string cosmology community and be extremely relevant
to this model. Additionally, the recent work [22] argues that the use of nonperturbative
e ects in the superpotential  needed to stabilize Kahler moduli  is not trustworthy in
the presence of supersymmetry breaking
ux. This analysis casts suspicion on all known
methods of Kahler moduli stabilization using nonperturbative e ects. While there are
no nogo's which prevent stabilization, further study of nonperturbative e ects in time
dependent backgrounds is clearly an important topic for future research.
The plan for the remainder of this paper is as follows: stabilize, in ate, and backreact.
In section 2 we specify the setup in which we will stabilize all moduli using both uxes at
treelevel, and also the leading 0 corrections in the LVS potential. In section 3 we explain
the in ationary mechanism and derive the in ationary potential. We also introduce the rst
e ect of backreaction, taking into account the presence of antibranes in reducing the depth
of the KS throat. In section 4 we collect the long list of constraints on parameter space that
{ 3 {
must be respected in order for our approximations to be valid. Then, we present explicit
realizations which respect these constraints and discuss their properties and observables.
In section 5 we discuss backreaction e ects at length, explicitly computing e ects due to
the evolution of the Kahler moduli, as well as estimating the results of supersymmetry
breaking on the asymptotic KS geometry. Lastly, we discuss possible avenues for future
research in section 6.
2
Stabilizing moduli
Before the in ationary dynamics of a fourdimensional lowenergy e ective theory
descending from a string compacti cation can be considered, one must ensure the stabilization of
the geometric moduli of the compact manifold. These moduli must receive masses above
the fourdimensional Hubble scale so that they can be safely integrated out of the
fourdimensional EFT. In this section we discuss the potentials which result in stable minima
for the complex structure and Kahler moduli. A discussion of the backreaction of the
in ationary dynamics on these moduli will be presented in section 5.
2.1
Complex structure moduli
To achieve a hierarchy of scales between the compact manifold and a fourdimensional
cosmology, we use the warped orientifold compacti cations of Giddings, Kachru and
Polchinski [
9
]. In [
9
] the authors show that all complex structure moduli can be stabilized by ux
via the GukovVafaWitten treelevel superpotential [23]:
Z
X
W0 =
G3 ^
;
R6
mcs
where G3 = F3
H3,
is the axiodilaton and
is the holomorphic (3; 0)form of the
internal manifold X. Furthermore, the axiodilaton,
= C0 + ie
, can be xed by
threeform
ux. We will take C0 = 0, and use a constant dilaton. The values of W0 and gs are
xed by global ux numbers and we will take them to be free parameters.
The masses of the complex structure moduli are estimated by [21] mcs
0=R3, where
V is a typical length scale of the compact geometry (a precise de nition of V is given
in (2.2)). We will require this mass to be larger than the Hubble constant during in ation:
H such that the complex structure moduli remain stabilized and nondynamical.
However, at treelevel, the compacti cations of [
9
] are `noscale' models, meaning that the
Kahler moduli are at directions in moduli space. In the next subsection we will add nonperturbative and 0 corrections in order to stabilize the overall volume.
2.2
Kahler moduli
In the previous string embedding of unwinding in ation [7], the KKLT model [21]
was employed to stabilize a single Kahler modulus corresponding to a fourcycle
volume. That treatment fell short in that the overall volume modulus, which determines
the fourdimensional Planck mass, was not explicitly tied to the single stabilized
fourcycle and was treated as an independent parameter.
Here, we will employ the minimal LVS scenario [10, 11] with two Kahler moduli and the leading 0 correction to the (2.1) { 4 {
warped bulk (hbulk
V
given by:
In this ansatz, g4 is the metric on fourdimensional spacetime, dsCY0 is the line element on
the compact CalabiYau rescaled so that it has unit volume,1 h(y) describes the warping,
and V is the dimensionless volume modulus related to physical volumes via vol6
V(2 )6 03. In what follows we assume that the compact manifold consists of a weakly
Vls6 =
2=3) and a small warped throat region (hKS
2=3). Using this
ansatz to reduce to a fourdimensional Einstein frame we nd that the Planck mass is
Mp2l =
2
1
We start by specifying a string frame metric in which the stabilized volume modulus
can be identi ed with a constant shift of the warp factor [25{27]:
where the approximation is good in the limit where warping can be neglected over the
majority of the compact manifold.
We will assume a `strong Swisscheese' structure for the CalabiYau threefold with
only two Kahler moduli:
V = big
3=2
3=2
b3i=g2 :
Generally one should allow for a linear combination of moduli V =
simplicity, we set ;
= 1. We view this as conservative since including
and
( b3i=g2
parameters only increases the exibility of the model. In lieu of an explicitly constructed
CalabiYau which accommodates our model2 this structure is motivated by the fact that
stabilization of several moduli has been explicitly carried out in Swiss cheese examples
using LVS [29]. Further, it is argued in [30] that in
bred CalabiYau, one expects to nd
vacua at large volume where a large number of moduli can be stabilized by higher order
corrections in gs. Therefore, we use the structure (2.5) as a proof of principle for this model
and assume that the specialization to an explicit example which accommodates a suitable
warped throat region is possible.
The string frame LVS Kahler and superpotential are given by:
(2.5)
3=2). For
as tunable
K =
2 log
V +
2
+ log gs + Kcs ;
2
W = W0 + X Aie ai i=gs :
(2.6)
1It is important to note a subtlety regarding (2.3): the rescaling using the volume modulus does not
rescale the radius of the S
3 at the tip of the KS throat which is a blowup cycle whose size is xed by
uxes [28].
Euler number.
2While there are explicit examples that match the structure (2.5), we additionally require a KS throat
The Kahler potential includes the leading 0 correction given by the =2 term. Here, is
a positive constant that depends on the Euler number of the CalabiYau threefold and
Kcs = log
i R
^
. In the following,
will be treated as a free parameter and Kcs = 0.
In the superpotential, Ai are complex structuredependent constants, and ai = 2 =Ni where
Ni = 1 if the nonperturbative e ect on the particular fourcycle i arises from a Euclidean
D3 brane, and Ni = ND7 if the nonperturbative e ect is gaugino condensation on a stack
of D7 branes wrapped on i
.
Because of the assumed hierarchy, big
, nonperturbative e ects are only
relevant on the smaller fourcycle. Then, the contribution of the LVS potential to the
fourdimensional supergravity action is:
where we choose to parameterize the moduli using the small fourcycle, , and the overall
3=2
big . The prefactor to the term in square brackets is necessary in
order for the usual Fterm potential arising from (2.6) to match the result of dimensional
h i
hVi
2
2
3
;
4aA
3gs W0 h ieah i=gs :
p
Here, the moduli are stabilized using only the potential in square brackets in (2.7), and
only after a stable minimum is found should the overall potential be scaled by the
factor gs4Mp4=(8 ).
We will now lift this AdS minimum by adding the e ect of the p antiD3 branes.
The antibranes contribute to the Kahler moduli potential via a positive `uplift' that scales
like V
2p=(gsh0V
4=3 [31]. The uplift potential is calculated in the next section; the result is Vup =
4=3). Then, the expectation values of the Kahler moduli in the presence of p
antibranes is given by minimizing the following potential:
V ( ; V) =
2p
gsh0V4=3 +
8 (aA)2 p e 2a =gs
3 g2
s
V
4aAW0
p antiD3 branes have annihilated against ux, is the most dangerous point for the stability
of V. Then, the modulus V becomes more stable throughout the cascade as can be seen
from
gure 1.
In order to treat this model as single eld in ation, we need the masses of the Kahler
moduli to be greater than the Hubble parameter, and the expectation values not to evolve
3The approximation here is due to neglecting order one factors relative to ags
good in the limit that higher order instanton e ects can be ignored.
1, and is therefore very
{ 6 {
cascade and at the end. Plot made using parameter set 1 given in table 1.
dramatically during in ation. Using the results of [30, 32], the canonically normalized mass
eigenstates corresponding to the two Kahler moduli of a strong Swiss cheese model are to
leading order in a large V expansion:
m2~
m2~b
8
8
g4eKcs 18a2gs3W02 p
s
2
g4eKcs 729gs6 W302 Mp2 ;
s
8p2a V
V
2
Mp2 ;
(2.11)
(2.12)
where by m~(m ~b ) we indicate the mass eigenstate that is nearly aligned with the
small(large) Kahler modulus.
These masses are useful for order of magnitude estimates, however are subject to
correction in several respects. First, they are computed in the AdS minimum; uplifting the
potential is expected to reduce m ~b by an order one factor and leave m~ nearly invariant [32].
Second, the numerical factors are sensitive to the factors ;
which we discuss under (2.5).
While we set ;
= 1, the moduli masses could change dramatically if ; 6 1. Third, and
most importantly, we will nd that for the cases studied in section 4.2 the mass of m~ is
very close to the estimated KaluzaKlein scale of the throat,4 mKK
0 1=2gs1=4
1=4. The
other moduli mass, m ~b , is safely below all estimates of mKK . When m~
mKK the
estimate (2.11) is no longer trustworthy and particularly this could indicate light KaluzaKlein
4Due to the large hierarchy between cycles in the strong Swisscheese model, the usual naive estimate
mKK / V 1=6 is no longer accurate and we should use a more local de nition based on individual
4cycles [11, 30].
{ 7 {
modes entering the e ective eld theory as predicted by the swampland conjecture [
3
]. In
lieu of an explicit geometry the estimates of m~ and mKK are only reliable up to order of
magnitude, however an explicit calculation of moduli and KaluzaKlein masses
demonstrating that this model cannot obey the required hierarchy would serve as additional evidence
for the swampland conjecture. However, this potential inconsistency in the hierarchy can
also be avoided due to the fact that the coupling of the KaluzaKlein modes to
elds in
the e ective theory should be additionally suppressed by powers of the volume [33]. It is
crucial to check this hierarchy where m~ and mKK can be reliably computed; given the
current lack of an explicit geometry which accommodates our model, this is beyond the
current scope of this work.
Additionally, we must check that the moduli's expectation values do not evolve
significantly over the in ationary epoch. If this were not the case backreaction e ects on the
in aton trajectory will be strong, potentially spoiling in ation or reducing the eld range.
We compute backreaction e ects due to the evolution of the moduli vevs in section 5.2.
3
3.1
The in ating sector
The unwinding mechanism
Having stabilized all moduli, we move on to the embedding of the unwinding in ation
mechanism into warped throat geometries. We refer the reader to [6] for more details on
the general mechanism, and [7] for a more indepth description of the speci c realization
used here. The recipe for in ation calls for p antiD3 branes in a warped throat region of
the compact geometry (we will study the scenario both in the KS throat and its sdual
(SDKS)). Due to the presence of F5
ux in the throat geometry, the antiD3 branes are
forced to the point of highest warping  the tip of the throat. Once at the tip of the
throat, the antibranes polarize via the Myers e ect [34] into a 5brane which wraps an S2
localized in the polar direction,
, of the S3 at the tip of the deformed conifold. This
5brane can be thought of as a bubble with decreased threeform
ux (under which it is
magnetically charged) in the interior. Through the process of brane ux annihilation [12],
the reduction of threeform
ux is accompanied by the reduction of antibrane charge.
We will be interested in the case where this brane ux annihilation has no xed point,
but rather results in an unstable potential for the 5brane. Then, the 5brane bubble will
expand in the
direction and move across the compact space, removing threeform
ux
and decreasing p as it goes. This repeated brane ux annihilation is an instance of a ux
cascade [13] which results in a slow roll potential. As in the original unwinding model,
the canonically normalized in aton is proportional to the distance in the
direction that
the 5brane has moved. A closely related model which uses a single instance of brane ux
annihilation (as opposed to ux cascade) to give rise to slow roll in ation can be found
in [
35
].
In this section we reproduce the local embedding of the unwinding in ation mechanism
in the KS geometry and the SDKS geometry that was presented in [7]. Furthermore, we
extend the previous work to incorporate a uni ed treatment of global volume modulus
stabilization with the in ationary dynamics. This results in an action describing the motion
{ 8 {
of the 5brane on the S3, i.e. our in aton action. Ultimately, we prefer to embed the
unwinding mechanism in SDKS because the stack of antiD3 branes polarize into a D5,
as opposed to an NS5 whose action is not known at weak string coupling. However, for
the sake of presentation, we begin with the familiar KS geometry, and then summarize the
SDKS result afterwards.
The literature on the KS geometry is both rich and vast, and we will refer the reader to
previous works for a full appreciation of the solution. Particularly, we have found [36] to
be a valuable resource and we will follow their conventions here. The KS solution describes
a warped product of a fourdimensional spacetime with the deformed conifold, de ned by
Pi4=1 zi2 = 2, where zi are coordinates on C4. In the limit
over T 1;1; nonzero
corresponds to blowing up the conical singularity on the S3 of T 1;1.
! 0 this is a singular cone
Striving towards a cosmologically viable model, we need to work in a compacti ed
geometry. Then, the KS throat is thought of as a localized region of a compact
CalabiYau [
9
]. In a compact manifold, the threeform
ux in the KS throat is quantized on both
the blowup S3 and its dual, S3:
HJEP05(218)7
The other relevant details of this solution are:
1
Probe antiD3 branes in this geometry will feel a force due to the presence of F5
ux which drives them to the region of highest warping  the bottom of the KS throat.
Therefore, the ux cascade is localized and con ned at the bottom of the throat where h(y)
V and the metric is of the form [36]:
dst2ip = V
1=3h0 1=2g4 dx dx + b02gsM 0(d 32 +
h( = 0)1=2
0
h1=2 =
b0gsM 0 4=3 ;
2
is a dimensionless radial coordinate which is equal to zero at the tip of the throat
0:933. Note that the size of the blowup S3 at the tip of the throat does not scale
{ 9 {
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
under (2.3)  that is, the size of a blowup cycle is xed by uxes, regardless of the size
of the manifold it is attached to.
In a compact manifold, the warped hierarchy between the tip (IR) and top (UV) of
the throat is controlled by the deformation parameter, . In the absence of probe branes,
this parameter is given by [
9
]:
(20=)3 = rUVe 2 K=(3gsM) ;
where the (0) subscript is used to refer to 0th order in the probe approximation. We
also introduce the radial coordinate, r, which is related to
at large
3 4=3 exp(2 =3)=25=3, and by rUV we denote the location where the throat transitions into
In the large r limit the KS metric is given by [36]:
ds2UV = V
1=3hUV1=2g4 dx dx +
h(r ! 1)
hUV =
L4
r4
r
log 2=3
1
Vh1U1==V23 (dr2 + r2dsT1;1 ) ;
log(1024=729)
12
;
L4 =
81(gsM 0)2
8
(3.6)
via r
2 =
(3.7)
: (3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
The numerical factors in hUV are insigni cant and will be dropped henceforth. This UV
expression matches the asymptotics of the singular KlebanovTseytlin (KT) solution [37]:
ds2KT = hKT1=2g4 dx dx + h1K=T2(dr2 + r2dsT1;1 ) ;
hKT(r) =
L4
r4 log
r
rs
=
L4
r4
log
r
rUV
+
which is a noncompact solution with a singularity at r = rs, containing both three form ux
and N D3branes. To match the asymptotics of the two solutions (neglecting insigni cant
numerical factors) one should associate the location of the singularity in KT with the scale
of the deformation in KS, rs !
2=3, and the number of branes in KT should be replaced
by the total D3 charge in the compact KS geometry, N = KM
p. This asymptotic charge
matching takes into account the presence of the probe antibranes as they lower the net D3
charge responsible for the depth of the throat. The resulting deformation parameter is [14]:
which reduces to (3.6) when the probe approximation is valid: p
KM . Following [38],
we de ne the place where the throat attaches to the bulk manifold, rUV, as the radial
position where the warp factor (3.8) is unity; this de nes:
2=3 = rUVe 3gs2M2 (KM p)
;
4
rUV =
Taking into account the presence of the probe branes in the warp factor constitutes
the leading order backreaction on the UV geometry and is crucial to the reliability of our
in ationary model. We will discuss this further, as well as additional backreaction e ects,
in section 5.
With these details in hand, we can compute the action for the
probe NS5brane formed via the polarization of p antiD3 branes [12]:
SNS5 =
d6 [ det(Gk) det(G? + 2 gsF2)]1=2
5
Z
B6 ;
with
The stack of p antiD3 branes polarize into a bubble that is extended in a fourdimensional
FLRW cosmology and the polar angle on the S3, while G
? describes the S2 on the S3 which
the bubble wraps. For the details of the computations see e.g. [
7, 12, 35
]. The resulting
action, taking into account the volume modulus for the compact geometry (3.4), is:
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
SNS5 =
=
M V
2=3h 1 Z
0
where we use 3 = (2 )2 0 5 = (2 ) 3 0 2 and de ne:
V2( ) =
qb40 sin4( ) + U 2( ) ;
C =
M h0 1
Here, we use (2.4) to pull out an overall factor Mp4gs4=8 in order to identify the uplift
term, Vup = 2CU ( ), in the Kahler potential (2.10).
Expanding the DBI kinetic term in (3.16) in both small velocity and large5 p=M we
identify the canonically normalized in aton, :
=
p
8
Mp2gs2 b0h0 1=4p
pM 0
V5=6
f (V) :
This expression assumes that the volume modulus V remains constant and so is only valid
in the regime of small backreaction. Taking into account the backreaction of in ation will
also alter the eld range  we compute the e ect of this backreaction in section 5.2. In
terms of the canonically normalized eld, we can write the full in aton potential, including
the LVS vacuum contribution (2.7):
Z
S =
The potential, which is linear with oscillations is shown in gure 2.
5Large p=M is the regime of parameter space where a ux cascade is possible.
Unwinding in the Sdual of KlebanovStrassler
Taking the Sdual of the KS geometry switches the positions of the F3 and H3 ux, such
that the stack of antiD3 branes polarize into a D5 brane as opposed to an NS5 brane. It
should be noted that while Sduality is a strongweak duality, taking gs ! gs 1, we use
it as a solution generating technique, and then study the new solution at weak coupling.
Therefore the two cases we look at, KS vs SDKS, are not identical as they are both studied
at weak coupling. Switching F3 and H3 takes (3.1) to:
1
Z
Using tildes to indicate quantities are given in the Sdual geometry, the form of the metric
at the tip of the SDKS throat is [39]:
Following steps identical to those of section 3.2 but now matching to the Sdual of the
KlebanovTseytlin geometry, the deformation parameter becomes:
and the warp factor at large r is equal to unity at:
~2=3 = r~UV e 32K2gs (KM p) ;
4
r~UV =
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
The action for a probe D5brane is:
with
SD5 =
gs
The D5 action in the Sdual of the KS geometry was calculated in [7], following closely
the original computation for an NS5brane in KS [12]. Again, accounting for the correct
treatment of the volume modulus (3.21), we nd:
with analogous de nitions:
of (3.18) and (3.19):
and
4
Realizing in ation
V~2( ) =
qb40 sin4( ) + U~ 2( ) ;
C~ =
Kh~0 1
Expanding the action for the canonically normalized eld, we nd the SDKS analogues
= p
8
Mp2gs2 b0h~0 1=4ppK 0
pgsV5=6
f~(V) ;
S =
Z
Mp4gs4 C~(V~2( ) + U~ ( )) +
In this section we will present several realizations of in ationary epochs. Before moving to
speci c scenarios, we will accumulate the various constraints on parameters that must be
satis ed for the consistency of the description we have presented thus far.
4.1
Constraints on parameter space
Validity of the string loop and 0 expansion: the assumptions of supergravity require:
M; K
gs
V
1 ;
1 ;
:
Validity of the probe approximation: to ensure the antibrane charge is not so large
that it backreacts signi cantly on the throat geometry we require:
We also need to ensure that the polarized 5brane does not signi cantly alter the S3
at the tip of the throat. This will be satis ed as long as the radius of backreaction
of the antibranes is small compared to the radius of the S3:
p
M K :
gsp
gsp
(gsM )2
K2
KS ;
SDKS:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
HJEP05(218)7
Then, in order to safely neglect the contribution of the warped throat to the total
warped volume in (2.4) we require:
V
V
2=3 Z rUV
0
8
Vthroat = vol(T 1;1)
V
dr r5hUV (r) ;
33=2 7=2pgs(KM
p)
<gs 8 (KM
p)
7gsM 2
8V2=3
: 8 (KM
p)
7K2
KS,
SDKS.
Validity of the nonperturbative expansion in the superpotential: in order to safely
truncate higher order nonperturbative e ects we require:
Although for simplicity of the presentation we estimate the warped volume of the
throat using the large r form of the metric, it is clear from the expression that the
majority of the volume comes from large r. The di erence between this estimate
and a numerical integration using the full KS warp factor has been checked and is
negligible.
Validity of single eld in ation (cosmological hierarchy): as mentioned in section 2
in order for the moduli to remain stabilized and nondynamical during in ation, it is
necessary that they be heavier than the Hubble scale:
Validity of the simplifying assumptions on the geometry: beginning with (2.2), we
assume a small, strongly warped throat and a large, unwarped bulk. The assumption
of strong warping implies:
where estimates for the Kahler moduli masses are given in (2.11) and (2.12).
1
V
> H2 ;
m2~ > H2 ;
m2~b > H2 ;
Validity of classical evolution: due to the dependence of the in aton potential on
quantized
ux, it can be viewed as a multibranched potential where transitions
between branches correspond to changing ux numbers. This structre is also present
in models of axion monodromy where it has been argued that tunneling between
branches can compete with the perturbative evolution of the in aton [40{46]. A
rough estimate of the size of nonperturbative e ects is given by the ColemanDe
Luccia thinwall approximation [47] of the tunneling rate:
exp
27 2T 4
2 V 3
;
(4.10)
where T is the e ective tension of the wall in 4dimensions and
V is the change
in potential energy. For the parameters sets given below this e ect is completely
negligible for our model:
exp( 109). We check that the size of the thin wall
bubble is safely below the Hubble radius. The characteristic size of the bubble is
several times the thickness of the brane, which is set by the string length.
Therefore, the thin wall treatment naively gives a good estimation of the tunneling rate.
However, we note that the application of the ColemanDe Luccia result to tunneling
from a perturbatively unstable point is not obviously justi able and a more careful
treatment may result in corrections to the tunneling rate [48].
4.2
Examples
We present here a few examples of speci c realizations of in ationary scenarios. We choose
to focus on the unwinding mechanism in SDKS where the antibranes polarize into a D5
as opposed to an NS5, whose action is less wellunderstood at weak coupling. To avoid
mixing KS and SDKS quantities in this section, we relegate an example that works in KS
to appendix A.
We nd that in order to have successful in ation we are driven to the regime of large
ux numbers, M and K, and also, as is usually the case in LVS, large W0. These large
values of M; K enter into (3.3) to imply Euler characteristics for the associated CalabiYau
fourfold that are larger than 1,820,448, which is the largest known [
49
]. Thus, we either
need to rely on the existence of currently unknown manifolds with large Euler
characteristic, or we need to assume that negative contributions to the left hand side of (3.3) exist
outside of the throat geometry. Neither of these are entirely satisfactory, however both are
technically possible.
In table 1 we present two sets of parameters which give rise to qualitatively di erent
in ationary scenarios. The
rst, set 1, is very similar to the
ndings in [7]. Although
of less phenomenological interest due to a very small power spectral amplitude, it has a
very large eld range. Meanwhile, set 2 represents a new twophase version of unwinding
in ation, which in addition to a superPlanckian eld excursion also reproduces the correct
amplitude of the scalar power spectrum. Here, most of the efolds of in ation come from a
single in ection point region of the potential near
= 0.6 Then, the unwinding mechanism,
6Caveat: because the in aton lingers for many efolds near one pole, this scenario could potentially receive
Set 1
Set 2
1.1
1
.23
still within the slow roll regime, exits in ation giving rise to the large eld excursion, but
only a few efolds. The di erent evolution of the in aton for these two sets can be seen in
the evolution of the Hubble parameter in gure 3. Their power spectra and eld ranges
are collected in table 2.
In both scenarios the initial conditions for in ation place the polarized 5 brane near
one pole with small velocity. We require that the canonically normalized eld displacement
at the initial condition is larger than H, as one would naturally expect thermal uctuations
to be of this order; for the results listed in table 2 we take (0)
107H and (0)
105H
for set 1 and 2 respectively. The unwinding scenario resulting from set 1 is insensitive to
initial conditions, whereas in set 2 the number of efolds will increase if the initial condition
is taken closer to the in ection point at
= 0.
It is extremely nontrivial that there remains a viable region in the parameter space
bounded by the highly nonlinear constraints listed in section 4.1 In table 3 we present the
degree to which sets 1 and 2 satisfy these constraints. Of course, seeing that a constraint
is satis ed does not necessarily mean that it is satis ed to a strong enough degree to
avoid catastrophic backreaction. Thus, in section 5.2 we compute backreaction e ects and
summarize their magnitudes for these parameter sets in table 4.
5
Backreaction
The ux cascade changes the number of threeform ux as well as the number of antibranes,
both of which contribute to the stabilization of the geometric moduli. Thus, through the
dependence ND3( ) and K( ) (or M ( ) in SDKS), the moduli vevs depend on the in aton.
This dependence will in turn lead to new forces on the in aton which could spoil the atness
large corrections beyond the probe approximation due to open questions regarding the IR backreaction of
antibranes, see section 5.
of the in aton potential. In this section we will discuss the backreaction on the geometry
due to the presence and depletion of antibranes. For simplicity, we will only consider
the backreaction of the swave, or 0th harmonic, an approximation that becomes better at
large radius. First, we will take into account the backreaction due the presence of antibrane
charge, but without taking into account supersymmetry breaking. This can be though of
as treating p antiD3 branes as
p D3 branes (see e.g. [
50
]). We will restrict ourselves to
an asymptotic charge matching along the lines of [14], without attempting to take
subleading corrections to the UV expansion of the metric into account. Then, we will compute
the contribution to the slow roll parameters due to the backreaction of the in aton on
the Kahler moduli vevs. Finally, we estimate the additional e ects of the supersymmetry
breaking on the geometry as well as discuss open questions in the literature regarding IR
backreaction e ects.
5.1
Backreaction of antibranes on net D3 charge
The rst backreaction e ect we wish to consider is that of the antibranes on the warping
of the throat geometry. As explained in section 3.2 and section 3.3, the matching of the
asymptotic charges results in (3.12) and (3.23), from which one can see that the presence
of the antibranes reduces the net D3 charge, which in turn reduces the `depth' or total
warping of the throat as seen from the UV. Because of the reduced warping, each antibrane
contributes a stronger uplift in the Kahler potential, so that not as many antiD3s can
be added before destabilizing the volume modulus.
This limitation on the number of
antibranes that we can add ultimately limits the amount of in ation that can be realized
in a given throat geometry.
However, this backreaction is also crucial in that now the total warping only depends
on the net D3 charge in the throat  a quantity that is conserved. Speci cally, the
warping, (3.12), (3.23), only depends on the combination KM
p which is exactly constant,
as required by the tadpole condition (3.3), throughout in ation.7 The fact that the
dependence cancels out of the warp factor means that no new terms in the in aton potential
will arise due the presence of the warp factor in the action. Furthermore, the results of [51]
7The total warp factor h0 (h~0) additionally depend on M (K), but this ux is constant throughout the
cascade in KS (SDKS).
show that beyond explicit dependencies on the warp factor in the in aton potential, the
nonperturbative e ects in the superpotential depend exponentially on the warped volume
of the fourcycle supporting the wrapped D7 or Euclidean D3 branes. Backreaction due
in aton dependence in this warped volume was computed for Dbrane in ation in [38]
and for axion monodromy in [
50
]. Because the warped volume of the fourcycle appears
exponentially in the superpotential, these e ects are is especially dangerous. The fact that
the leading UV behavior of the warp factor is independent of the in aton protects this
model from potentially serious backreaction e ects.
While in this approximation the in aton dependence exactly cancels, this only indicates
that the leading contribution to the
dependence of the metric will enter as a higher
harmonic. Not only are higher harmonics generally suppressed exponentially with distance,
but the magnitude of this backreaction is not expected to grow throughout in ation due to
charge conservation. Although the leading backreaction beyond the swave approximation
is beyond the scope of this work, charge conservation implies that it will not be a cumulative
e ect and should not signi cantly alter our results. Furthermore, unlike large eld in ation
models in more complicated geometries, the calculation of this backreaction using harmonic
functions in KS [52] is relatively straightforward.
Backreaction on Kahler moduli
Despite the cancelation of in aton dependence in the warp factor, there will still be
backreaction due to the dependence on the number of antibranes  as opposed to the total D3
charge  in the Kahler moduli potential (2.10). We will now check the backreaction on
the slow roll parameters:
" =
Mp2
2
V 0( ) 2
V ( )
and
= Mp2 V 00( )
;
V
(5.1)
due to the dependence Mp(V), C~(V), f~(V) and
LV S (V) and the backreaction V( ) and
( ) for the in aton potential (3.30). For simplicity we will only worry about the average
contribution to the slow roll parameters. Speci cally, because the potential is linear with
oscillations (see gure 2),
oscillates around zero with a large amplitude. This oscillating
part may give rise to resonant features in the power spectrum or nonGaussianity, but is not
important to the question of backreaction because it averages to zero over several efolds.
In order to compute the backreaction e ects we need functions V( ) and ( ), which
are computed by numerically solving for the minimum of (2.10) at each point in the cascade.
Because the Kahler potential (2.10) is rather messy, we will not bother writing the analytic
expressions for the slow roll parameters, but rather refer to gures 4 and 5 for the results.
We nd that while the backreaction indeed increases h i > 0, it remains true that h i; h"i
1 until the end of in ation, which we de ne by8 h"i = 1. Here, the angle brackets indicate
averaging over oscillations in the potential. While " was found to be small for all scenarios
with p < M K, the small contribution to h i is far less trivial and considerably tightened
the constraints on parameter space.
8Strictly speaking we use the slow roll parameter "H = H_ =H2 to de ne the end of in ation as it
additionally accounts for changes in the eld's kinetic energy.
1.0
0.8
slow roll parameter averaged over oscillations in the potential and without the contribution due
to the backreaction on the Kahler moduli, while h"i is the averaged slow roll parameter taking
backreaction into account.
HJEP05(218)7
0
10
20
30
40
50
60
40
45
50
55
60
65
70
for set 1 (left) and set 2 (right). While the averaged
parameter is zero
in the absence of backreaction, including the dependence of the Kahler moduli on the in aton gives
rise to a small but nonzero average.
In addition to potentially dangerous contributions to the potential, it is also important
to calculate the e ect of backreaction on the eld excursion, which could spoil the claim
of a transPlanckian
eld range. Speci cally, the V dependence in (3.29) should not be
treated as constant and the canonical eld should be corrected:
d
= f~(V( ))d :
(5.2)
Taking this backreaction into account we indeed
nd that the eld range is decreased,
although not dramatically when the problem is avoided. Actually, the fact that " is
slightly larger when backreaction is included, resulting in a slightly earlier end of in ation
makes a larger impact on the eld range than the correction (5.2). The backreacted eld
ranges are collected in table 4.
5.3
Breaking supersymmetry
The backreaction of antibranes on the KS geometry is a vast topic. Particularly, the
backreaction in the deep IR  at the tip of the throat  has been the subject of longstanding
Set 1
Set 2
bk=Mp
second slow roll parameter is evaluated at the observationally relevant window, 60 efolds before the
end of in ation, and will alter the value of ns given in table 2
debates. Both pointlike and smeared antiD3 brane sources have been shown to give rise
to illbehaved singularities [15{17, 53]. Recently, [54] showed (building upon [55, 56]) that
localized NS5branes carrying antiD3 charge avoid previous nogo arguments given against
the existence of regular IR boundary conditions, and therefore provide a viable candidate
for a regular supersymmetry breaking solution.9 In this work we will only consider the
supersymmetry breaking backreaction in the UV, which should be enough for the purposes
of the backreaction on the warped volume of the fourcycle supporting nonperturbative
e ects, and the subleading backreaction of the warp factor on the moduli.
The asymptotic analysis of [14], together with the amendments of [15{17] and further
insights of [18, 19], nds that antibranes at the tip of the KS throat break supersymmetry,
squash the KS geometry and induce a radial running on the dilaton. A full calculation
of backreaction for this model would include calculating the warped volume of the
fourcycle relevant for nonperturbative e ects using the backreacted metric, which depends on
the antibrane charge and therefore on our in aton. We will merely dip our toes into this
full calculation by noting that the strength of the perturbations to the KS geometry and
running of the dilaton is expected to be proportional to [14]:10
S
~
S
p
p
KM
KM
p
p
e
8 (KM p)
e 3gsM2
8 gs(KM p)
3K2
KS ;
SDKS : (5.3) (5.4)
Very schematically, the warp factor is a ected as h ! h(1 + S=r4), and similarly for the
dilaton. Thus the change to the metric in the UV should be small as long as these quantities
can be tuned to be small. Furthermore, the in aton dependence will arise only linearly via
p( ) in (5.3) and (5.4) and with a strong exponential suppression (again, the exponent is
in aton independent due to charge conservation.) We collect a summary of the e ects of
backreaction and the size of the supersymmetry breaking e ects in table 4.
6
Conclusions
In this work we have taken steps towards an explicit model of large eld in ation. We
achieve an in ationary epoch with a superPlanckian eld excursion by placing a stack of
antiD3 branes in a throat geometry. The use of simple and wellstudied ingredients, such
as antiD3 branes and throat geometries in ux compacti cations, make this one of the
9See also [57, 58].
10The holographic interpretation of this parameter was recently clari ed in [59].
most tractable examples of large eld in ation in string theory. Thus, we were able to
compute leading backreaction e ects and
nd scenarios for which neither the in ationary
epoch nor the eld excursion is strongly a ected by the backreaction on Kahler moduli.
These promising results point to avenues for future research. First, in order to move
towards a fully explicit model, we will need a CalabiYau threefold which contains a
warped throat region and either 1) contains large negative sources of D3 charge outside
the unwinding throat, or 2) lifts to a CalabiYau fourfold in Ftheory that has a very large
Euler number. The di culty in nding such an example could place strong constraints on possible scenarios. Second, even without a fully explicit manifold, additional UV backreaction e ects beyond the swave approximation and taking into account the e ects of
supersymmetry breaking discussed in section 5.3 are straightforward and tractable
calculations. Meanwhile, a full analysis beyond the probe approximation in the IR remains a
challenging but interesting topic for future research.
As opposed to computing in ationary observables with many signi cant gures, we
would like to highlight the relative simplicity of this model as an ideal setting to further
investigate the important questions of what is allowed in the string landscape.
Furthermore, while it is clear that the string cosmology community would bene t from a better
understanding of the basic ingredients necessary to nd de Sitter phases in string theory,
strong observational evidence suggests that current technical barriers to rigorously
achieving an in ationary epoch will be overcome. Therefore, it is important to understand simple
and
exible in ationary scenarios which can adapt to our evolving understanding of
positive vacuum energies in string theory. We hope that antibranes, brane ux annihilation,
and warped throats will prove robust enough that unwinding in ation can continue to be
relevant in the study of string cosmology.
Acknowledgments
We would like to thank Thomas Bachlachner, Fridrik Freyr Gautason, Oliver Janssen,
Matthew Kleban, Liam McAllister, Thomas Van Riet and Bert Vercnocke for useful
discussions. Additionally, we thank Fridrik Freyr Gautason and Thomas Van Riet for detailed
feedback on the manuscript. JDD is supported by the National Science Foundation of
Belgium (FWO) grant G.0.E52.14N Odysseus. MS is supported by the European Union's
Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant
agreement No. 656491.
We present here two in ationary sets in KS with roughly the same characteristics as the
sets discussed in the main text.
Set 1
Set 2
a
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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