Towards an explicit model of large field inflation

Journal of High Energy Physics, May 2018

Abstract The unwinding inflation mechanism is studied in a type IIB flux compactification where all moduli are stabilized using flux, non-perturbative effects, and the leading α′ corrections of the large volume scenario. We consider the backreaction on the geometry due to the presence of anti-D3 branes as well as the backreaction of inflation on the Kähler moduli, and compute the resulting corrections to the slow-roll potential. By taking large flux numbers, we are able to find inflationary epochs where backreaction effects are under control, the inflaton traverses a super-Planckian field range, and the resulting amplitude of scalar perturbations is consistent with observation.

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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 anti-D3 branes as well as the backreaction of in ation on the Kahler moduli, and compute the resulting corrections to the slow-roll 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 super-Planckian of scalar perturbations is consistent with observation. Cosmology of Theories beyond the SM; Flux compacti cations; D-branes - 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 Klebanov-Strassler Unwinding in the S-dual of Klebanov-Strassler 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 Klebanov-Strassler 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 super-Planckian 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 UV-completable 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 Calabi-Yau three-folds [ 9 ] where all complex structure moduli are stabilized at tree level. Additionally, we use the large volume scenario (LVS) [10, 11] to break the no-scale 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 anti-D3 branes. This is mediated by a 5-brane bubble which expands, crossing many times over the S3 at the tip of the KS geometry. As the 5-brane bubble moves across the S3, it removes both three-form ux and anti-D3 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 four-dimensional 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 anti-D3 brane charge on the warping of the KS geometry, and 2) the evolution of the Kahler moduli's potential due to the depletion of anti-D3 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 anti-D3 branes in the KS geometry is extremely well-studied [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 well-studied setting, this model serves as an ideal setting to further solidify or rule out the claim of a trans-Planckian 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 three-form 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 Calabi-Yau four-fold that is larger than known examples. Consequently, we do not have an explicit global example of a Calabi-Yau three-fold 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 non-Gaussianity and observable tensors. However, the details of these observables will be sensitive to the explicit details of the global Calabi-Yau three-fold through the highly nonlinear constraints that Calabi-Yau 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 Swiss-cheese 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 non-perturbative 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 non-perturbative e ects. While there are no no-go's which prevent stabilization, further study of non-perturbative 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 set-up in which we will stabilize all moduli using both uxes at tree-level, 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 four-dimensional low-energy 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 four-dimensional 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 four-dimensional 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 Gukov-Vafa-Witten tree-level superpotential [23]: Z X W0 = G3 ^ ; R6 mcs where G3 = F3 H3, is the axio-dilaton and is the holomorphic (3; 0)-form of the internal manifold X. Furthermore, the axio-dilaton, = 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 non-dynamical. However, at tree-level, the compacti cations of [ 9 ] are `no-scale' models, meaning that the Kahler moduli are at directions in moduli space. In the next sub-section we will add non-perturbative 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 four-cycle volume. That treatment fell short in that the overall volume modulus, which determines the four-dimensional 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 four-dimensional spacetime, dsCY0 is the line element on the compact Calabi-Yau 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 four-dimensional 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 Swiss-cheese' structure for the Calabi-Yau three-fold 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 Calabi-Yau 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 Calabi-Yau, 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 blow-up 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 Calabi-Yau three-fold and Kcs = log i R ^ . In the following, will be treated as a free parameter and Kcs = 0. In the superpotential, Ai are complex structure-dependent constants, and ai = 2 =Ni where Ni = 1 if the non-perturbative e ect on the particular four-cycle i arises from a Euclidean D3 brane, and Ni = ND7 if the non-perturbative e ect is gaugino condensation on a stack of D7 branes wrapped on i . Because of the assumed hierarchy, big , non-perturbative e ects are only relevant on the smaller four-cycle. Then, the contribution of the LVS potential to the fourdimensional supergravity action is: where we choose to parameterize the moduli using the small four-cycle, , and the overall 3=2 big . The pre-factor to the term in square brackets is necessary in order for the usual F-term 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 anti-D3 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 anti-D3 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 Kaluza-Klein 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 Kaluza-Klein 4Due to the large hierarchy between cycles in the strong Swiss-cheese 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 Kaluza-Klein 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 Kaluza-Klein 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 in-depth description of the speci c realization used here. The recipe for in ation calls for p anti-D3 branes in a warped throat region of the compact geometry (we will study the scenario both in the KS throat and its s-dual (SDKS)). Due to the presence of F5 ux in the throat geometry, the anti-D3 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 5-brane which wraps an S2 localized in the polar direction, , of the S3 at the tip of the deformed conifold. This 5-brane can be thought of as a bubble with decreased three-form ux (under which it is magnetically charged) in the interior. Through the process of brane- ux annihilation [12], the reduction of three-form 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 5-brane. Then, the 5-brane bubble will expand in the -direction and move across the compact space, removing three-form 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 5-brane 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 5-brane on the S3, i.e. our in aton action. Ultimately, we prefer to embed the unwinding mechanism in SDKS because the stack of anti-D3 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 four-dimensional 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; non-zero 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 three-form ux in the KS throat is quantized on both the blow-up S3 and its dual, S3: HJEP05(218)7 The other relevant details of this solution are: 1 Probe anti-D3 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 blow-up 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 blow-up 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 Klebanov-Tseytlin (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 non-compact solution with a singularity at r = rs, containing both three form ux and N D3-branes. 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 NS5-brane formed via the polarization of p anti-D3 branes [12]: SNS5 = d6 [ det(Gk) det(G? + 2 gsF2)]1=2 5 Z B6 ; with The stack of p anti-D3 branes polarize into a bubble that is extended in a four-dimensional 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 S-dual of Klebanov-Strassler Taking the S-dual of the KS geometry switches the positions of the F3 and H3 ux, such that the stack of anti-D3 branes polarize into a D5 brane as opposed to an NS5 brane. It should be noted that while S-duality is a strong-weak 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 S-dual 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 S-dual of the Klebanov-Tseytlin 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 D5-brane is: with SD5 = gs The D5 action in the S-dual of the KS geometry was calculated in [7], following closely the original computation for an NS5-brane 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 5-brane 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 non-perturbative expansion in the superpotential: in order to safely truncate higher order non-perturbative 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 non-dynamical 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 multi-branched 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 non-perturbative e ects is given by the Coleman-De Luccia thin-wall 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 4-dimensions 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 Coleman-De 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 well-understood 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 Calabi-Yau four-fold 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 two-phase version of unwinding in ation, which in addition to a super-Planckian 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 non-trivial that there remains a viable region in the parameter space bounded by the highly non-linear 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 three-form 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 s-wave, 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 anti-D3 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 anti-D3s 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 non-perturbative e ects in the superpotential depend exponentially on the warped volume of the four-cycle supporting the wrapped D7 or Euclidean D3 branes. Backreaction due in aton dependence in this warped volume was computed for D-brane in ation in [38] and for axion monodromy in [ 50 ]. Because the warped volume of the four-cycle 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 s-wave 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 non-Gaussianity, 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 non-zero 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 trans-Planckian 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 point-like and smeared anti-D3 brane sources have been shown to give rise to ill-behaved singularities [15{17, 53]. Recently, [54] showed (building upon [55, 56]) that localized NS5-branes carrying anti-D3 charge avoid previous no-go 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 four-cycle supporting non-perturbative e ects, and the sub-leading 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 non-perturbative 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 super-Planckian eld excursion by placing a stack of anti-D3 branes in a throat geometry. The use of simple and well-studied ingredients, such as anti-D3 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 Calabi-Yau three-fold 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 Calabi-Yau four-fold in F-theory 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 s-wave 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 Sklodowska-Curie 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. 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Juan Diaz Dorronsoro, Marjorie Schillo. Towards an explicit model of large field inflation, Journal of High Energy Physics, 2018, 75, DOI: 10.1007/JHEP05(2018)075