Towards multi-field D-brane inflation in a warped throat
Heng-Yu Chen, Jinn-Ouk Gong, Kazuya Koyama, Gianmassimo Tasinato
aa r X i v : . [ h e p - t h ] N ov Preprint typeset in JHEP style - HYPER VERSION
Towards multi-field D-brane inflation in a warped throat
Heng-Yu Chen a , Jinn-Ouk Gong b , Kazuya Koyama c and Gianmassimo Tasinato d a Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA b Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, 2333 CA Leiden, The Netherlands c Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK d Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, 69120 Heidelberg, Germany
Abstract:
We study the inflationary dynamics in a model of slow-roll inflation in warped throat.Inflation is realized by the motion of a D-brane along the radial direction of the throat, and at laterstages instabilities develop in the angular directions. We closely investigate both the single fieldpotential relevant for the slow-roll phase, and the full multi-field one including the angular modeswhich becomes important at later stages. We study the main features of the instability process,discussing its possible consequences and identifying the vacua towards which the angular modes aredriven. ontents
1. Introduction and summary 12. Warped conifold as an inflationary playground 4
3. Angular stability analysis 94. Slow-roll inflation in the Ouyang embedding 115. Explicit multi-field potential from warped conifold 17
6. Future Directions 21A. Properties of warped deformed conifold 23
A.1 Parametrization of the angular coordinates 23A.2 The metric of the warped deformed conifold 24
B. Extremal trajectory and angular mass matrix 25
B.1 Linear expansion: Identifying the extremal trajectory 27B.2 Quadratic expansion: Mass matrix 27B.3 Euler-Rodriguez versus Cayley-Kline parametrization of SU (2) group 30 C. Stabilized value of σ
1. Introduction and summary
Inflation is our most widely accepted paradigm of the very early universe that enables us to explain theobserved properties of the cosmic microwave background anisotropies. Present data are well describedby the simplest inflationary model, consisting of a single, slowly rolling light scalar field [1]. However,data are also consistent with more complex models, for example characterized by non-canonical kineticterms, or involving the dynamics of multiple scalar fields during inflation. Future observations areexpected to enable us to distinguish among the various scenarios, provided that we can computedistinctive observational consequences of different setups.From the theoretical point of view, building models of inflation provides both opportunities andchallenges. Inflation gives a unique opportunity to probe high energy physics since, in its most naturalrealizations, the characteristic energy scale is well beyond the current or planned particle acceler-ators. Also, inflation is highly sensitive to its ultraviolet completion: higher dimensional operatorscontributing to the inflationary potential play a crucial role in determining inflationary dynamics. This– 1 –mplies that inflation is able to probe properties of the theory that underlies a given model. On theother hand, this also means that a detailed knowledge of the setup under consideration is necessary,when embedding models of inflation in fundamental theories like string theory (or its supergravitylimit). Usually, one consequence of this ultraviolet sensitivity is that a successful inflationary setuprequires careful tunings of the available parameters to avoid the corrections due to higher dimensionaloperators, which can spoil the delicate inflationary dynamics and lead to the so-called η problem [2, 3].Despite these challenges, there have been several attempts to embed consistent models of inflationwithin string theory: see Ref. [4] for representative examples, and Ref. [5] for recent reviews. Theabundance of moduli fields in the string compactifications, both closed and open, in principle providesus a wide range of possible inflaton candidates. On the other hand, precisely due to the fact that manymoduli are usually involved in a given string model, there is the danger that light moduli would interferewith the inflationary process. This is due to the fact that light moduli can gravitationally couple withthe inflaton candidate(s), generically leading to the aforementioned η problem. Fortunately, over thepast ten years new methods have been provided to stabilize undesired moduli within string theory,by means of fluxes and non-perturbative effects, starting with the seminal work of Giddings, Kachruand Polchinski [6]. However, explicitly calculable setups are generally scarce where various requiredingredients such as the background geometric fluxes or non-perturbative superpotential are known.Given these reasons, the warped deformed conifold [9] holds a rather special place as an idealplayground for string inflationary model building, leading to the framework of brane inflation in warpedthroats. To be specific, following the original work [10], one considers mobile D3-brane(s) moving insuch a background geometry, that are attracted by an anti D3-brane located at the tip of the conifold.The setup is sufficiently well understood that the metric, the background geometric fluxes and themoduli stabilization effects are known in detail, and the potential governing the entire inflationarytrajectory can be constructed explicitly. Moreover, the parameter space of such a model is also richenough that semi-realistic inflationary trajectories can be found and compared with observational data.Finally, D3-brane inflation in warped deformed conifold is a rare example among string inflationarymodels which has a holographic dual description [8, 9, 11]. This offers us a new perspective on thevarious contributions to the D3-brane motion from dual field theory, and allows us to employ thepowerful and highly developed computational techniques of gauge/string duality.It has been shown that, by suitably tuning the ingredients responsible for stabilizing moduli, itis possible to find examples of inflection point inflation in this scenario [12, 13]. Inflation occursin regions of the warped throat in which different competing forces act on the moving D3-brane,compensating each other in such a way that the resulting potential is sufficiently flat around aninflection point (see Refs. [14, 15, 16] for subsequent developments along these lines). In the originalmodel, a Kuperstein-embedded D7-brane on the warped conifold [17] was considered, and the non-perturbative gaugino condensation on its worldvolume contributes to stabilize all the moduli besidesthe inflaton. A sufficient number of e -foldings of single field, slow-roll inflation can then be obtained,which can be geometrically interpreted as a D3-brane moving along the radial direction of the warpedthroat. All the moduli that do not take part in the inflationary dynamics are made sufficiently massiveby fluxes and non-perturbative effects.In this work, we revisit such an explicit framework, but instead focus on the so-called the Ouyangembedding [18, 19]. The main motivation here is to enrich the properties of the inflationary dynamics,in particular obtaining a framework in which more than one field play a role in the inflationary process.Indeed, in Ref. [13] (see also Refs. [20, 21]), it was anticipated that, for this embedding, moduli fields We however also note that, there have been interesting proposals to geometrize the non-perturbative superpotentialin the four dimensional effective field theory, i.e. to replace them by explicit ten-dimensional geometric fluxes [7, 8]. – 2 –ssociated with the D3-brane motion along the angular directions of the warped manifold becometachyonic towards the tip of the throat. This property is very interesting for realizing a scenarioin which the scalar fields associated with angular directions take part in the inflationary dynamics.Indeed, it is known that the evolution of the curvature perturbation after horizon exit, in multiplefield scenarios, can have important consequences for the spectrum of the curvature perturbation andnon-Gaussianity. The dynamics of light angular fields in brane inflation, in particular in the contextof DBI models with non-canonical kinetic terms [22, 23], have been widely studied over the past years,starting from Ref. [24], mainly due to the fact that they can lead to a peculiar pattern of non-Gaussianspectrum of the curvature perturbation [23, 25].In this paper we mainly focus on the slow-roll inflation. We show that an expansion of more than60 e -foldings can be obtained in this model, by again realizing inflection point inflation. Namely, a D3-brane can slow-roll along the radial direction of the throat, spanning for a sufficiently long range beforethe angular directions become tachyonic. The amplitude and spectral tilt of the power spectrum ofthe curvature perturbation, produced during the epoch of slow-roll inflation, are compatible with thepresent data. The instability develops along the transverse angular directions when the inflaton reachesa region near the tip of the throat, and may make inflation ending `a la hybrid inflation [2, 26]. Allthe process is fully under control from the supergravity point of view. We present the necessary toolsto follow the dynamics explicitly, and estimate the masses of the fields involved along the inflationarytrajectory. For our particular model of inflation, we find that angular directions become very light,and eventually tachyonic, only towards later stages of inflation. The angular directions then roll downthe potential towards their true minima. The fact that the dynamics remain single field for mostof the inflationary trajectory renders the predictions of our particular model similar to the ones ofstandard slow-roll inflation. On the other hand, the mass of the angular mode during inflation is neversignificantly larger than the Hubble scale. This can render the instability process particularly slow,with important observational consequences for the power spectrum as discussed in other contexts inRef. [27] (see also Ref. [28] for recent analysis of the dynamics of cosmological perturbations due tothe waterfall field in hybrid inflation).To conclude this introduction, it is important to mention that hybrid inflation scenarios wereconsidered in previous realizations of D-D inflationary models, as discussed in some of the papers ofRef. [4]. In those contexts, inflation ends when an instability of the tachyon field connecting the D-Dsystem arises. This can lead to formation of cosmic strings at the end of inflation, which is a subjectthat has raised a large research activity starting with Ref. [29] (see for example Ref. [30] for a review).An advantage of our model is that, as we mentioned, the development of instabilities along the angulardirections can be described within a supergravity framework, without involving string theory tachyons.On the other hand, as we will discuss, after the instability process takes places the system does notimmediately fall into a global minimum. Consequently, it is not possible to make definite statementsabout the production and stability of topological defects at the end of inflation in our setup withoutrelying on more detailed analysis.The paper is organized as follows. In Section 2, after introducing some basic facts of branepotentials in the warped deformed conifold, we present the explicit form of the potential in the Ouyangembedding. In Section 3, we discuss the kinetic term of a mobile D3-brane and identify the properlynormalized angular fields, and briefly address features of the angular masses near the tip of the throat.In Section 4, we show that, with an explicit example, that an expansion of 60 e -foldings of slow-rollinflation is possible in the Ouyang embedding. In Section 5, we give an analysis of the multi-fieldpotential of both radial and angular fields, and provide the necessary formulae for further study of themulti-field dynamics. In Section 6, we conclude concisely discussing interesting issues to be analysed– 3 –n the future. We relegate calculational details to three appendices.
2. Warped conifold as an inflationary playground
In generic string compactifications, a smooth warped deformed conifold throat can easily develop ina bulk compact Calabi-Yau manifold near the conifold singularity. Such a deformation was done inthe context of IIB string theory by the backreaction of localized imaginary self dual (ISD) three formfluxes [6]. To study the D3 trajectory in such a throat, an elegant systematic way to parametrizethe potential experienced by a D3-brane was recently given in Refs. [8, 11], which can be summarizedsuccintly as V D3 = T Φ − , (2.1)where T = 1 / [(2 π ) ( α ′ ) ] is the D3-brane tension, and Φ − = e A − α is a combination of the throatwarp factor and the five form field strength given in Ref. [11]. The mode Φ − characterizes theperturbation away from the ISD background such as warped deformed conifold where Φ − = G − = 0.Here G − = ( ⋆ − i ) G is the IASD component of the complex three form flux. At low orders inperturbative expansion, D3-branes only couple to Φ − field, and do not couple to metric or dilatonfluctuations, thus giving the simple form (2.1). The equation of motion of Φ − can be derived explicitlyfrom IIB supergravity action [8] as ∇ Φ − = e A + φ | G − | + R + e − A |∇ Φ − | + S local , (2.2)where e φ = g s is the string coupling, R is the four dimensional Ricci scalar and S local denotes thelocalized sources. The Laplacian operator ∇ is defined with respect to the unperturbed backgroundmetric. As the warped throat is attached to the bulk Calabi-Yau at some large radius r UV , Φ − captures the bulk perturbations such as distant supersymmetry breaking fluxes or quantum effects. Inthe language of holography, at these large radius, ultraviolet perturbations are packaged into the so-called “ non-normalizable modes ”. Moreover, Φ − can also receive contributions from the “ normalizablemodes ” of supergravity fields, which encode small radius, infrared perturbations. A particular exampleis the perturbation due to the D3 at the tip of warped deformed conifold which induces the D3-D3Coulomb attraction [31]. In other words, Φ − can in principle parametrize the potential for the entiretrajectory of a D3-brane in the warped deformed conifold throat.The parametric solution to (2.2) have been extensively investigated in Refs. [8, 11]. The resultscan be expressed in a simple form as V D3 ( φ ) = V D3-D3 ( φ ) + X i c i H i (Ψ) φ ∆ i , (2.3)where c i are constants and φ is the canonically normalized scalar field describing the radial motion ofthe D3-brane in the deformed conifold. The parameter Ψ = { θ , θ , φ , φ , ψ } collectively denotes thefive different angular coordinates of the conifold , and H i (Ψ) is expressible in terms of the angularharmonic functions. We have isolated the D3-D3 Coulomb interaction as it is a normalizable pertur-bation, while the polynomial series consists of the contributions from non-normalizable perturbations.It should also be noted that there can be additional φ -independent constants added to V D3 ( φ ) due See Appendix A for the explicit metric and angular coordinates of warped deformed conifold. – 4 –o the coupling with the four dimensional Ricci scalar R which can also receive contributions fromdistant sources. In Refs. [8, 11], much of effort was devoted to enumerating the discrete spectrumof the scaling dimensions { ∆ i } . The angular harmonic function H i (Ψ) is related to ∆ i and can alsobe computed in principle from the expansion of (2.2) . The values of the expansion coefficients c i however are model dependent, and can only be specified when quantities such as particular modulistabilization effects, supersymmetry breaking fluxes and other microscopic quantities are known.The parametrization (2.3) makes it apparent that the radial and angular motions of a D3-braneare coupled in warped throat, and offer rich landscape for inflationary model building. On one hand, toobtain a single field inflationary model where φ plays the role of the canonical inflaton, it is necessary torestrict to a special trajectory in the parameter space where all the angular modes are either stabilizedat their minima or decoupled from φ : a good example was presented in Ref. [13]. We, on the otherhand, should expect multi-field inflation being rather generic in the warped throat. If we are to realizethis and construct an explicit potential, it is necessary to study how D3 angular modes Ψ couple tothe radial mode φ . In particular, as we demonstrate in an example in Section 4 after taking intoaccount moduli stabilization, the angular masses can change sign for certain values of φ , and becometachyonic. We have thus a situation where single field inflation is connected to a system with multiplefield dynamics. We consider the motion of a D3-brane on a warped deformed conifold, whose geometrical propertiesare summarized in Appendix A. One of the basic quantities that characterize the geometry, and thatplays an important role in what follows, is the deformation parameter ǫ . We include a D3-brane at thetip of the cone, and we take into account stabilization effects that are needed for providing masses toundesired light moduli. In this setup, we calculate the potential experienced by the moving D3-brane.In this and the following sections we mostly present our main results, while details can be found inthe appendices to which we will refer in due course. In order to calculate the D3-brane potential, wedo not directly solve (2.2). Instead, we follow a supergravity approach, that allows us to apply wellestablished techniques and results on the properties of the warped deformed conifold with embeddedD7-branes.More specifically, we work in the framework of the KKLT moduli stabilization mechanism [32],where in order to stabilize the K¨ahler moduli ρ = σ + iχ , (2.4)whose real part corresponds to the overall volume of the compact Calabi-Yau space, we consider thesuperpotential W ( z α , ρ ) consisting of two contributions W ( z α , ρ ) = W + A ( z α ) e − aρ . (2.5)Here, z , · · · z parameterize the complex coordinates of the internal manifold. The first term W = R G ∧ Ω is the perturbative Gukov-Vafa-Witten flux superpotential [33], which, at least in principle,can stabilize the complex structure moduli and the dilaton-axion combination. Without loss of gen-erality, we shall assume W ∈ R − . It is well known that the tree level K¨ahler potential for ρ exhibits To be more specific, the computations in Refs. [8, 11] were done in the asymptotic, singular conifold limit. Toextend the analysis into the deformed conifold region, we expect logarithmic corrections to the scaling dimensions { ∆ i } ,and modifications to the angular harmonic function H i (Ψ) as the U (1) subgroup of the SU (2) × SU (2) × U (1) singularconifold isometry group is broken down to discrete Z . – 5 –o scale structure and leaves ρ unfixed. A mechanism for stabilizing ρ is therefore to include thenon-perturbative gaugino condensate on a stack of space-filling D7-branes (or a Euclidean D3-brane)wrapping a holomorphic four cycle in the Calabi-Yau space, as appears in the second term of (2.5). Inthe presence of a mobile D3-brane, the one-loop determinant A ( z α ) picks up dependence on the D3-brane position moduli, which appear through the holomorphic D7 embedding function f ( z α ) = 0 [34] .We can rewrite A ( z α ) as A ( z α ) = A (cid:20) f ( z α ) f (0) (cid:21) /n . (2.6)Here A is a complex constant whose exact value depends on other stabilized complex structure moduli,and n is the number of D7s (or n = 1 for a Euclidean D3) giving the gaugino condensate (or instantoncorrection), which also enters in the definition of the exponent a in (2.5) as a = 2 π/n . In other words,the non-perturbative gaugino condensate on D7-branes not only stabilizes K¨ahler modulus ρ , but alsogenerates potential for the mobile D3-brane, echoing an earlier result of Ref. [35]. This also fits wellwith the general parametrization (2.3), as it was also discussed in Ref. [8] that gaugino condensate onD7-branes can act as local sources for IASD flux, and make contribution to Φ − .In this paper we focus on the so-called Ouyang embedding in warped deformed conifold [18, 19], f ( w ) = µ − w , (2.7)where w = ( z + iz ) / √ µ is a complex constant . Note that | µ | heuristically measures the depthwhich D7-branes enter the conifold throat. The D7 embedding (2.7) breaks the SU (2) × SU (2) × U (1)isometry of the singular conifold down to U (1) × U (1), and can generate potential for the D3 angularmodes associated with the broken isometries. The explicit potential for the w -embedding (2.7) can beobtained by substituting (2.5), (2.6) and (2.7), along with the warped K¨ahler potential [36] describingthe kinetic terms for ρ and D3-brane κ K ( z α , ¯ z α , ρ, ¯ ρ ) = − ρ + ¯ ρ − γk ( z α , ¯ z α )] ≡ − U ( ρ, ¯ ρ, z α , ¯ z α ) , (2.8)into the standard N = 1 supergravity expression of the F -term scalar potential. Here κ = M − =8 πG , γ = σ T / (3 M ) with σ being the stabilized value of σ at the tip of the warped deformedconifold, and k ( z α , ¯ z α ) is the geometric K¨ahler potential for the deformed conifold given in (A.22). Thedetailed calculations are similar to those for the Kuperstein embedding [17] as given in Refs. [13, 16],and required inverse deformed conifold metric is given in (A.23), (A.24) and (A.25). After somecalculations, we can obtain V F as V F = AG /n (cid:20) B − ae aσ | W || A | G − / (2 n ) (cid:21) + A n G /n − (cid:18) k ′ k ′′ coth τ C + ǫ cosh τaγnµ k ′′ D (cid:19) , (2.9) The complex coordinates defining the deformed conifold are given in Appendix A. The quantities w i ( i = 1 , · · ·
4) correspond to an alternative coordinate system for the internal manifold underinvestigation. For the relation between { w α } and { z α } , see Appendix A. The residual U (1) × U (1) isometry gets further broken to U (1) in the deformed conifold. For the subtleties ofsupersymmetric D7 embeddings in warped deformed conifold, see Ref. [19]. – 6 –here A ≡ aκ e − aσ | A | U , (2.10) B ≡ a U + γ k ′ k ′′ ! + 6 , (2.11) C ≡ w + w µ − | w | µ + (cid:18) w + w µ − w w + w w µ (cid:19) sech τ , (2.12) D ≡ − | w | ǫ cosh τ + coth τ (cid:18) k ′′ k ′ − coth τ (cid:19) (cid:20) tanh τ − w w + w w ǫ cosh τ − | w | + | w | ǫ cosh τ (cid:21) , (2.13) G ≡ − w + w µ + | w | µ . (2.14)We can see that V F depends on the volume modulus σ , the radial coordinate τ and various combi-nations of the D3-brane coordinates (cid:8) w + w , | w | , w + w , | w | , w w + w w (cid:9) . The coordinate τ is associated with the radial coordinate r , frequently discussed when analysing the singular warpedthroat, by r = ǫ cosh τ : see (A.26). In the limit A ( w ) → A , V F reduces to the scalar potentialconsidered in Ref. [10]. We stress that similar computations have been done in the singular conifoldlimit of (2.7) [20, 21]. Here we complete the computation for the full warped deformed conifold, as itwill be crucial for studying the angular instability at small radius in a fully under control setup.As the isometries of the conifold are partially broken by the D7 embedding (2.7), some of theangular directions can pick up effective masses through V F , and their values are localized along theextremal trajectory which satisfies ∂V F ∂ Ψ i = 0 . (2.15)In writing the above equation, we separate the radial coordinate τ from the five angular coordinatesof the warped conifold, collectively denoted with Ψ i . For the embedding (2.7), two such trajectoriesexist and are given by w (0)1 = ± ǫ √ e τ/ , w (0)2 = ∓ ǫ √ e − τ/ , (non-delta-flat) (2.16) w (0)1 = ± ǫ √ e − τ/ , w (0)2 = ∓ ǫ √ e τ/ . (delta-flat) (2.17)The detailed steps are given in Appendix B. Notice that these extremal trajectories do not fix theoverall sign of w and w , but only the relative sign between the two quantities. The choice of theoverall sign has important physical consequences that we will discuss in the following sections. Theresultant F -term scalar potential along these extremal trajectories is given by V F ( τ, σ ) = A ( τ, σ ) G /n ( τ ) (cid:20) B ( τ, σ ) − ae aσ | W || A | G − / (2 n )0 ( τ ) + F ( τ ) (cid:21) , (2.18)– 7 –here A ( τ, σ ) = aκ e − aσ | A | U ( τ, σ ) , (2.19) U ( τ, σ ) =2 σ − / / an β Z τ dτ ′ [sinh(2 τ ′ ) − τ ′ ] / , (2.20) B ( τ, σ ) = a (cid:26) U ( τ, σ ) + 3 / / βan [sinh(2 τ ) − τ ] / sinh τ (cid:27) + 6 , (2.21) F ( τ ) = 1 n G ( τ ) (cid:20)
34 sinh(2 τ ) − τ sinh τ coth τ C ( τ ) + 3 / / α β cosh τ sinh τ [sinh(2 τ ) − τ ] / D ( τ ) (cid:21) , (2.22)and the functions which explicitly depend on the choice of the extremal trajectories are C ( τ ) = ( ± αe τ/ tanh τ p G ( τ ) , (non-delta-flat) ∓ αe − τ/ tanh τ p G ( τ ) , (delta-flat) (2.23) D ( τ ) = e τ τ , (non-delta-flat) e − τ τ , (delta-flat) (2.24) G ( τ ) = ((cid:0) ∓ αe τ/ (cid:1) , (non-delta-flat) (cid:0) ∓ αe − τ/ (cid:1) . (delta-flat) (2.25)Here, we have defined two dimensionless parameters α and β by α ≡ ǫ √ µ , (2.26) β ≡ πγk ′′ | τ =0 = 2 / / anγǫ / . (2.27)They have the following geometrical meaning. α measures the depth which the D7-branes extend intothe deformed conifold, while β is inversely proportional to the four dimensional Planck mass M Pl , andhence to the ultraviolet cutoff r UV of the warped throat, i.e. β ∼ ǫ / /r . We therefore deduce that r UV & µ / , and equivalently α & β . In fact, when we take into account the contribution to M Pl dueto the compact Calabi-Yau to which the throat is connected, we generally expect α ≫ β . Also, thesign choice depends on the sign chosen in the extremal trajectories. Choosing the upper sign impliesto choose the upper sign in the expressions above, and vice versa. In obtaining (2.18), we have alsostabilized the axion χ in the K¨ahler modulus ρ . The scalar potential now appears as a function ofonly two variables τ and σ , which we also necessarily need to stabilize to prevent decompactifcation.Before we conclude this section, let us make some remarks. First, as noted in Ref. [13], the non-delta-flat trajectory represents an unstable minimum. While at large radius, the angular mass matrixnormally has positive eigenvalues, at small radius the trajectory becomes unstable and the angularmodes become light and tachyonic. In the next sections we shall investigate in detail such instabilityand the multi-field potential. Second, we should note that in addition to the F -term scalar potential,the mobile D3-brane also experiences a Coulomb attraction coming from the D3 at the tip of warpeddeformed conifold, whose form is given by V D3-D3 ( τ, σ ) = D U ( τ, σ ) (cid:20) − D π T (∆ y ) (cid:21) , (2.28)– 8 –here D = 2 T a and ∆ y parameterizes the D3-D3 separation. This provides an additional contribu-tion to the inflationary potential. Notice that, as shown in Ref. [10], if the separation between braneand antibrane is sufficiently large, the Coulomb potential does not depend on the angular coordinates.Third, let us also briefly comment on the possible relevance of worldvolume fluxes on the D7-branesthat can in principle induce a force on the moving D3-brane. As shown in Ref. [13], this force isgenerically suppressed with respect to the other contributions to the D3 potential. Consequently,worldvolume flux effects are not expected to qualitatively change the analysis of the inflationary dy-namics that we are going to discuss.To conclude, let us once again discuss the difference between the supergravity approach adoptedin this section and in the rest of the paper, and the more direct approach introduced in the previoussection based on the dynamics of the field Φ − . The two approaches should provide the same results,as long as one neglects the effects of the compact Calabi-Yau manifold to which the throat is attachedat large radius. The method of Section 2.1 is in principle suitable also for taking into account theeffects of the bulk Calabi-Yau that are instead hard to describe within the supergravity approach ofthis section. On the other hand, as we will discuss in due course, the phenomena that we analyse inthis paper are not particularly sensitive to these bulk effects, and our results are expected to remaincorrect also when we include these contributions.
3. Angular stability analysis
Naively, after we further stabilize the volume modulus σ in (2.18) at a given value of τ , we obtain asingle field potential in τ . However for this statement to hold for the entire deformed throat, we needto ensure that the broken angular isometries are much heavier than the radial mode and effectivelystabilized at their minima along the entire extremal trajectories. In this section, we investigate thisissue for all broken isometries by examining their effective masses which depend on the radial direction.The first step is to identify the canonically normalized fields associated with the angular coordinates.As we discuss in Appendix B, when considering small displacements of the angular directions from theextremal trajectories, it is convenient to assemble the angular degrees of freedom into three variablesthat we call P , Q and R . We concentrate on fields representing small displacements from the extremaltrajectory since, as we will discuss in more detail in what comes next, the angular masses duringslow-roll inflation are large and positive. So angular modes remain localized nearby the extremaltrajectory. Only at later stages of slow-roll inflation, as we will see, angular modes become light andlarge displacements from the extremal trajectory can occur. We will discuss the consequences of thisfact in Section 5.The kinetic term of a mobile D3-brane can be deduced from the K¨ahler potential (2.8) as L kin = − T σ σ ⋆ ( τ ) d ˆ s . (3.1)Here, d ˆ s denotes the pull-back of the warped deformed metric (A.16) and σ ⋆ ( τ ) is the solution to thestabilization condition ∂ ( V F + V D3-D3 ) ∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ = σ ⋆ = 0 . (3.2)Note that using σ ⋆ , we can write σ = σ ⋆ ( τ = 0). We can obtain the analytic expression of σ ⋆ ( τ ) forsmall τ region as in the case of the Kuperstein embedding: see Appendix C. But for general τ wehave to solve the stabilization condition numerically. Considering small angular displacements around– 9 –he extremal trajectories, the pull-back of the metric is given by (B.65) as d ˆ s (cid:12)(cid:12) = ǫ / K ( τ ) (cid:26) dτ + 4 dP K ( τ ) + 4 h cosh (cid:16) τ (cid:17) dR + sinh (cid:16) τ (cid:17) dQ i(cid:27) . (3.3)From this form, we can canonically normalize the radial direction by defining a scalar field φ ( τ ) as φ ( τ ) ≡ r T ǫ / Z τ dτ ′ K ( τ ′ ) r σ σ ⋆ ( τ ′ )= 3 / / r βanσ M Pl Z τ dτ ′ K ( τ ′ ) r σ σ ⋆ ( τ ′ ) , (3.4)where the function K ( τ ) is given by (A.17). Also we rescale angular fluctuations including constantfactors as b P ≡ r T ǫ / P = 2 / / r βanσ M Pl P , (3.5) b R ≡ p T ǫ / R = 2 / / r βanσ M Pl R , (3.6) b Q ≡ p T ǫ / Q = 2 / / r βanσ M Pl Q , (3.7)so that they pick up mass dimension 1. We can then write the kinetic term Lagrangian as L kin = −
12 ( ∂φ ) − K ( τ )2 σ σ ⋆ ( τ ) (cid:16) ∂ b P (cid:17) K ( τ ) + cosh (cid:16) τ (cid:17) (cid:16) ∂ b R (cid:17) + sinh (cid:16) τ (cid:17) (cid:16) ∂ b Q (cid:17) , (3.8)where τ should be regarded as a function of φ by inverting relation (3.4), i.e. τ = τ ( φ ). Kinetic termswith field-dependent coefficients, as the previous ones, have been used in the past in the context ofmultiple field inflation. They arise when discussing models characterized by a non-trivial metric infield space: see Ref. [37] for some literature. We will use this form of kinetic terms in our numericalanalysis in Section 4.Let us now discuss the signs of the mass eigenvalues of the angular fields P , Q and R in thesmall τ limit. A similar analysis was performed in Refs. [13, 20] in the singular conifold limit, and theresults showed that some of the angular directions become generically tachyonic at small values of theradial direction. Here we complete the analysis to the case of warped deformed conifold, and analysein detail the range of parameters leading to instabilities.In the limit τ →
0, the two different extremal trajectories, (2.16) and (2.17) coincide. After somecalculations (see Appendix B for details), we can find that the mass eigenvalues for the angular modes P , Q and R , that we denote respectively with X P , X Q and X R , are given by X P = ∓ α AG /n − n (cid:20) s − α β (1 ∓ α ) − (cid:21) , (3.9) X Q =0 , (3.10) X R =2(1 ∓ α ) X P + 4 α AG /n − n (cid:18) β − (cid:19) . (3.11)– 10 –t this point, we can understand whether the eigenvalues are negative or positive, when approachingthe tip of the cone ( τ → X P , with uplifting ratio s (C.4) being large enough, it is easyto make sure that both X P and X R are positive at the tip of the warped throat . We focus here onthis case, that will be the one analysed in more detail in the following sections. On the other hand,since X Q → τ →
0, suppose it approaches zero from below: since it is positive at large τ , bycontinuity, this means that it changes sign in an intermediate region, and develops an instability. Asufficient condition to ensure that the eigenvalue approaches zero from below is to demand that it hasnegative first derivative along τ , in the limit τ →
0. A simple calculation provides ∂X Q ∂τ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = 2 α A G /n − n (cid:20) (2 s −
3) (1 + α ) − α − α β (1 + α ) (cid:16) − α (cid:17)(cid:21) . (3.12)When the ratio α/β is sufficiently large, this is negative, but α/β is generally much larger than one:see the discussion after (2.27). We see explicitly that X Q becomes negative for sufficiently small τ while it vanishes at τ = 0. These analytic considerations will be confirmed by the numerical analysisof the following section.
4. Slow-roll inflation in the Ouyang embedding
In the previous section, we found that there appears an instability at least in one or more angularmasses along the extremal trajectories for natural choices of parameters. On the other hand, bysuitably tuning the parameters, it might be possible to realize slow-roll inflation that lasts for morethan 60 e -folds well before the instability develops. In this section, we show that this is indeed thecase. We specifically focus on the lower branch solution of the non-delta-flat direction (2.16), i.e. w = − ǫe τ/ / √ w = ǫe − τ/ / √
2. The discussion on the application of the delta-flat trajectoryto the multi-field inflation is postponed in the next section. The choice of the parameters is summarizedin Table 1. α β n s | W | A . × − M . M Table 1:
The parameters used in the numerical computations.
The single field potential along the radial direction τ is shown in Fig. 1 in terms of the canonicallynormalized field φ ( τ ). Here we plot the potential in terms of φ ( τ ) /φ µ , where φ µ ≡ anσ (cid:18) α (cid:19) / βM . (4.1) φ µ corresponds to the maximal value for φ in the warped throat. We see that there is a point wherethe second derivative of the potential with respect to φ vanishes. Slow-roll inflation can be realizedaround this inflection point . In order to understand whether inflation can be realized or not, we define Notice that this does not imply that X P and X R are always positive at τ > – 11 –he slow-roll parameters ε ≡ − ˙ HH ≈ M (cid:18) V ′ V (cid:19) , (4.2) η ≡ − M V ′′ V . (4.3)These slow-roll parameters are shown in Fig. 2. As expected, ε remains small but | η | quickly becomeslarge once the field moves away from the inflection point. Φ (cid:144) Φ Μ ´ V Φ (cid:144) Φ Μ ´ V Figure 1:
The full potential versus normalized canonical field φ/φ µ , which runs from 0 to 1. We set M Pl = 1.In the left panel, we simply show the entire potential, while in the right panel we concentrate on the inflectingregion, where after the instability of Q develops we show the potential with a dotted line. Φ (cid:144) Φ Μ ´ ¶ - - Φ (cid:144) Φ Μ Η Figure 2:
The “potential” slow-roll parameters (left) ε and (right) η . Note that while | η | ∼ Q develops, ε remains smaller than 1: at the moment of instability, we have ε ≈ . × − . Next we consider the masses of the angular directions. In Fig. 3, we show X P , X Q and X R . For ourchoice of parameters, we find that all the angular directions become unstable at small radius. Howeverwe should note that once instability develops, we cannot trust anymore the analysis of the mass matrix Notice that we write the first slow-roll parameter as ε to distinguish it from the deformation parameter ǫ usedelsewhere. – 12 –round the extremal trajectory in Appendix B, as the angular displacements need not to be small sinceangular fields start to roll towards their true minimum. Thus we only study slow-roll inflation beforeone of the angular directions becomes unstable: in this case the first among the angular directionswhose mass squared vanishes is Q . From this point on, we need to consider multi-field dynamics,which is the subject of the next section. Φ (cid:144) Φ Μ ´ X P Φ (cid:144) Φ Μ ´ H X Q , X R L Figure 3:
The angular masses (left panel) X P , (right panel) (dotted line) X Q and (solid line) X R . Thedifference between X Q and X R is only noticeable near the tip region. We can estimate the number of e -folding number from this instability point as N = r β anσ (cid:18) α (cid:19) / Z φφ inst dφ √ ε , (4.4)which is shown in Fig. 4. As we can see, the number of e -folds blows up near the inflection point. Thatis, we expect an indefinitely large number of e -folds if the classical initial condition is such that the fieldstarts from very near the inflection point. Of course, in reality quantum fluctuations will push the fieldaway from the inflection point. In any case, we learn that it is not difficult to realize slow-roll inflationthat lasts for more than 60 e -folds before instability develops along one of the angular directions. Aninteresting, separated issue is the problem of initial conditions that can lead to inflation: we will notdiscuss it in this context, but see for example Ref. [38].Also, we calculate the power spectrum of the curvature perturbation P R and the correspondingspectral index n R , under the slow-roll approximation given by P R = V π εM , (4.5) n R =1 − ε + 2 η . (4.6)In Fig. 5, we show log P R and n R versus the number of e -folds. For our choice of parameters, weobtain P R = 2 . × − and n R = 0 . N = 60, which are well within the 2 σ range of thecurrent observations [1].The observational predictions from slow-roll inflation is very similar to those in the Kupersteinembedding [16]. This suggests that the inflection point inflation is quite generally realized regardlessof the choice of the embeddings. In fact, as discussed in Section 2, it was suggested that the D3-branepotential can be expressed as V = X i c i φ △ i H i (Ψ) , (4.7)– 13 – .10 0.12 0.14 0.16 0.18 0.20 0.22020406080100 Φ (cid:144) Φ Μ N Figure 4:
The number of e -folds N versus φ/φ µ counted from the instability point . As we approach theinflection point N diverges, which indicates that we have indefinitely large N near the inflection point. - - - - - - - N l og P R N n R Figure 5: (Left panel) log P R and (right panel) n R versus N . At N = 60, the values of P R and n R areclose to the observed values. where Ψ is the angular coordinates, c i are constants and ∆ i are given by∆ i = 1 , , , , , · · · . (4.8)In our setup, we find that the potential along the extremal trajectory is well fitted by the polynomialform VM = V " . (cid:18) φM Pl (cid:19) + 0 . (cid:18) φM Pl (cid:19) / − . (cid:18) φM Pl (cid:19) + 6 . (cid:18) φM Pl (cid:19) / − . (cid:18) φM Pl (cid:19) , (4.9)where V = 2 . × − . This constant term arises from the uplifting contribution that is necessaryto obtain a quasi de Sitter solution. This parametrization emphasizes the fact that inflation is obtainedaround an inflection point region, that results from a delicate cancelation among the various termsin the previous expansion. Notice that although the previous expression provides a potential for the– 14 –anonically normalized inflaton along the radial direction φ , the dependence on the angular coordinatesis contained in the coefficients H i (Ψ). Their form can be deduced from the explicit scalar potentialderived earlier, which also gives mass eigenvalues X P , X Q and X R along the extremal trajectories.The knowledge of these quantities is necessary to understand whether or not the angular masses areheavier than the Hubble parameter during inflation: if not, angular directions are not stabilized ontheir extremal value and their dynamics can have interesting observational consequences.To estimate the effective masses for the angular directions, as can be read from (3.3), we notice thekinetic terms induce a mixing between the radial field φ and the angular fields b P , b Q and b R . Preciselyspeaking, we need to properly take into account this non-trivial field space metric to analyze the massesof the angular fields and we will explain the procedure in more detail in the next section. However,during inflation, the motion along the radial direction is suppressed by the slow-roll parameter. Thusat the leading order in the slow-roll approximation, we can assume that the radial field is constant, aswell as the coefficients of the kinetic terms. We are then interested in the effective masses, that takeinto account the non-trivial τ -dependent coefficients in the kinetic term, as m b P , eff = X P M an / / β σ ⋆ ( τ ) K ( τ ) , (4.10) m b Q, eff = X Q M an / / β σ ⋆ ( τ ) K ( τ ) sinh ( τ / , (4.11) m b R, eff = X R M an / / β σ ⋆ ( τ ) K ( τ ) cosh ( τ / . (4.12)For a given φ , these masses characterize the behaviour of the isocurvature perturbations along theangular directions. We found at N = 60, their ratios to the Hubble parameter are m b P , eff H =5 . , (4.13) m b Q, eff H =2 . , (4.14) m b R, eff H =1 . , (4.15)and the behaviours of these ratios are shown in Fig. 6. We learn that the effective masses of theangular fields are comparable to the Hubble parameter during inflation: they are stabilized on theirextremal values until the instability region is reached, at which the fields become tachyonic.This conclusion is not unexpected. In supergravity, the typical mass of the moduli fields duringinflation is of O ( H ). This is the reason of the η problem that we mentioned in Section 1: only bycarefully tuning the parameters, one can obtain a flat potential for the inflaton field in some regionof parameter space. In our framework, we can also understand more explicitly why the masses ofthe angular modes turn out to acquire these values during inflation, using the large τ expressions for X P,Q,R . These quantities, in this limit, are found to be X P → ± A n (cid:16) ∓ αe τ/ (cid:17) − /n − αe τ/ (cid:26) aσ + 92 − ae aσ | W || A | (cid:16) ∓ αe τ/ (cid:17) /n − (cid:18) − n (cid:19) (cid:16) ∓ αe τ/ (cid:17) − (cid:20) ± αe τ/ (cid:16) ∓ αe τ/ (cid:17) + 3 / α β e τ/ (cid:21)(cid:27) , (4.16) X Q , X R → (cid:16) ∓ αe τ/ (cid:17) X P + 2 A / n α β e τ/ (cid:16) ∓ αe τ/ (cid:17) − − /n ) (cid:18) − / βe τ/ (cid:19) . (4.17)– 15 – .00 0.05 0.10 0.15 0.20 0.25 0.30 - - Φ (cid:144) Φ Μ m P ` , e ff (cid:144) H - - Φ (cid:144) Φ Μ H m Q ` , e ff , m R ` , e ff L (cid:144) H Figure 6:
The plots of (left panel) m b P, eff /H and (right panel) (solid line) m b Q, eff /H and (dotted line) m b R, eff /H versus φ/φ µ . Note that we assume H is completely dominated by the potential, which should be agood enough approximation near the flat region of the potential. With our choice of parameters, the stabilisation condition for σ can be approximated by (C.10), whichgives σ ⋆ ∼ σ even for large τ . Then substituting (C.10) into the above expressions, we find that theterm proportional to aσ cancels out where aσ ⋆ ≫ X P,Q,R as X P,Q,R = N P,Q,R anσ ⋆ V, (4.18)where N P,Q,R are functions of αe τ/ , α e τ/ /β and s . In the example we studied here, αe τ/ ≪ , α e τ/ /β ≪ N P,Q,R = O ( αe τ/ ) at N = 60. Then the effective masses can be estimated as m b P , b Q, b R, eff H = O (cid:18) αβ e − τ/ (cid:19) . (4.19)This is generally larger than one as α ≫ β , but they are not significantly large with our choices ofparameters: the mass eigenvalues during inflation turn out to be of the order of the Hubble scale.This can have important consequences for the field dynamics right at the end of inflation. Indeed, indifferent contexts, mainly motivated by hybrid inflation, it has been shown that when the waterfallfields have not too large masses the instability process might take longer than one Hubble time tocomplete [27]. This implies that slow-roll motion along the radial direction can continue, while theangular modes roll, not necessarily too rapidly, towards their true minima. It would be interestingto investigate these issues in the present context, although the analysis has to take into account thespecific form of both the potential and the non-canonical kinetic terms.Let us finish this section discussing whether the results of this section can be modified after takinginto account the effects of the bulk Calabi-Yau attached to the throat at large radius. As shownin Ref. [39], the main point to notice is that bulk contributions to the angular potential give riseto suppressed by powers of the warp factor. Then, modifications of the angular potential due tobulk effects are generally subleading with respect to the potential produced by moduli stabilizationeffects on the throat. On the other hand, bulk effects could play a role in the infrared region wherethe angular instabilities develop, and where the overall force on the angular modes due to modulistabilization effects vanish. But bulk contributions are subdominant away from this particular region,– 16 –eing suppressed by the warp factor. Consequently, we expect that their net effect is at most to inducea tiny shift of the position at which the instability occurs along the radial direction.
5. Explicit multi-field potential from warped conifold
In our previous analysis, we mostly focused on studying the homogeneous dynamics of the inflaton fieldalong the extremal trajectories. On the other hand, since towards the later stages of slow-roll inflationan instability develops in the angular directions, in this region the masses of the corresponding fieldsbecome small, comparable to that of the radial direction, and eventually tachyonic. To follow thedynamics from this stage onwards, we need to consider in more detail the properties of the angularfields, significantly deviating from their extremal values. With this purpose, in this section we studymore closely the potential governing the angular fields that develop an instability towards the endof inflation, as well as the corresponding kinetic terms. This analysis represents the necessary firststep for discussing in detail the field evolution in this system at the homogenous level as well as thedynamics of fluctuations. Simple considerations, based on the properties of the potential, allow us toobtain important information about the instability process. In particular, we show that the instabilityconnects one of the extremal trajectories followed along during inflation to another, e.g. non-delta-flattrajectory to delta-flat or non-delta-flat one. This is studied in Section 5.2 where we also discussthe fact that the actual dynamics of the system is characterized by non-standard kinetic terms whichcontain cross terms involving derivatives of the fields.
The fields P , Q and R that we used in Sections 3 and 4 are given by combinations of the angularcoordinates α i , β i and γ i in the warped conifold, using the so-called Euler-Rodriguez parametrizationdiscussed in Appendix B. Recall that, in Section 4, we found that the angular field P remains heavierthan Q and R during the most interesting phase of inflation, and does not take part in the instabilityprocess . Thus we set it at its extremal value P = 0 in the following discussion. From its definition(B.33), this gives α + α = 0. We will restrict ourselves to the case α = α = 0, which greatlysimplifies the following calculations. We then concentrate on the remaining angular modes Q and R inthe previous discussion. We have seen in the previous section that the masses of these modes becometachyonic at almost the same point along the inflationary region, see Fig. 3. For this reason we areinterested in determining the complete potential that describes these fields together, as well as theradial direction. As can be read from (B.31) and (B.32), they are actually formed by a combinationsof two pairs of angular variables. It turns out that, while for small perturbations around the extremaltrajectories the combinations of β , β , γ and γ giving Q and R are the most convenient to dealwith, this is no longer true when describing arbitrary displacements from the extremal points. Todescribe this last case, it is useful to define another basis of fields.From (B.3) and (B.12), we can easily find that starting from a point along the extremal trajectories, w and w are given by w = w (0)1 cos | χ | cos | χ | − w (0)2 sin | χ | sin | χ | (cid:18) γ | χ | − i β | χ | (cid:19) (cid:18) γ | χ | − i β | χ | (cid:19) , (5.1) w = w (0)2 cos | χ | cos | χ | + w (0)1 sin | χ | sin | χ | (cid:18) γ | χ | + i β | χ | (cid:19) (cid:18) γ | χ | + i β | χ | (cid:19) , (5.2) Notice that this fact holds only for the specific inflationary model we have considered. There could be otherinflationary trajectories, with different choices of the parameters, for which the behavior of P is different. – 17 –ith χ i given by (B.11) as χ i = β i + γ i . The actual range of the quantities β i and γ i depends on theextremal trajectory one considers, as discussed in Appendix B.3 (see also the discussion in the nextsection), but we can take them positive. Along with χ and χ , it is natural to define a field ξ as ξ + ξ =2 ξ , (5.3) ξ = cos − (cid:18) γ | χ | (cid:19) = sin − (cid:18) β | χ | (cid:19) , (5.4) ξ = cos − (cid:18) γ | χ | (cid:19) = sin − (cid:18) β | χ | (cid:19) . (5.5)It is convenient to consider ξ rather than ξ and ξ separately since, as can be read from (5.1) and(5.2), only the combination ξ + ξ appears in the potential. With these new fields, and using (2.16)and (2.17), we can write w and w as w = ± ǫ √ e τ/ L −− ± i ǫ √ e − τ/ sin | χ | sin | χ | sin(2 ξ ) , (non-delta-flat) ± ǫ √ e − τ/ L + − ± i ǫ √ e τ/ sin | χ | sin | χ | sin(2 ξ ) , (delta-flat) (5.6) w = ∓ ǫ √ e − τ/ L + − ± i ǫ √ e τ/ sin | χ | sin | χ | sin(2 ξ ) , (non-delta-flat) ∓ ǫ √ e τ/ L −− ± i ǫ √ e − τ/ sin | χ | sin | χ | sin(2 ξ ) , (delta-flat) (5.7)where we have defined L ± + ≡ cos | χ | cos | χ | + exp ( ± τ ) sin | χ | sin | χ | cos(2 ξ ) , (5.8) L ±− ≡ cos | χ | cos | χ | − exp ( ± τ ) sin | χ | sin | χ | cos(2 ξ ) . (5.9)In the limit of vanishing χ , χ and ξ , we recover (2.16) and (2.17). Note that these expressions havean interesting property: starting from one extremal trajectory, we can move to another by continuouslyvarying χ , χ and ξ : for example, as will be demonstrated momentarily, from the lower branch of thenon-delta-flat extremal trajectory obtained by χ = χ = ξ = 0, we can obtain both branches of thedelta-flat one by taking | χ | = | χ | = π/ ξ = 0 or ξ = π/
2. We will discuss some consequencesof this fact in what follows.
Having found w and w as (5.6) and (5.7), we can straightforwardly calculate the scalar potential.After some computations, we can find that the form (2.9) does not change, but the functions G , C ,– 18 –nd D become different. The results are G = ((cid:0) ∓ αe τ/ L −− (cid:1) + α e − τ [sin | χ | sin | χ | sin(2 ξ )] , (non-delta-flat) (cid:0) ∓ αe − τ/ L + − (cid:1) + α e τ [sin | χ | sin | χ | sin(2 ξ )] , (delta-flat) (5.10) C = τ n ± αe τ/ L − + − α e τ L −− L − + + α e − τ [sin | χ | sin | χ | sin(2 ξ )] o , (non-delta-flat)2 tanh τ n ∓ αe − τ/ L ++ + α e − τ L + − L ++ − α e τ [sin | χ | sin | χ | sin(2 ξ )] o , (delta-flat) (5.11) D = − sech τ ( e − τ L + − + e τ | χ | sin | χ | sin(2 ξ )] ) + k ′′ k ′ − coth τ ! × ( tanh τ + 2csch(2 τ ) " (cos | χ | cos | χ | ) + (sin | χ | sin | χ | ) + 12 cosh τ sin(2 | χ | ) sin(2 | χ | ) cos(2 ξ ) − sech τ e τ L −− + e − τ L + − + cosh τ [sin | χ | sin | χ | sin(2 ξ )] !) , (non-delta-flat)1 − sech τ ( e τ L −− + e − τ | χ | sin | χ | sin(2 ξ )] ) + k ′′ k ′ − coth τ ! × ( tanh τ + 2csch(2 τ ) " (cos | χ | cos | χ | ) + (sin | χ | sin | χ | ) + 12 cosh τ sin(2 | χ | ) sin(2 | χ | ) cos(2 ξ ) − sech τ e − τ L + − + e τ L −− + cosh τ [sin | χ | sin | χ | sin(2 ξ )] !) . (delta-flat) (5.12)Plugging these expressions into (2.9), we can obtain the full potential including angular directions.This is the complete form of the potential that is needed in order to study the dynamics of the angularand radial directions in the instability region. While in Section 4 we focussed on the lower branch ofthe non-delta flat trajectory, the previous formulae allow us to also obtain the potential in all othercases. In the limit of small χ i , or equivalently of small Q and R , this potential coincides with the onewe used in Appendix B to determine the eigenvalues X Q and X R .The potential is periodic along the angular directions χ , χ and ξ , as it is clear given our expres-sions for w and w . There are “ extremal points ”, defined as the positions for which derivatives of thepotential along χ i and ξ vanish. These extremal points coincide with the extremal points along thedirections Q and R : namely they provide the same non-delta-flat and delta-flat trajectories determinedin Appendix B. They are shown in Table 2, but others that are periodically identifiable with theseones are also extremal points. NDF, L NDF, U NDF, U DF, U DF, L | χ | π π/ π/ | χ | π π/ π/ ξ π/ π/ π/ Table 2:
The values of χ , χ and ξ for extremal points. “NDF” and “DF” stand for non-delta-flat anddelta-flat, and “U” and “L” for upper and lower branches respectively. For example, the first column indicatesthe lower branch in the non-delta flat trajectory, the case on which we focussed in Section 4. – 19 –hese observations are sufficient to extract important qualitative information about the outcome ofthe instability process. For definiteness, let us consider the explicit example we studied in the previoussection, which corresponds to the first column of Table 2. For the choice of parameters summarised inTable 1, all of the extremal points have at least one unstable angular mode except for the upper branchof the delta-flat direction ( | χ | = | χ | = π/ χ i and ξ start to roll down the potential. In this case, the range of the variables are foundto be ξ ∈ [0 , π ] and χ i ∈ [ − π/ , π/
2] (or equivalently | χ i | ∈ [0 , π/ χ = χ and keeping ξ = 0. The plot clearly exhibits an instability along the non-delta-flat direction( χ = χ = 0), and we can more easily see that the potential connects the false vacuum correspondingto the lower branch of the non-delta-flat trajectory with the true vacuum corresponding to the upperbranch of the delta-flat trajectory ( | χ | = | χ | = π/
2) in the angular directions. Φ (cid:144) Φ Μ -Π- Π Π Π Χ ´ V -Π - Π Π Π Χ Φ (cid:144) Φ Μ Figure 7:
The multi-field potential as a function of the radial direction and a particular choice of angularcoordinates χ = χ and ξ = 0. We can see that other stable trajectories exist at | χ | = | χ | = π/ χ = χ = 0. The potential is periodic along the direction χ , with period π . Thisis consistent with our observation that | χ i | ∈ [0 , π/ As we have seen from the previous arguments, the angular instability connects different extremaltrajectories in the warped deformed conifold. Depending on initial conditions, after the field falls fromone trajectory to the other, a non-trivial dynamics continues along the radial and, possibly, along theangular directions. This implies that, at the later stages of slow-roll inflation, the inflaton field doesnot generically fall into a global minimum, but into other extremal points that further evolve due– 20 –o non-trivial motion in the radial direction. In this setup, consequently, it is not possible to obtaina quantitative understanding of the process of formation and evolution of topological defects at theend of inflation, based on the symmetries of the potential. Instead, a detailed numerical analysis isnecessary, which takes into account of the post-inflationary evolution along the radial direction.The analysis of the actual dynamics that develops at the instability is further complicated by thenon-standard form of the kinetic terms characterizing the fields. In order to obtain them, one canfollow an identical procedure to the one discussed in Section 3 and Appendix B. After straightforwardcalculations, we can find that the form of the kinetic term is given by (3.1) where the pull-back d ˆ s given by (A.16), with g (1) = sin(2 χ ) dξ + cos(2 ξ ) sin(2 χ ) dξ + 2 sin(2 ξ ) dχ , (5.13) g (2) =2 dχ + sin(2 ξ ) sin(2 χ ) dξ − ξ ) dχ , (5.14) g (3) = sin(2 χ ) dξ − cos(2 ξ ) sin(2 χ ) dξ − ξ ) dχ , (5.15) g (4) =2 dχ − sin(2 ξ ) sin(2 χ ) dξ + 2 cos(2 ξ ) dχ , (5.16) g (5) = [1 − cos(2 χ )] dξ + [1 − cos(2 χ )] dξ . (5.17)The kinetic terms are then characterized by the cross terms involving derivatives of the fields. Thesecross terms have very small coefficients when considering small deviations around the extremal tra-jectory. But they play an important role when large angular displacements are considered, which isexpected during the instability process. The range of the angular fields, in the previous equations,depends on the initial trajectory one is considering, as discussed in the first part of this section.In this section, we have provided all the necessary tools that are needed in order to study thedynamics of the instability process at later stages of inflation. We have obtained important qualitativeinformation about the field dynamics and the instability process. These ingredients are the startingpoint to perform a more detailed analysis of the field evolution after the instability develops. Forexample, it would be interesting to study how the annihilation of D3-D3 branes occurs, the role ofthe angular directions in this process, and whether moduli stabilization induced by the presence ofD7-branes can have some effects in this phenomenon. Indeed, in Ref. [39] it has been shown that nearthe tip of the throat the force between brane and antibrane can acquire significant angular dependence.Another important issue is to investigate possible formation and evolution of topological defects afterinflation ends. We hope to return to discuss these issues elsewhere.
6. Future Directions
In this work, we have considered a model of D3-brane inflation in a warped throat. Different forcesact on the D3-brane, due to the presence of an anti D3-brane at the tip of the throat, and of aOuyang-embedded D7-brane in the warped deformed conifold. We have shown that these forcescompensate each other, and can obtain 60 e -foldings of slow-roll inflation around an inflection point ofthe potential along the radial direction. At later stages of single field slow-roll inflation, the angulardirections develop instabilities after which more complicated dynamics follows. Our work provides anew explicit example for obtaining slow-roll inflation in warped throats, other than using Kupersteinembedded D7-brane [13, 16]. Moreover, it is an example where angular directions play an active rolefor determining the inflationary dynamics while the model is fully under control from the supergravitypoint of view and the complete potential for the relevant modes is known.There are various topics, motivated by our study, that deserve further analysis. We have shownthat the masses of the angular modes remain comparable to the Hubble scale during most of the– 21 –nflationary period, becoming light towards the later stages of slow-roll inflation. The fact that theangular masses are not much larger than the Hubble scale can lead to a situation in which the slow-roll motion continues along the radial direction for few e -foldings, even after the development of theinstability in the angular directions. When the slow-roll conditions are violated during the instabilityprocess, however, the DBI nature of the D-brane action shows up, since the brane velocity increases.This fact can also play an important role in determining the detailed dynamics governing the instability.It would be interesting to study the consequences of these features for the spectrum of the curvatureperturbation, taking into account the presence of the non-canonical kinetic terms for the fields involved,and the particular form of the potential.Another important direction is to study the formation and stability of topological defects producedafter inflation. As we have argued, the instability process connects different extremal trajectories.Consequently, after the onset of instability, the system does not reach a global minimum, and furthermotion of the brane along the radial direction takes place. The study of implications of this branemotion, for a possible development of topological defects, needs a detailed numerical investigation.Finally, it would also be interesting to understand whether the energy stored in the inflaton field canbe transmitted into the matter sector, while the angular fields roll off their false minima. We hope toreturn to analyse these issues in the near future. Acknowledgments
We would like to thank Ruth Gregory, Liam McAllister, Peter Ouyang, Gary Shiu, and Ivonne Zavalafor very useful discussions. HYC would like to thank PCTS, Princeton University, where this work wasinitiated. He is supported in part by NSF CAREER Award No. PHY-0348093, DOE grant DE-FG-02-95ER40896, a Research Innovation Award and a Cottrell Scholar Award from Research Corporation,and a Vilas Associate Award from the University of Wisconsin. JG is grateful to the Yukawa Institutefor Theoretical Physics, Kyoto University for hospitality during the long-term workshop “Gravityand Cosmology 2010 (GC2010)” (YITP-T-10-01) and the YKIS symposium “Cosmology – The NextGeneration –” (YKIS2010) where this work was being finished. He is partly supported by a VIDI and aVICI Innovative Research Incentive Grant from the Netherlands Organisation for Scientific Research(NWO). KK would like to thank the Yukawa Institute for Theoretical Physics, Kyoto Universityand the Royal Society for two workshops, “Non-linear cosmological perturbations” (YITP-W-09-01)and “The non-Gaussian universe” (YITP-T-09-05) where he is benefitted from many stimulatingdiscussions. He is supported by European Research Coucil, Research Councils UK and STFC.– 22 – ppendicesA. Properties of warped deformed conifold
A.1 Parametrization of the angular coordinates
Here we collect some facts concerning the various coordinates parameterizing the deformed conifoldfollowing Ref. [40]. It can be defined as a submanifold C via the complex embedding equation X α =1 ( z α ) = ǫ , (A.1)where { z α } are complex coordinates in C . They can be expressed in terms of a 2 × W = (cid:18) − w w − w w (cid:19) = 1 √ (cid:18) z + iz z − iz z + iz − z + iz (cid:19) . (A.2)The deformed conifold relation (A.1) can then be written asdet W = w w − w w = − ǫ . (A.3)The complex embedding coordinates of deformed conifold { z , z , z , z } can also be expressed in termsof the real global coordinates { τ ∈ R , ψ ∈ [0 , π ] , θ , ∈ [0 , π ] , φ , ∈ [0 , π ] } , Ξ = τ + iψ as z = ǫ (cid:20) cosh (cid:18) Ξ2 (cid:19) cos (cid:18) θ + θ (cid:19) cos (cid:18) φ + φ (cid:19) + i sinh (cid:18) Ξ2 (cid:19) cos (cid:18) θ − θ (cid:19) sin (cid:18) φ + φ (cid:19)(cid:21) , (A.4) z = ǫ (cid:20) − cosh (cid:18) Ξ2 (cid:19) cos (cid:18) θ + θ (cid:19) sin (cid:18) φ + φ (cid:19) + i sinh (cid:18) Ξ2 (cid:19) cos (cid:18) θ − θ (cid:19) cos (cid:18) φ + φ (cid:19)(cid:21) , (A.5) z = ǫ (cid:20) − cosh (cid:18) Ξ2 (cid:19) sin (cid:18) θ + θ (cid:19) cos (cid:18) φ − φ (cid:19) + i sinh (cid:18) Ξ2 (cid:19) sin (cid:18) θ − θ (cid:19) sin (cid:18) φ − φ (cid:19)(cid:21) , (A.6) z = ǫ (cid:20) − cosh (cid:18) Ξ2 (cid:19) sin (cid:18) θ + θ (cid:19) sin (cid:18) φ − φ (cid:19) − i sinh (cid:18) Ξ2 (cid:19) sin (cid:18) θ − θ (cid:19) cos (cid:18) φ − φ (cid:19)(cid:21) . (A.7)One can readily verify that the constraints (A.1) and (A.3) are satisfied. We can also encode thecoordinates (A.4)-(A.7) compactly in the matrix form as W = L c W s R † c , (A.8)where the 2 × W s , L c and R c are given by W s = ǫ √ e τ/ ǫ √ e − τ/ , (A.9) L c = cos θ ! exp " i ψ + φ ) − sin θ ! exp " − i ψ − φ ) sin θ ! exp " i ψ − φ ) cos θ ! exp " − i ψ + φ ) , (A.10)– 23 –nd similarly for R c with the subscript 1 replaced by 2. L c and R c are two independent SU (2) groupelements, which generate the entire deformed conifold by acting on the special point W s . Notice thatthe coordinates ψ and ψ only appear in the combination ψ = ψ + ψ . A.2 The metric of the warped deformed conifold
It is convenient to define the following basis of one forms [9, 41], g (1) ≡ − sin θ dφ − (cos ψ sin θ dφ − sin ψdθ ) √ , (A.11) g (2) ≡ dθ − (sin ψ sin θ dφ + cos ψdθ ) √ , (A.12) g (3) ≡ − sin θ dφ + (cos ψ sin θ dφ − sin ψdθ ) √ , (A.13) g (4) ≡ dθ + (sin ψ sin θ dφ + cos ψdθ ) √ , (A.14) g (5) ≡ dψ + cos θ dφ + cos θ dφ . (A.15)The explicit metric of the six dimensional internal part of the geometry, the deformed conifold, canbe expressed as ds = ǫ / K ( τ ) (cid:26) K ( τ ) (cid:20) dτ + (cid:16) g (5) (cid:17) (cid:21) + cosh (cid:16) τ (cid:17) (cid:20)(cid:16) g (3) (cid:17) + (cid:16) g (4) (cid:17) (cid:21) + sinh (cid:16) τ (cid:17) (cid:20)(cid:16) g (1) (cid:17) + (cid:16) g (2) (cid:17) (cid:21)(cid:27) , (A.16)where K ( τ ) ≡ [sinh(2 τ ) − τ ] / / sinh τ . (A.17)We can also write the full ten dimensional warped metric as ds = e A ( y ) η µν dx µ dx ν + e − A ( y ) ds . (A.18)The warping is obtained by turning M units of F flux through the A -cycle of the deformedconifold, and − K units of H flux through the dual B -cycle, when we put a ultraviolet cutoff of thewarped throat at e − A ∼
1. The warp factor is given by the expression [9] e − A ( τ ) = 2 / ( g s M α ′ ) ǫ − / I ( τ ) , (A.19)where I ( τ ) ≡ Z ∞ τ dx x coth x − x [sinh(2 x ) − x ] / . (A.20)The internal, deformed conifold metric (A.16) can be obtained from the so-called “little” K¨ahlerpotential k ( z α , ¯ z ¯ β ) as ˜ g α ¯ β = ∂ α ∂ ¯ β k . (A.21)As the angular directions of the warped deformed conifold are isometries, they do not appear explicitlyin the little K¨ahler potential. It thus only depends on the radial coordinate τ . An expression for k ( τ )is [40] k ( τ ) = ǫ / / Z τ dτ ′ [sinh(2 τ ′ ) − τ ′ ] / , (A.22)– 24 –here without loss of generality we set the integration constant to zero. For the calculation of the F -term scalar potential, the unwarped inversed K¨ahler metric in terms of the holomorphic embeddingcoordinates { w α } is also needed and is given by k ¯ αβ = r k ′′ (cid:20) R ¯ αβ + coth τ (cid:18) k ′′ k ′ − coth τ (cid:19) L ¯ αβ (cid:21) , (A.23) R ¯ αβ = δ ¯ αβ − c αα ′ c ββ ′ w α ′ w β ′ r , (A.24) L ¯ αβ = (cid:18) − ǫ r (cid:19) δ ¯ αβ + ǫ r c ββ ′ w α w β ′ + c ββ ′ w α w β ′ r − w α w β + c α ′ α c β ′ β w α ′ w β ′ r . (A.25)Here c αα ′ is a 4 × c = c = − c = c = 1. Notice thatthe metric (A.23) is indeed six dimensional as the deformed conifold constraint w w − w w = − ǫ / w and w .To recover the familiar singular conifold limit, we can define the radial coordinate r as r = X α =1 | z α | = ǫ cosh τ . (A.26)In the asymptotic limit e τ → ∞ , the metric (A.16) reduces to usual conical form ds → (cid:0) dr + r ds T , (cid:1) , (A.27)where the metric ds T , for the five dimensional base T , can be easily deduced from (A.16). Fur-thermore the warp factor e − A (A.19) acquires an AdS form e − A ( r ) ≈ πg s N ( α ′ ) r , (A.28)where N is the quantized five form flux. B. Extremal trajectory and angular mass matrix
In Section 2, we derived the complete potential controlling the dynamics of the D3-brane along thethroat. It explicitly depends on the D3-brane coordinates, in particular on the angular directions. Inthis section, we determine extremal trajectories of the D3-brane that extremize the potential alongthe angular directions. These extremal trajectories, along which the brane moves only in the radialdirection, correspond to stable points in the moduli space of angular directions, at least for sufficientlylarge radius r = ǫ cosh τ . For small values of r , instead, extremal trajectories become unstable forsome of the angular directions. The consequences of this fact for inflation is discussed in Sections 4and 5.We denote the five angular coordinates of deformed conifold by { Ψ i } . We then look for extremaltrajectories that satisfy ∂V∂ Ψ i = 0 . (B.1)The chain rule ensures that finding points satisfying (B.1) is equivalent to identifying points such that ∂ ( w + w ) ∂ Ψ i = ∂ | w | ∂ Ψ i = ∂ ( w + w ) ∂ Ψ i = ∂ | w | ∂ Ψ i = ∂ ( w w + w w ) ∂ Ψ i = 0 . (B.2)– 25 –his is because, as we can read from (2.9), the F -term scalar potential V F only depends on thesecombinations of coordinates of the deformed conifold. Thus, our next task is to relate the branecoordinates w α to the angular coordinates Ψ i .To parametrize the angular coordinate Ψ i , we can rewrite the matrix W as [13] W ≡ L e W R † e . (B.3)Here W is given by W = − w (0)3 w (0)2 − w (0)1 w (0)4 ! , (B.4)and denotes a fiducial point, which we consider to be angularly stable. The matrices L e and R e aretwo independent SU (2) group elements. Using the generators of the SU (2), i.e. the Pauli matrices σ = (cid:18) (cid:19) , (B.5) σ = (cid:18) − ii (cid:19) , (B.6) σ = (cid:18) − (cid:19) , (B.7) L e and R e can then be identified as L e = e iT , (B.8) R e = e iT , (B.9)where T and T are parametrized by two real three-vectors { α i , β i , γ i } ( i = 1 ,
2) as T i = β i σ + γ i σ + α i σ = (cid:18) α i β i − iγ i β i + iγ i − α i (cid:19) . (B.10)Denoting χ i ≡ α i + β i + γ i , (B.11)a simple calculation provides L e = cos | χ | + iα sin | χ || χ | i ( β − iγ ) sin | χ || χ | i ( β + iγ ) sin | χ || χ | cos | χ | − iα sin | χ || χ | , (B.12)and similarly for R e .The two SU (2) group elements L e and R e take a point W on deformed conifold to the entiremanifold. Notice that here we have adopted different parametrization of SU (2) group element (Euler-Rodriguez) from the one used in (A.8) (Cayley-Kline), as the former parametrization is particularlysuitable to study the properties of extremal trajectories. The parameters { α i , β i , γ i } ( i = 1 ,
2) of(B.12) and { θ i , φ i , ψ i } ( i = 1 ,
2) of (A.10) are usually related non-linearly. We will discuss therelation between these two different parametrizations in Appendix B.3.– 26 – .1 Linear expansion: Identifying the extremal trajectory
As we want angular stability near w (0) α satisfying (B.2), expanding (B.8) and (B.9) and only keepingthe linear order, then we find W = ( iT ) W ( − iT ) = W + i ( T W − W T ) . (B.13)The changes in w and w are then given by δw = − i ( α + α ) w (0)1 + ( iβ − γ ) w (0)3 + ( iβ − γ ) w (0)4 , (B.14) δw = i ( α + α ) w (0)2 + ( iβ + γ ) w (0)4 + ( iβ + γ ) w (0)3 . (B.15)Then, it is trivial to find, by demanding the linear variations of the combinations of w and w whichappear in the potential vanish, that n w (0)1 , w (0)2 o ∈ R and w (0)3 = w (0)4 = 0. Thus, the conifoldconstraint equation now gives w (0)1 w (0)2 = − ǫ . (B.16)Combining with (A.26), which now reads (cid:12)(cid:12)(cid:12) w (0)1 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) w (0)2 (cid:12)(cid:12)(cid:12) = (cid:16) w (0)1 (cid:17) + (cid:16) w (0)2 (cid:17) = ǫ cosh τ , (B.17)we determine the following extremal trajectories w (0)1 = ± ǫ √ e τ/ , w (0)2 = ∓ ǫ √ e − τ/ , (B.18)and w (0)1 = ± ǫ √ e − τ/ , w (0)2 = ∓ ǫ √ e τ/ . (B.19)In the large radius limit e τ/ → ∞ , the trajectories (B.18) and (B.19) correspond respectively to theso-called non-delta-flat and delta-flat trajectories determined in Ref. [13]. While we present an explicitrealization of slow-roll inflation using the non-delta-flat trajectory, we will also discuss the role of thedelta-flat trajectory for the multi-field inflationary models. B.2 Quadratic expansion: Mass matrix
We now develop the tools necessary to investigate the stability of the extremal trajectories (B.18) and(B.19) along the angular directions. To find the mass matrix associated with the angular coordinates,we expand (B.8) and (B.9) up to second order in perturbations. Explicitly, we have W = (cid:18) iT − T (cid:19) W (cid:18) − iT − T (cid:19) = W + i ( T W − W T ) + (cid:20) T W T − (cid:0) T W + W T (cid:1)(cid:21) . (B.20)– 27 –sing w (0)3 = w (0)4 = 0 along the extremal trajectories, we can explicitly calculate the combinations (cid:8) w + w , | w | , w + w , | w | , w w + w w (cid:9) , and find w + w =2 w (0)1 − (cid:2) ( α + α ) + β + β + γ + γ (cid:3) w (0)1 − β β − γ γ ) w (0)2 , (B.21) w + w =2 w (0)2 − (cid:2) ( α + α ) + β + β + γ + γ (cid:3) w (0)2 − β β − γ γ ) w (0)1 , (B.22) | w | = (cid:16) w (0)1 (cid:17) − (cid:0) β + β + γ + γ (cid:1) (cid:16) w (0)1 (cid:17) − β β − γ γ ) w (0)1 w (0)2 , (B.23) | w | = (cid:16) w (0)2 (cid:17) − (cid:0) β + β + γ + γ (cid:1) (cid:16) w (0)2 (cid:17) − β β − γ γ ) w (0)1 w (0)2 , (B.24) w w + w w =2 w (0)1 w (0)2 − (cid:0) β + β + γ + γ (cid:1) w (0)1 w (0)2 − β β − γ γ ) (cid:20)(cid:16) w (0)1 (cid:17) + (cid:16) w (0)2 (cid:17) (cid:21) . (B.25)As required, the departures from the fiducial point vanishes at linear order. We can thus write themass matrix as ∂ V∂ Ψ i ∂ Ψ i (cid:12)(cid:12)(cid:12)(cid:12) = " ∂V∂ ( w + w ) ∂ ( w + w ) ∂ Ψ i ∂ Ψ j + ∂V∂ | w | ∂ | w | ∂ Ψ i ∂ Ψ j + ∂V∂ ( w + w ) ∂ ( w + w ) ∂ Ψ i ∂ Ψ j + ∂V∂ | w | ∂ | w | ∂ Ψ i ∂ Ψ j + ∂V∂ ( w w + w w ) ∂ ( w w + w w ) ∂ Ψ i ∂ Ψ j , (B.26)where the linear partial derivatives of the brane coordinates do not appear since we have chosen theextremal fiducial trajectory. We can see that in the present angular coordinates { α i , β i , γ i } , the massmatrix is not diagonal because of the terms which mix different coordinates, such as ( α + α ) , β β and γ γ . Let us rewrite the angular coordinates β i and γ i in terms another set of real coordinates q i and r i as β , = q ± r √ , (B.27) γ , = r ± q √ . (B.28)Then we immediately find β + β + γ + γ = q + q + r + r , (B.29) β β − γ γ = q + q − r + r . (B.30)We can further define shorthand notations q + q ≡ Q , (B.31) r + r ≡ R , (B.32)and similarly α + α ≡ P , (B.33)– 28 –ince only these combinations appears. Using these new coordinates { P, Q, R } , we obtain w + w =2 w (0)1 − (cid:0) P + Q + R (cid:1) w (0)1 − (cid:0) Q − R (cid:1) w (0)2 , (B.34) w + w =2 w (0)2 − (cid:0) P + Q + R (cid:1) w (0)2 − (cid:0) Q − R (cid:1) w (0)1 , (B.35) | w | = (cid:16) w (0)1 (cid:17) − (cid:0) Q + R (cid:1) (cid:16) w (0)1 (cid:17) − (cid:0) Q − R (cid:1) w (0)1 w (0)2 , (B.36) | w | = (cid:16) w (0)2 (cid:17) − (cid:0) Q + R (cid:1) (cid:16) w (0)2 (cid:17) − (cid:0) Q − R (cid:1) w (0)1 w (0)2 , (B.37) w w + w w =2 w (0)1 w (0)2 − (cid:0) Q + R (cid:1) w (0)1 w (0)2 − (cid:0) Q − R (cid:1) (cid:20)(cid:16) w (0)1 (cid:17) + (cid:16) w (0)2 (cid:17) (cid:21) . (B.38)These new coordinates therefore make the mass matrix diagonal, with rows and the columns corre-sponding to { P , q , q , r , r } , as ∂ V∂ Ψ i ∂ Ψ i (cid:12)(cid:12)(cid:12)(cid:12) = X P X Q X Q X R
00 0 0 0 X R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pq q r r (B.39)where X P ≡ − (cid:20) w (0)1 ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) + w (0)2 ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) , (B.40) X Q ≡ − (cid:16) w (0)1 + w (0)2 (cid:17) (cid:20) ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) + ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) + 2 ǫ (1 − cosh τ ) ∂V∂ ( w w + w w ) (cid:12)(cid:12)(cid:12)(cid:12) + ∂V∂ | w | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) − (cid:16) w (0)1 (cid:17) + ǫ (cid:21) + ∂V∂ | w | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) − (cid:16) w (0)2 (cid:17) + ǫ (cid:21) , (B.41) X R ≡ − (cid:16) w (0)1 − w (0)2 (cid:17) (cid:20) ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) − ∂V∂ ( w + w ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) + 2 ǫ (1 + cosh τ ) ∂V∂ ( w w + w w ) (cid:12)(cid:12)(cid:12)(cid:12) + ∂V∂ | w | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) − (cid:16) w (0)1 (cid:17) − ǫ (cid:21) + ∂V∂ | w | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) − (cid:16) w (0)2 (cid:17) − ǫ (cid:21) . (B.42)It is consequently straightforward to substitute the derivatives of the potential, as well as theexpressions for w (0) i in the previous formulae. Explicitly, we can calculate the angular masses associated– 29 –ith them as X P = ( ∓ α (cid:0) e τ/ V w + w − e − τ/ V w + w (cid:1)(cid:12)(cid:12) , (non-delta-flat) ∓ α (cid:0) e − τ/ V w + w − e τ/ V w + w (cid:1)(cid:12)(cid:12) , (delta-flat) (B.43) X Q = X P + 2 α (cid:0) V w w + w w − e τ V | w | − e − τ V | w | (cid:1) (non-delta-flat)+ (cid:2) α (cid:0) V | w | + V | w | (cid:1) − α cosh τ V w w + w w ∓ α (cid:0) e τ/ V w + w − e − τ/ V w + w (cid:1)(cid:3)(cid:12)(cid:12) ,X P + 2 α (cid:0) V w w + w w − e − τ V | w | − e τ V | w | (cid:1) (delta-flat)+ (cid:2) α (cid:0) V | w | + V | w | (cid:1) − α cosh τ V w w + w w ∓ α (cid:0) e − τ/ V w + w − e τ/ V w + w (cid:1)(cid:3)(cid:12)(cid:12) , (B.44) X R = X P + 2 α (cid:0) V w w + w w − e τ V | w | − e − τ V | w | (cid:1) (non-delta-flat) − (cid:2) α (cid:0) V | w | + V | w | (cid:1) − α cosh τ V w w + w w ∓ α (cid:0) e τ/ V w + w − e − τ/ V w + w (cid:1)(cid:3)(cid:12)(cid:12) ,X P + 2 α (cid:0) V w w + w w − e − τ V | w | − e τ V | w | (cid:1) (delta-flat) − (cid:2) α (cid:0) V | w | + V | w | (cid:1) − α cosh τ V w w + w w ∓ α (cid:0) e − τ/ V w + w − e τ/ V w + w (cid:1)(cid:3)(cid:12)(cid:12) , (B.45)where V w + w = − A n G /n − (cid:20) B − ae aσ | W || A | G − / (2 n )0 − n − n G ( K coth τ C + R ◦ D ) − K coth τ (cid:21) , (B.46) V w + w = A n G /n − K sinh τ , (B.47) V | w | = − V w + w − A n G /n − coth τ (cid:20) K + R × (cid:18) K − coth τ (cid:19)(cid:21) , (B.48) V | w | = − A n G /n − R × (cid:20) τ (cid:18) K − coth τ (cid:19)(cid:21) , (B.49) V w w + w w = − A n G /n − csch τ (cid:20) K + R × (cid:18) K − coth τ (cid:19)(cid:21) , (B.50)and K ( τ ) ≡ k ′ k ′′ = 34 sinh(2 τ ) − τ sinh τ , (B.51) R ◦ ( τ ) ≡ r anµ γk ′′ = 2 α cosh τ R × ( τ ) , (B.52) R × ( τ ) ≡ anγk ′′ = 3 / / β [sinh(2 τ ) − τ ] / sinh τ . (B.53)These results are the basic tools for studying the stability of inflationary trajectories along the angulardirections. B.3 Euler-Rodriguez versus Cayley-Kline parametrization of SU (2) group We now discuss in more detail the connection between the Euler-Rodriguez and the Cayley-Klineparametrizations of SU (2) group elements. In particular, we are interested in the relation betweenthe matrices L e and R e , and the matrices L c and R c . The analysis here also leads to the derivationof the kinetic terms used in the multi-field analysis in the main text.– 30 –ecall that W is obtained separately by using two different parametrizations, (A.8) and (B.3).Since, the expression for W (B.3) depends on which extremal trajectory one chooses, also the relationbetween L e and L c depends on this choice. From the similarity transformation between W and W s ,we can find that one possible realization of the relation between L c and L e is L c = L e iσ , (non-delta-flat, lower branch) L e iσ , (non-delta-flat, upper branch) L e , (delta-flat, lower branch) L e σ , (delta-flat, upper branch) (B.54)and precisely the same for R c and R e . But other realizations work as well. For the remaining partof this appendix, we focus on the lower branch of the non-delta flat trajectory: this is the case weexplicitly considered in Section 4, and the other cases can be treated in an identical way. The relationbetween the angles θ i , ψ i and φ i used in the Cayley-Kline representation and α i , β i and γ i in theEuler-Rodriguez representation is obtained from the first equation in (B.54), and reads, both for i = 1and 2, cos (cid:18) θ i (cid:19) cos (cid:18) ψ i + φ i (cid:19) = − γ i sin | χ i || χ i | , (B.55)cos (cid:18) θ i (cid:19) sin (cid:18) ψ i + φ i (cid:19) = − β i sin | χ i || χ i | , (B.56)sin (cid:18) θ i (cid:19) cos (cid:18) ψ i − φ i (cid:19) = − cos | χ i | , (B.57)sin (cid:18) θ i (cid:19) sin (cid:18) ψ i − φ i (cid:19) = α i sin | χ i || χ i | . (B.58)By means of these relations, we see that the extremal trajectory corresponding to α i = β i = γ i = 0is obtained, for example, by choosing θ (0) i = π , ψ (0) i − φ (0) i = 2 π . By expanding the above expressionsaround this point, we find δα i = 12 ( δφ i − δψ i ) , (B.59) δβ i = δθ i ψ (0) i + φ (0) i ! , (B.60) δγ i = δθ i ψ (0) i + φ (0) i ! . (B.61)These expansions allow to obtain the relation between { P, Q, R } and { θ i , φ i , ψ i } . Choosing for defi-niteness ψ (0) i = 2 π , φ (0) i = 0, the identifications (B.31), (B.32) and (B.33) yield P = 12 δ ( φ + φ − ψ ) , (B.62) Q = − √ δ ( θ − θ ) , (B.63) R = − √ δ ( θ + θ ) (B.64)– 31 –t is easy to check that, near this extremal trajectory, expanding up to quadratic order in angularfluctuations, the pull back-metric (A.19) reduces to d ˆ s (cid:12)(cid:12) = ǫ / K ( τ ) (cid:26) dτ + 4 dP K ( τ ) + 4 h cosh (cid:16) τ (cid:17) dR + sinh (cid:16) τ (cid:17) dQ i(cid:27) . (B.65)This result plays an important role in Section 4. We have checked that the same expression holds forthe expansion around all the extremal trajectories.To conclude, it is interesting to consider in more detail the case α = α = 0, that plays animportant role in the discussion of Section 5. In this case, in order to satisfy (B.58), we can choose ψ i = 2 π + φ i . Then (B.55), (B.56) and (B.57) simplify, and provide | χ i | = π − θ i , (B.66) β i = | χ i | sin ψ i , (B.67) γ i = | χ i | cos ψ i . (B.68)They are valid for arbitrary values of the angles, inside their interval ranges. Since θ i ∈ [0 , π ], we thenlearn that | χ i | ∈ [0 , π/ χ i ∈ [ − π/ , π/ ξ = ψ in this specific case. This implies that ξ liesin the range [0 , π ]. The potential is periodic in these angular variables, with periods set by the rangeswe have just found. However, let us emphasize once again that the identifications (B.66), (B.67) and(B.68) and the relative field range, are only valid starting from the lower branch of the non-delta-flattrajectory. Making other choices in (B.54) lead to different identifications. On the other hand, it issimple to work out them following the same procedure as above. C. Stabilized value of σ In this Appendix, we analyse the stabilisation of σ in the limit τ →
0. In the absence of uplifting, thestabilized value of σ at τ = 0 is given by the condition ∂V F (0 , σ ) ∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ = σ F = 0 . (C.1)From (2.18), one easily finds3 ae aσ F | W || A | G − / (2 n )0 (0) = 3 + 2 aσ F + 1 + aσ F aσ F F (0) . (C.2)Then, we add to V F an uplifting term of the form V D ( τ, σ ) = D ( τ ) U ( τ, σ ) , (C.3)from which we define the uplifting ratio s ≡ D (0) /U (0 , σ F ) | V F (0 , σ F ) | . (C.4)In order to avoid a runaway decompactification, s should be of order one, 1 ≤ s ≤ σ ≡ σ F + δσ [13], isfound from the condition ∂V∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ = 0 ≈ ∂ V F ∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ F δσ + ∂V D ∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ , (C.5)where ∂V D ∂σ (cid:12)(cid:12)(cid:12)(cid:12) σ ≈ − V D σ F (cid:18) − δσσ F (cid:19) . (C.6)Then in the limit δσ/σ F ≪
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