Statistical Anisotropy from Vector Curvaton in D-brane Inflation
PPreprint typeset in JHEP style - HYPER VERSION
Statistical Anisotropy from Vector Curvaton in D-brane Inflation
Konstantinos Dimopoulos a ∗ , Danielle Wills b † , Ivonne Zavala b ‡ a Cosnortium for Fundamental Physics, Physics Department, LancasterUniversity, Lancaster LA1 4YB, UK. b Bethe Center for Theoretical Physics and Physikalisches Institut derUniversit¨at Bonn, Nußallee 12, D-53115 Bonn, Germany.
Abstract:
We investigate the possibility of embedding the vector curvatonparadigm in D-brane models of inflation in Type IIB string theory in a simple toymodel. The vector curvaton is identified with the U(1) gauge field that lives onthe world volume of a D3-brane, which may be stationary or undergoing generalmotion in the internal space. The dilaton is considered as a spectator field whichmodulates the evolution of the vector field. In this set up, the vector curvaton isable to generate measurable statistical anisotropy in the spectrum and bispectrumof the curvature perturbation assuming that the dilaton evolves as e − φ ∝ a where a ( t ) is the scale factor. Our work constitutes a first step towards exploring how suchdistinctive features may arise from the presence of several light fields that naturallyappear in string theory models of cosmology. Keywords:
Vector Curvaton, D-brane Inflation, Statistical Anisotropy,non-Gaussianity. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] a r X i v : . [ h e p - t h ] N ov ontents
1. Introduction 1
2. The General Set-up 7
3. Stationary brane 11 γ A > γ A ∼
4. Statistical Anisotropy 21
5. Moving brane 316. Conclusions 37A. Mass generation mechanism for U (1) field 39
1. Introduction
Ever-improving galaxy surveys have revealed to us that structure on the largestcosmological scales has an intricate net-like configuration, aptly named the Cosmic– 1 –eb. According to the current model of structure formation, the gravitationallyentangled strands that we see today have evolved from small density perturbationsthat left their imprint on the oldest source that is directly observed: the CosmicMicrowave Background (CMB). To understand how these “seeds of structure” cameto be is an important challenge for cosmology.The CMB also provides us with the strongest observational evidence for thehomogeneity and isotropy of the universe on the largest scales, as it is uniform toabout one part in a hundred thousand [1]. We can then ask the question: What sortof processes or events in the earliest cosmological moments were able to make theuniverse so uniform, and yet still implant the tiny deviations from uniformity thatcould subsequently grow into the rich galaxy systems that we observe today?The most successful framework for answering this question is the idea that theuniverse underwent a period of rapid, exponential or quasi-exponential expansionearly on in its history, which on the one hand drove all classical perturbations to zero,but on the other hand was able to amplify fluctuations of the vacuum to set up theinitial conditions for structure growth. This framework, dubbed cosmic inflation, alsoprovides compelling insight into other seemingly unrelated problems in cosmology,such as the problem of the flatness of space, the problem of superhorizon correlationsand the topological defects problem.The amplification of vacuum fluctuations is of particular interest because, whilethe other problems can be overcome or at least ameliorated by any models that areable to generate sufficiently long-lasting inflation, producing a spectrum of amplifiedfluctuations with the correct properties (as dictated by observations), necessitatesthe use of a more stringent approach. Such properties are manifest in the curva-ture perturbation ζ which is generated by the dominant fluctuations, and typicallyincludes a high degree of scale invariance, of Gaussianity and of statistical isotropyand homogeneity. As a first test, all viable models of inflation must be able to re-produce these features. However, precision measurements of the CMB have revealedthe presence of finer-grained deviations from these basic properties which appearrobust against foreground removal. For example, the latest observations favour aslightly red spectrum for ζ , while there are hints of non-Gaussian features at 1- σ .Furthermore, the low multipoles of the CMB appear to be in alignment, which mightimply the existence of a preferred direction on the microwave sky and therefore mayconstitute a violation of statistical isotropy [2]. The capacity to provide a concreteexplanation for the appearance of these deviations forms the finest sieve for modelsof inflation.Many inflation models assume that the energy density for inflating the universeand the vacuum fluctuations for seeding the galaxies must be provided by one andthe same field, making it difficult to find a candidate field with all the right traits.There is however no a priori reason to assume that this should be the case, given thatthe two jobs are rather independent. Indeed, any field that is around at the time of– 2 –he expansion may end up having its vacuum fluctuations amplified, so long as it issufficiently light and conformally non-invariant, regardless of which field is driving theexpansion. If such a field is subsequently able to affect the expansion of the Universethen its own perturbation spectrum may be imprinted in the form of the curvatureperturbation. One way for a field to do so is by dominating the energy density ofthe Universe after inflation, an idea that has been fruitfully expounded upon in thecurvaton paradigm [3]. Such a field, which has nothing to do with expanding theUniverse but instead does the job of generating the dominant contribution to ζ , isthen referred to as the curvaton field.Scalar fields, in addition to their wide use as inflatons, have also been popularlystudied as curvatons, because if such fields are able to dominate the energy density ofthe Universe to imprint their spectrum, they will do so in an inherently isotropic way.A vector field on the other hand features an opposite sign for the pressure along itslongitudinal direction relative to that along its transverse directions. Hence, if sucha field (homogenised by inflation) were able to dominate the radiation backgroundso as to imprint its own spectrum onto the curvature perturbation, it is likely toinduce substantial anisotropic stress leading to excessive large-scale anisotropy whenits pressure is non-vanishing, which is not observed. However, a massive homoge-neous vector field undergoes coherent oscillations, much like a scalar field, and it hasbeen shown that an oscillating vector field behaves like pressureless isotropic matter.Therefore, as long as the vector field begins to oscillate before its density parameterbecomes significant, it may indeed dominate the radiation background in an obser-vationally consistent way [4]. Thus, a vector field can, in principle, play the role ofthe curvaton.The process by which the fluctuations of the vacuum are amplified in an ex-panding space is known as gravitational particle production, for the appearance ofreal particles in such a background is interpreted as the creation of particles by thechanging gravitational field [5]. A second problem which arises when using a vectorfield as a curvaton is related to this particle production process. A field must besufficiently light to undergo gravitational particle production. However, a masslessvector field is conformally invariant, therefore it does not respond to the expandingbackground and its vacuum fluctuations do not become amplified. This is an oldproblem which plagued the efforts to generate a primordial magnetic field duringinflation. To overcome it many proposals introduced couplings to the vector fieldwhich break its conformality explicitly [6]. A non-zero mass also breaks the confor-mal invariance of a vector boson field and was indeed employed in models of scalarelectromagnetism to generate a primordial magnetic field [7].In Ref. [4] the consequences of including a small non-zero vector mass were ex-amined, and it was found that a vector field may indeed undergo particle productionand obtain a scale invariant superhorizon spectrum of perturbations if the mass sat-isfies m ≈ − H ∗ during inflation, where H ∗ gives the inflationary Hubble scale.– 3 –his work pioneered the vector curvaton scenario, demonstrating for the first timethat it is possible for the curvature perturbation to be affected or even generated bya vector field. The idea was implemented by coupling the vector field non-minimallyto gravity through a RA term in Refs. [8, 9]. However, this model was criticisedfor suffering from instabilities such as ghosts [10] (but see Ref. [11]).The vector curvaton scenario was further developed in Refs. [12, 13], where thesupergravity motivated case of a vector field with a varying gauge kinetic function isexplored as an alternative to demanding the afore-mentioned mass-squared condition,in order to avoid the issue of instabilities. There, it was shown that, under certainconditions, both the transverse and the longitudinal components of such a vectorfield can indeed obtain a scale invariant spectrum of superhorizon perturbations.As demonstrated in Ref. [9] the contribution of a vector field to the curvatureperturbation is in general statistically anisotropic. Thus, in the context of the vectorcurvaton mechanism, it then becomes possible to generate a statistically anisotropiccontribution to the curvature perturbation, at once providing an explanation forthe appearance of such a feature within a concrete paradigm (For a recent reviewsee Ref. [14] ). In the vector curvaton model of Refs. [12, 13], the gauge kineticfunction and mass of the vector field are modulated by a scalar degree of freedomthat varies during inflation, which may be the inflaton itself (e.g. see Ref. [16] fora model realisation in supergravity). In the case that the statistical anisotropy inthe spectrum generated by the vector field is large, the dominant contribution to ζ must come from an isotropic source, such as the scalar inflation. The vector curvatonthen affects the curvature perturbation such that it acquires a measurable degree ofstatistical anisotropy. The idea that more than one field might have a role to play in the evolution of theearly universe is highly motivated by string theory, which generically contains manyfields, even at energies far below both the string and compactification scales. Instandard inflation scenarios, the focus is often placed purely upon the behaviour ofthe candidate inflaton, while the other fields present in the set-up are consideredto be either negligible or stabilised at the minima of their respective potentials. Inthe light of the curvaton scenario, it is interesting to consider how these fields mightimpact the evolution of the universe if they themselves undergo evolution. Thesefields appear with very precise couplings in the string set-up, and in this work webegin to explore how this precise structure dictated by string theory can enrich thecosmological picture. In particular, in open string D-brane models of inflation, theinflaton is typically identified with the position coordinate of a D p -brane moving in In these works the vector field is assumed to be Abelian. For non-Abelian vector fields andtheir potential contribution to ζ see Ref. [15]. – 4 – warped throat [17]. Such a brane features a world volume two form field F MN ,associated with the open strings that end on it. The components of such a field whichcorrespond to Wilson lines have been studied as potential inflaton candidates, bothin the unwarped and warped cases in Refs. [18, 19]. Thus it is natural to investigatethe role of the other components, in particular the 4D components, of such a fieldduring the cosmological evolution, taking into account the precise way that thesecomponents couple to the various closed string modes. For example, these couplingscan lead to a St¨uckelberg mass for the vector field. St¨uckelberg masses are ubiquitousin string theory and so it is interesting to consider whether these massive vector fieldsmay give rise to cosmological signals, and one goal of the our work is to begin thatline of enquiry. In addition, scalar moduli fields usually enter in the mass and gaugekinetic function for the vector field, and these can in principle vary during inflation.Such moduli include the dilaton as well as volume moduli for the case of wrappedbranes. This indeed suggests the possibility to embed the vector curvaton paradigmin D-brane models of inflation.In what follows we consider the simplest case of vector fields on D3-branes, asa first attempt at demonstrating the vector curvaton mechanism in string theory,which may then be used as a starting point for constructing concrete cases. Vectorfields on D3-branes may couple to the four-dimensional components of the bulk two-form C which can lead to a St¨uckelberg mass for these fields. While it is true thatin compactifications of Type IIB string theory with O3/O7-planes, as opposed toO5/O9-planes, the four-dimensional components of C are projected out, we do notspecify a compactification at this point, but aim at illustrating the mechanism ratherthan to provide a full model. We note however that for realistic compactificationswith O3/O7-planes, one would need to consider D( p > very steep ratherthan very flat.In analogy to the DBI inflaton, while the U(1) vector field whose kinetic term– 5 –ppears in the DBI action is often considered to be of canonical form, keeping thegeneral Born-Infeld form may lead to new features if this field plays a role in thecosmological evolution. A single vector field is unsuitable for the role of the inflaton,however the vector curvaton paradigm has demonstrated that it can indeed affector even generate the curvature perturbation. In view of the fact that the vectorcurvaton paradigm considers, thus far, only the canonical vector field, part of thiswork is devoted to computing the equations of motion for the quantum fluctuationsof the (inherently stringy) non-canonical field.The majority of this work deals with the simpler case of a canonical vector fieldon a D3-brane, except Sections 3.1 and 3.2, where the brane in question may eitherbe stationary or undergoing general motion in the internal space. For this case,we provide a full cosmological analysis detailing how such a field may play the roleof a vector curvaton and thus give rise to measurable statistical anisotropy in thespectrum and bispectrum of the curvature perturbation. As the dilaton appears inthe St¨uckelberg mass and gauge kinetic function, we allow the possibility that itmay undergo time evolution and thus treat it as a “modulon”: a degree of freedomthat varies during inflation and modulates the mass and gauge kinetic function ofthe vector field [13]. In order for the modulated vector field to give rise to a scaleinvariant spectrum, the gauge kinetic function and the mass must obey f ∝ a − ± and m ∝ a respectively, where a is the scale factor. In the case of the D3-brane vector fieldmodulated by the dilaton, we found that these conditions are precisely met assuming e − φ ∝ a while the cosmological scales exit the horizon. While this is promising fromthe cosmology point of view, it might be challenging to realise concretely in stringtheory if the brane featuring the vector curvaton is static. This is because at tree-level the dilaton is expected to be stabilised at the energies in question. While itis plausible that the dilaton could be perturbed from its minimum, the effectivepotential would be quadratic, whereas the specified time evolution requires a linearpotential. Away from the minimum, the potential for the dilaton is exponential,which is effectively linear for small displacements, therefore the required behaviourmay be achievable if the dilaton is able to roll towards its minimum along theseregions. On the other hand, for a moving brane in a generic warped backgroundwhere the dilaton has a non-trivial profile, this will depend on the inflaton, and thuson time. In such a case, the required dependence on the scale factor might indeedbe realised, but this will involve a careful study that goes beyond the scope of thiswork. In what follows we merely assume that this is possible.We point out however that we have chosen to use the dilaton as a modulonbecause it appears generically in the gauge kinetic function and mass for the D3-brane vector field, and, strikingly, with the precise powers in f and m that make itpossible for the D3-brane vector field to generate a scale invariant spectrum: as wewill discuss in what follows, if we impose f ∝ a then we automatically arrive at m ∝ a , a non-trivial relation between the mass and gauge kinetic function that is– 6 –pecified by the vector curvaton paradigm, whereas it is not obvious at all that thisrelation should arise in string theory. Therefore, this simple picture nicely capturesthe key features of a stringy implementation of the vector curvaton paradigm, andwe may readily compute the cosmological implications of these D-brane vector fields,to see how they may affect the curvature perturbation in the universe.It should also be noted that in more complicated scenarios with branes of higherdimensionality, there may be other moduli which exhibit the appropriate powers in f and m that lead to scale invariance, and which are more attractive to use as modulonsfrom the string theory point of view. Along these lines, the problems that arise fromusing the dilaton as a modulon are useful as guiding principles for selecting suitablemoduli in realistic models.Based on the assumption that the dilaton is able to evolve accordingly, we showthat distinctive features in the curvature perturbation may arise from the intrinsicpresence of several light fields in string theory models of cosmology, and notably fromvector fields with St¨uckelberg masses. In particular, we discuss a scenario in whichthe dominant contribution to the curvature perturbation is given by the inflaton field,and the vector field can contribute measurable statistical anisotropy. Throughout ourpaper we use natural units for which c = (cid:126) = k B = 1 and Newton’s gravitationalconstant is 8 πG = M − P , with M P = 2 . × GeV being the reduced Planck mass.Sometimes we revert to geometric units where M P = 1.
2. The General Set-up
In this section we discuss the set up which will be the basis for our investigation ofthe D-brane vector curvaton with non-standard kinetic terms.We consider a warped geometry in type IIB theory [22, 23]. Thus, the tendimensional metric takes the following form (in the Einstein frame) G MN dx M dx N = h − / ( y A ) g µν dx µ dx ν + h / ( y A ) g AB dy A dy B . (2.1)Here h is the warp factor which depends on the compact coordinates y A , and g AB is the internal metric which may also depend on the compact coordinates. Thisgeometry is the result of having all types of fluxes present in the theory turned on:RR forms F n +1 = dC n for n = 0 , , n = 6 , F is self dual), as well as NSNS flux H = dB . These fluxes have only compactinternal components, therefore their duals have legs in all four infinite dimensionsplus the relevant components in the internal dimensions.We now consider a probe D p -brane (or anti-brane) embedded in this background.This has four of its dimensions lying parallel to the four infinite dimensions, and( p −
3) spatial dimensions wrapped along an internal ( p − S DBI = − µ p (cid:90) d p +1 ξ e ( p − φ (cid:113) − det( γ ab + e − φ F ab ) , (2.2)where the tension of a D p -brane in the Einstein frame is T p = µ p e ( p − φ with µ p = (2 π ) − p ( α (cid:48) ) − ( p +1) / , where φ is the dilaton, which in general depends also on y A . Furthermore, F ab = B ab +2 πα (cid:48) F ab , with B the pullback of the NSNS background 2-form field on the brane, F is the world volume gauge field we are interested in, and γ ab = G MN ∂ a x M ∂ b x N is thepullback of the ten-dimensional metric on the brane in the Einstein frame. Finally, α (cid:48) = (cid:96) s is the string scale and ξ a are the brane world-volume coordinates. Theindices M, N = 0 , , ..., a, b = 0 , , ..., p ; µ, ν = 0 , , ..., A, B = 4 , ..., ∂ a x M , so longas these are themselves slowly varying in spacetime; that is, for small accelerationscompared to the string scale (equivalently, for small extrinsic curvatures of the braneworldvolume). In addition, the string coupling should be small in order for theperturbative expansion to hold, i.e. g s (cid:28)
1, where g s = e φ .The Wess-Zumino action describing the coupling of the D p -brane to the RR formfields is given by S WZ = q µ p (cid:90) W p +1 (cid:88) n C n ∧ e F , (2.3)where C n are the pullbacks of the background RR C n forms present, F = B + 2 πα (cid:48) F as before, and the wedge product singles out the relevant terms in the exponential.Furthermore W p +1 is the world volume of the brane and q = 1 for a probe D p -brane,while q = − p -antibrane.We now discuss these two actions in detail for our case of interest, a D3-brane. Let us consider a probe D3-brane positioned such that its three axes are alignedwith the axes of the three extended spatial dimensions. We consider the brane to be In the string frame, the DBI action is given by S DBI = − µ p (cid:90) d p +1 ξ e − φ (cid:112) − det( γ ab + F ab ) . In D dimensions, the Einstein and string frames are related by G EMN = e − D − φ G sMN . Notice that for a D3-brane, T p = µ p in the Einstein frame. – 8 –oving in the internal space and so the internal coordinates become functions of theworld volume coordinates, y A = y A ( ξ µ ). In typical single-field inflation scenarios withD3-branes/D3-antibranes, the inflaton field is identified with the position coordinateof the D3-brane moving radially in the potential of the antibrane. Taking the pullbackof the NSNS 2-form field B to vanish, the DBI action for a D3-brane is given by S DBI = − T (cid:90) d x (cid:113) − det( γ µν + e − φ/ F µν ) , (2.4)where γ µν = h ( r ) − / g µν + h ( r ) / ∂ µ y A ∂ ν y B g AB , F µν = 2 πα (cid:48) F µν , (2.5)with F µν = ∂ µ A ν − ∂ ν A µ , and the warp factor depends only on the radial direction r . We expand the determinant as follows: − det[ h − / g µν + h / ∂ µ y A ∂ ν y A +2 πα (cid:48) e − φ/ F µν ] = − h − det[ g µβ ]det[ δ βν + h∂ β y A ∂ ν y A + lF βν ] , (2.6)with l = h / πα (cid:48) e − φ/ . The DBI action in Eq. (2.4) then becomes S DBI = − T (cid:82) d xh − √− g (cid:113) det( δ βν + h∂ β y A ∂ ν y A + lF βν ) . (2.7)We now proceed to calculate the determinant:det[ δ βν + h∂ β y A ∂ ν y A + lF βν ] = 1 + l F αβ F αβ − l F αβ F βγ F γδ F δα + l F αβ F αβ F γδ F γδ + hy αA y Aα + h y [ αA y Aα y β ] B y Bβ + h y [ αA y Aα y βB y Bβ y γ ] C y Cγ + h y [ αA y Aα y βB y Bβ y γC y Cγ y δ ] D y Dδ + 3 hl y [ αA y Aα F βγ F γ ] β + 3 h ly [ αA y Aα y γB y Bβ F β ] γ + 4 h ly [ αA y Aα y βB y Bβ y γC y Cδ F δ ] γ + 4 hl y [ αA y Aα F βγ F γδ F δ ] β + 6 h l y [ αA y Aα y βB y Bβ F γδ F δ ] γ , (2.8)where y αA ≡ ∂ α y A and the antisymmetrisation takes place over the Greek indices only. The general Wess Zumino (WZ) action for a D3-brane is given by (see Eq. (2.3)) S WZ = qµ (cid:90) W (cid:18) C + C ∧ (2 πα (cid:48) ) F + C (2 πα (cid:48) ) F ∧ F (cid:19) , (2.9)where we recall that the C n are the pullbacks of the background C n forms present inthe flux background, W is the world volume, and q gives the charge of the brane( q = +1 for a brane and q = − C = √− g h − dx ∧ dx ∧ dx ∧ dx andthus this term is essentially given by the warp factor. The last term in Eq. (2.9)is the coupling of the axion field C to the vector field we are interested in. Inthe present case where the C axion has been stabilised, this is just a topologicalterm. Finally, the second term which couples the vector field to the two-form, is anon-trivial term which is responsible for generating a mass for the U (1) field via theSt¨uckelberg Mechanism, which is a standard mass generation mechanism in stringtheory according to which non-anomalous U (1) vector fields may acquire masses .The details of how such a mass is generated are given in appendix A. We may now write down the complete expression for the three fields we are consid-ering. This includes the total action for the gauge field A µ living on the D3-brane(containing the DBI and WZ pieces), as well as the actions for the position coordinate r and the dilaton field φ . Using the results in appendix A and considering the braneto be moving in the radial direction only (generalisation to multi-field scenarios willbe commented upon later), the final D3-brane action, in terms of the canonicallynormalised vector field A µ = A µ / ˜ g with ˜ g = T − (2 πα (cid:48) ) − (see appendix A), is givenby S D3 = − (cid:90) d x √− g (cid:32) h − √ Λ − qh − + q m A µ A µ − q C (cid:15) µνλβ F µν F λβ (cid:33) , (2.10)whereΛ = 1 + he − φ F αβ F αβ + h e − φ (cid:0) F αβ F αβ F γδ F γδ − F αβ F βγ F γδ F δα (cid:1) + h ∂ α ϕ∂ α ϕ +3 h l (cid:0) ∂ α ϕ∂ α ϕ F βγ F βγ − ∂ α ϕ F αβ ∂ γ ϕ F γβ (cid:1) . In this expression we have introduced the canonically normalised (fixed) position fielddefined by ϕ = √ T r associated to the (radial) coordinate brane position r , and the Were the axion not stabilised, the axial term might have been the source of parity violatingstatistical anisotropy as explored in Ref. [24]. As mentioned in the introduction, in a realistic type IIB flux compactification with O3/O7planes, the four dimensional components of the RR-form C are projected out. However, in whatfollows we stick with the St¨uckelberg mass mechanism, which is a ubiquitous mechanism in stringtheory and thus our work can readily be generalised to realistic models. In a concrete scenario, onecould for example consider wrapped D5 or D7-branes in place of D3-branes. In such case there isa four dimensional 2-form that is not projected out by the orientifold action, and which couples tothe vector field appropriately such that a mass is generated via the St¨uckelberg mechanism. Thusfor an explicit realisation, one would have to consider branes of different dimensionality. On theother hand, one may stick with D3-branes but consider a more standard mass mechanism for suchbranes, such as the Higgs mechanism. – 10 –orresponding warp factor is defined as h ( ϕ ) = T − h ( r ). The dilaton dependentmass is given in string units by m = e − φ (2 π ) M s V , (2.11)where the dimensionless (warped) 6D volume is defined as V = V M s (see ap-pendix A). Furthermore (cid:15) µναβ is the Levi-Civita tensor, such that (cid:15) = √− g .Coupling this action to four dimensional gravity, and including the necessaryterms for an evolving dilaton as well as the potential for the brane’s position, whichwill arise due to various effects such as fluxes and presence of other objects, we canwrite S = M P (cid:90) d x √− g (cid:20) R − ∂ µ φ ∂ µ φ − V ( φ ) (cid:21) − (cid:90) d x √− g (cid:20) h − √ Λ + V ( ϕ ) − qh − + q m A µ A µ (cid:21) , (2.12)where M P = 2 V / [(2 π ) α (cid:48) ] is the Planck mass.
3. Stationary brane
We are now ready to study the cosmological implications of the U (1) gauge field whichlives on a D3-brane world volume. We start by considering the case in which thebrane whose world volume hosts the vector field of interest is stationary in the internalspace. Inflation is considered to be driven by a different D3-brane or any other work-ing inflationary mechanism. Therefore the “curvaton brane” is just a D3-brane thatmay be present in the bulk at the time of inflation, for which V ( ϕ ) = constant, ˙ ϕ =0.In what follows we look at two possibilities for the dilaton: either it is fixed duringinflation or it is able to evolve.For a stationary D3-brane, the action in Eq. (2.12) then simplifies to S = M P (cid:90) d x √− g (cid:20) R − ∂ µ φ ∂ µ φ − V ( φ ) (cid:21) + (cid:90) d x √− g (cid:40) h − (cid:34) q − (cid:114) hf ( φ )2 F αβ F αβ + h f ( φ )8 ( F αβ F αβ F γδ F γδ − F αβ F βγ F γδ F δα ) (cid:35) − q m ( φ )2 A µ A µ (cid:41) , (3.1) We drop the topological term at this stage since C is stabilised and therefore we may use thefact that dA ∧ dA = d(A ∧ dA), hence this term constitutes a total derivative and thus will not giveany effect in the cosmological evolution. – 11 –here we have defined f ( φ ) = e − φ and m ( φ ) = e − φ ˜ m . (3.2)We now focus on the equations of motion for the vector field, derived from the actionabove. These are given by (from now on we take q = 1) G µν = γ A (cid:18) f F µβ F βν + hf F µκ F κν F − hf F σν F σδ F δκ F κµ (cid:19) + m e φ A µ A ν + g µν h − (cid:18) − γ − A − m A µ A µ (cid:19) , (3.3) √− g m A ν = ∂ µ (cid:20) √− g γ A (cid:18) f F µν − hf M νµ + hf N µν (cid:19)(cid:21) , (3.4)where we have defined: γ − A ≡ (cid:114) hf F + h f F − F αβ F βγ F γδ F δα ) , (3.5) M νµ = F νβ F βγ F γµ , (3.6) N µν = F µν F αβ F αβ (3.7)and G µν = R µν − g µν R is the Einstein tensor. In a FRW universe, the four dimensionalmetric takes the usual form ds = − dt + a ( t ) δ ij dx i dx j , (3.8)where a ( t ) is the scale factor. Moreover, we can expect inflation to homogenise thevector field, and therefore for the background solution ∂ i A µ = 0 . (3.9)Using this condition, one can check that the factor γ A associated with the vectorfield is given simply by γ A = 1 (cid:113) hf F = 1 (cid:113) − hf a − ˙ A · ˙ A . (3.10)Moreover, the ν = t component of the vector field equation implies that A t = 0 , (3.11)and we thus have A µ = (0 , A ( t )). Using this, the ν = i component of the equationof motion becomes γ A ¨ A + γ A ˙ A (cid:32) H + ˙ γ A γ A + ˙ ff (cid:33) + m f A = 0 , (3.12)– 12 – A ... γ A AN N
Figure 1:
The qualitative behaviour for the vector background with an evolving dilaton.The solid lines correspond to initial conditions on the vector field such that γ A ini ∼
22 andthe dashed lines to γ A ini ∼ where H ≡ ˙ aa is the Hubble parameter and we have used the fact that the M and N terms cancel each other in the background solution.From the form of Eq. (3.12) we see that the effective mass of the vector M ≡ m √ f isconstant and given by M = √ π (2 π ) M P V . (3.13)As we demonstrate below, the desired behaviour of our system is attained when f ∝ a . Solving Eqs. (3.12) and (3.10) in the case that M (cid:28) H and e − φ ∝ a ( t ), weobtain the behaviour of the background A µ and γ A during the inflationary period.The results are plotted in Fig. 1, in which we indicate the qualitative behaviourof the vector background and its time derivatives as well as that of γ A . The solidlines correspond to initial conditions on the vector field such that γ A ini ∼
22 andthe dashed lines to γ A ini ∼ N . We see that the background soon freezes out at constant amplitudeduring inflation, while γ A converges very quickly to 1. This is in agreement with thefindings of Ref. [13], when f ∝ a . We now calculate the perturbations of the vector field during the cosmological evo-lution to see what sort of new terms arise in the most general case of varying dilatonand non-standard vector kinetic terms.We perturb the vector field around the homogeneous value A µ ( t ) as follows: A µ ( t, x ) = A µ ( t ) + δ A µ ( t, x ) ⇒A t ( t, x ) = δ A t ( t, x ) & A ( t, x ) = A ( t ) + δ A ( t, x ) . (3.14)– 13 –hen, the equation of motion for the temporal component ν = t becomes a m δ A t + f γ A (cid:104) ∇ · δ ˙ A − ∇ δ A t (cid:105) + hf a − γ A ˙ A · ∇ (cid:104) ˙ A · δ ˙ A − ˙ A · ∇ ( δ A t ) (cid:105) = 0 . (3.15)The spatial component takes a more complicated form. Combining it with theintegrability condition: ∂ ν (3.4), gives: − am δ A − af γ A (cid:32) δ ¨ A + (cid:34) H + ˙ γ A γ A + ˙ ff (cid:35) δ ˙ A − a − ∇ δ A − (cid:34) ˙ γ A γ A + ˙ ff − H − mm (cid:35) ∇ δ A t (cid:33) − hf a − γ A ˙ A ˙ A · (cid:32) δ ¨ A + (cid:34) − H + 3 ˙ γ A γ A + 2 ˙ ff (cid:35) δ ˙ A (cid:33) − hf a − γ A (cid:110) ˙ A ( ¨ A · δ ˙ A ) + ¨ A ( ˙ A · δ ˙ A ) − ˙ A [ ¨ A · ∇ ( δ A t )] − ¨ A [ ˙ A · ∇ ( δ A t )] (cid:111) + hf a − γ A ˙ A ( ˙ A · ∇ ) (cid:32)(cid:34) γ A γ A − H + 2 ˙ ff − mm (cid:35) δ A t + a − ∇ · δ A (cid:33) + hf γ A a − (cid:110) − ˙ A ( ˙ A · ∇ )( ∇ · δ A ) + ˙ A ∇ ( ˙ A · δ A ) − ˙ A · ∇ [ ∇ ( ˙ A · δ A )]+( ˙ A · ∇ )( ˙ A · ∇ ) δ A − ( ˙ A · ˙ A ) ∇ δ A + ( ˙ A · ˙ A ) ∇ ( ∇ · δ A ) (cid:111) = 0 . (3.16)We now pass to momentum space by performing a Fourier expansion of theperturbations as follows: δ A µ ( t, x ) = (cid:90) d k (2 π ) / δ A µ ( t, k ) exp ( i k · x ) . (3.17)Plugging this into (3.15) and (3.16), we can write the equations of motion forthe transverse and longitudinal components as follows. Making the definitions: δ A || ≡ k ( k · δ A ) k , δ A ⊥ ≡ δ A − δ A || ,k ≡ k · k , γ A = 1 (cid:114) − hf (cid:16) ˙ A a (cid:17) , Q = hf γ A a (cid:34) ˙ A ( ¨ A · k ) + ¨ A ( ˙ A · k ) + ˙ A ( ˙ A · k ) (cid:32) γ A γ A + 2 ˙ ff − H − mm (cid:33)(cid:35) + k (cid:32) ˙ γ A γ A + ˙ ff − H − ˙ mm (cid:33) ,R = k + ( am ) γ A f + hf γ A a ( ˙ A · k ) , – 14 –he transverse component becomes ∂ t + (cid:32) H + ˙ γ A γ A + ˙ ff (cid:33) ∂ t + m γ A f + (cid:18) ka (cid:19) + hfa (cid:32) ˙ A · k a (cid:33) − ˙ A (cid:18) ka (cid:19) δ A ⊥ + hf (cid:16) γ A a (cid:17) (cid:40) ¨ A + (cid:32) γ A γ A + 2 ˙ ff − H (cid:33) ˙ A + Q R ( ˙ A · k ) (cid:41) ˙ A · δ ˙ A ⊥ + hfa (cid:104) ˙ A k − ( ˙ A · k ) k (cid:105) ˙ A · δ A ⊥ + hf (cid:16) γ A a (cid:17) (cid:110) ˙ A ( ˙ A · δ ¨ A ⊥ ) + ˙ A ( ¨ A · δ ˙ A ⊥ ) (cid:111) = 0 , (3.18)while the longitudinal component is: ∂ t + (cid:34) H + ˙ γ A γ A + ˙ ff + 1 R (cid:32) H + 2 ˙ mm − ˙ γ A γ A − ˙ ff (cid:33) k (cid:35) ∂ t + m γ A f + (cid:18) ka (cid:19) + hfa (cid:32) ˙ A · k a (cid:33) δ A || + hfa (cid:34) γ A (cid:18) ka (cid:19) ˙ A − ( ˙ A · k ) k a (cid:35) ˙ A · δ A || + hf (cid:16) γ A a (cid:17) (cid:26)(cid:20) − k R (cid:21) ¨ A − Q R ( ˙ A · k )+ (cid:34) γ A γ A + 2 ˙ ff − H − (cid:32) γ A γ A + 2 ˙ ff − H − mm (cid:33) k R (cid:35) ˙ A (cid:41) ˙ A · δ ˙ A || + hf (cid:16) γ A a (cid:17) (cid:26) ˙ A ( ˙ A · δ ¨ A || ) + (cid:20) − k R (cid:21) ˙ A ( ¨ A · δ ˙ A || ) (cid:27) = 0 . (3.19)At this point we can compare these equations to the standard kinetic term case[12, 13]. In fact, the first lines in equations (3.18) and (3.19) have the same formas the standard kinetic term case for the vector field with f F . However in thepresent case, we obtain extra terms coming from the vector ‘Lorentz’ factor γ A . Thisfactor acts in a way analogous to the gauge kinetic function, but it is independentof the dilaton field. Moreover the Lorentz factor also adds second order terms in thebackground, coming from M , N in the equations of motion [cf. Eqs. (3.6) and (3.7)].These terms cannot be neglected once we consider γ A > .2 Non standard kinetic term, γ A > A = A || + A ⊥ . (3.20)For the perturbations δ A = δ A || + δ A ⊥ we define δ A ⊥ ≡ δ A ⊥ || + δ A ⊥ ⊥ (3.21)such that δ A ⊥ || · A ⊥ = δ A ⊥ || A ⊥ δ A ⊥ ⊥ · A ⊥ = 0 . In this way, δ A ⊥ || and δ A ⊥ ⊥ are the modes of δ A ⊥ that are parallel and perpendic-ular to A ⊥ respectively.Now taking k · (Eq. (3.18)) and using the decomposition in Eq. (3.21) we obtainan equation for δ A ⊥ || , δ ¨ A ⊥ || = − (cid:34) ¨ A ⊥ ˙ A ⊥ + ¨ A || ˙ A || + 3 ˙ γ A γ A + 2 ˙ ff − H + Q || R k (cid:35) δ ˙ A ⊥ || . (3.22)where Q || = k · Q . So we see that this mode is not oscillating on any scales andtherefore it will not give rise to particle production.To obtain an equation for δ A ⊥ ⊥ , we now take δ A ⊥ ⊥ · (Eq. (3.18)) which yields δ ¨ A ⊥ ⊥ + (cid:32) H + ˙ γ A γ A + ˙ ff (cid:33) δ ˙ A ⊥ ⊥ + (cid:40) m γ A f + (cid:18) ka (cid:19) (cid:20) − hfa (cid:16) ( ˙ A ⊥ ) + 2 ˙ A ⊥ ˙ A || (cid:17)(cid:21)(cid:41) δ A ⊥ ⊥ = 0 . (3.23)Taking the small-scale limit of Eq. (3.23), we see that we have particle production atearly times and the Bunch-Davies vacuum is well-defined, as soon as the mass termbecomes negligible in the UV limit. However the sound speed is reduced comparedto the canonical oscillator. This equation does not give rise to instabilities because γ − A is always positive.Now we turn to the equation for the longitudinal component δ A || . Taking k · (Eq. (3.19)) and using the decomposition in Eq. (3.21) we obtain– 16 – ¨ A || + (cid:110) H + ˙ γ A γ A + ˙ ff + k R (cid:16) H + 2 ˙ mm − ˙ γ A γ A − ˙ ff (cid:17) + 2 hf (cid:0) γ A a (cid:1) (cid:16) − k R (cid:17) ¨ A || ˙ A || + hf (cid:0) γ A a (cid:1) ( ˙ A || ) (cid:104) − Q || R + 3 ˙ γ A γ A + ˙ ff − H − k R (cid:16) ˙ γ A A + ˙ ff (cid:17)(cid:105)(cid:111) δ ˙ A || B + (cid:104) m Bγ A f + (cid:0) ka (cid:1) (cid:105) δ A || = 0 , (3.24)where B = 1 + hf (cid:16) γ A a (cid:17) ( ˙ A || ) . (3.25)Taking the small-scale limit we obtain a canonical oscillator equation, δ ¨ A || + (cid:18) ka (cid:19) δ A || = 0 , (3.26)therefore the Bunch-Davies vacuum is well-defined and we once again have particleproduction at early times. Therefore we see that particle production can take placefor two of the three modes only, namely in the δ A ⊥ ⊥ and δ A || directions, and henceis inherently anisotropic. After inflation, both of these modes will oscillate due tothe mass term in Eqs. (3.23) and (3.24) and could affect the curvature perturbationvia the vector curvaton mechanism. Note in particular that the mass terms aresuppressed by γ A and therefore these modes are indeed effectively massless at earlytimes while γ A is large.Taking the late-time limit of these equations, i.e. γ A → A → const., weobtain δ ¨ A ⊥ ⊥ + (cid:32) H + ˙ ff (cid:33) δ ˙ A ⊥ ⊥ + (cid:34) m f + (cid:18) ka (cid:19) (cid:35) δ A ⊥ ⊥ = 0 , (3.27)and δ ¨ A || + (cid:34) H + ˙ ff + (cid:0) ka (cid:1) (cid:0) ka (cid:1) + m f (cid:32) H + 2 ˙ mm − ˙ ff (cid:33)(cid:35) δ ˙ A || = 0 . (3.28)As we will see in the following section, these equations give rise to exactly scale-invariant superhorizon spectra as long as f ∝ a while the cosmological scales exitthe horizon. However, as a consequence of the intermediate regimes in Eqs. (3.23)and (3.24), it is not clear that the resultant spectra will be scale-invariant. Due tothe fact that the regime in which γ A (cid:54) = 1 is short relative to the regime in which γ A = 1, this could add some scale-dependence to the spectrum.We can get further inside into the behaviour of the perturbations by looking atthe their superhorizon limits. In order to see this cleare, we pass to conformal time– 17 –nd make some useful definitions: (cid:90) dta ( t ) = τ, (cid:48) = d/dτ, H = a (cid:48) a , α = 1 √ f γ A , z = α − , (3.29) δ A = δ W / (cid:112) f γ A , (3.30) R = k (cid:20) am ) k γ A f + hf γ A k a ( A (cid:48) · k ) (cid:21) = k (1 + r ) , (3.31)¯ Q = hf γ A a (cid:20) A (cid:48) ( A (cid:48)(cid:48) · k ) + A (cid:48)(cid:48) ( A (cid:48) · k ) + A (cid:48) ( A (cid:48) · k ) (cid:18) γ (cid:48) A γ A + 2 f (cid:48) f − H − m (cid:48) m (cid:19)(cid:21) + k (cid:18) γ (cid:48) A γ A + f (cid:48) f − H − m (cid:48) m (cid:19) . (3.32)Using this information, the new equations for the perturbations take the form: δ W (cid:48)(cid:48) ⊥ + (cid:20) m a γ A f + k (cid:18) − hfa ( A (cid:48) ) sin θ A k (cid:19) − z (cid:48)(cid:48) z (cid:21) δ W ⊥ + hfa k A (cid:48) ⊥ (cid:16) A (cid:48) ⊥ · δ W ⊥ (cid:17) + hf γ A a (cid:26)(cid:20) A (cid:48)(cid:48) + (cid:18) γ (cid:48) A γ A + 2 f (cid:48) f − H + 2 α (cid:48) α (cid:19) A (cid:48) + ¯Q R ( A (cid:48) · k ) (cid:21) A (cid:48) · δ W (cid:48) ⊥ + A (cid:48) (cid:20) α (cid:48) α A (cid:48)(cid:48) · δ W ⊥ + A (cid:48)(cid:48) · δ W (cid:48) ⊥ + A (cid:48) · δ W (cid:48)(cid:48) ⊥ (cid:21) + (cid:20) α (cid:48) α A (cid:48)(cid:48) + (cid:18) α (cid:48)(cid:48) α + α (cid:48) α (cid:18) γ (cid:48) A γ A + 2 f (cid:48) f − H (cid:19)(cid:19) A (cid:48) + ¯ Q R α (cid:48) α ( A (cid:48) · k ) (cid:21) A (cid:48) · δ W ⊥ (cid:27) = 0 , (3.33)while the longitudinal component is: – 18 – W (cid:48)(cid:48) || + 21 + r (cid:18) H − m (cid:48) m + α (cid:48) α (cid:19) δ W (cid:48) || + hf k a (cid:16) γ A A (cid:48) − A (cid:48) || (cid:17) A (cid:48) · δ W || + (cid:20)
21 + r (cid:18) H − m (cid:48) m + α (cid:48) α (cid:19) α (cid:48) α − z (cid:48)(cid:48) z + m a γ A f + k (cid:18) hfa ( A (cid:48) ) cos θ A k (cid:19)(cid:21) δ W || + hf γ A a (cid:40)(cid:34)(cid:18) r r (cid:19) A (cid:48)(cid:48) + (cid:18) γ (cid:48) A γ A + 2 f (cid:48) f − H + 2 α (cid:48) α −
11 + r (cid:18) γ (cid:48) A γ A + 2 f (cid:48) f − H − m (cid:48) m (cid:19)(cid:19) A (cid:48) − ¯Q R ( A (cid:48) · k ) (cid:35) A (cid:48) · δ W (cid:48) || + A (cid:48) (cid:34) A (cid:48) · δ W (cid:48)(cid:48) || + A (cid:48)(cid:48) · δ W (cid:48) || (cid:18) −
11 + r (cid:19) (cid:35) + (cid:34)(cid:18) −
11 + r (cid:19) α (cid:48) α A (cid:48)(cid:48) + (cid:18) α (cid:48)(cid:48) α + α (cid:48) α (cid:20) γ (cid:48) A γ A + 2 f (cid:48) f − H −
11 + r (cid:20) γ (cid:48) A γ A + 2 f (cid:48) f − H − m (cid:48) m (cid:21)(cid:21)(cid:19) A (cid:48) − ¯Q R ( A (cid:48) · k ) α (cid:48) α (cid:35) A (cid:48) · δ W || + α (cid:48) α (cid:18) −
11 + r (cid:19) A (cid:48) (cid:16) A (cid:48)(cid:48) · δ W || (cid:17) (cid:35)(cid:41) = 0 . (3.34)Taking the limit k/aH → γ A ∼ γ A ∼
1. We begin by outlining theconditions which should be placed on the evolution of the dilaton in order to achievea scale-invariant spectrum of vector perturbations. It was clearly demonstrated inRef. [13] that, for a light vector field evolving during quasi-de Sitter expansion witha gauge kinetic function f ∝ a α and mass m ∝ a β , the power spectrum of vectorperturbations is exactly scale invariant when α = − ± β = 1, as long asthe vector field remains light when the cosmological scales exit the horizon. In thecase that f ∝ a − (and m ∝ a ), the power spectra for the transverse and longitudi-nal components of the vector perturbations can become roughly equal if the vector– 19 –eld becomes heavy by the end of inflation. This allows for (approximately) isotropicparticle production and entails that the vector field can provide the dominant contri-bution to the curvature perturbation, which is known to be predominantly isotropic.In the other case, i.e. f ∝ a (and m ∝ a ), the power spectrum for the longitudinalcomponent of the vector perturbations is much larger than that of the transversecomponent, therefore particle production is strongly anisotropic and the dominantcontribution to the curvature perturbation must come from some other, isotropicsource, e.g. a scalar field. However, the important point is that, in this case, thevector field can contribute measurable statistical anisotropy.In our scenario, f ∝ e − φ and m ∝ e − φ/ = √ f [cf. Eqs. (2.11)], such that ifwe demand m ∝ a we must have f ∝ a . This suggests an explicit realisation ofthe latter of the above two possibilities, which leads to scale invariance of the vectorperturbations as shown in Ref. [13], as long the D-brane vector field is light andevolving during quasi-de Sitter expansion, and e − φ ∝ a .The equations of motion for the vector perturbations in this case simplify to( am ) f δ A − ∇ δ A + ∇· δ ˙ A = 0 , (3.35) m f δ A + Hδ ˙ A + δ ¨ A + (cid:16) δ ˙ A − ∇ δ A (cid:17) ˙ ff − a − ∇ δ A + 2 (cid:18) H + ˙ mm (cid:19) ∇ δ A = 0 . (3.36)Moving to Fourier space using Eq. (3.17) as before, the equations for the pertur-bations (3.35) become δ A + i∂ t ( k · δ A ) (cid:104) k + ( am ) f (cid:105) = 0 , (3.37) m f δ A + (cid:32) H + ˙ ff (cid:33) δ ˙ A + δ ¨ A + (cid:18) ka (cid:19) δ A + (cid:32) H + 2 ˙ mm − ˙ ff (cid:33) ∂ t ( k · δ A ) (cid:104) k + ( am ) f (cid:105) = 0 . (3.38)We can now compute the equations for the longitudinal and transverse compo-nents as before. Using the relations: δ A || ≡ k ( k · δ A ) k , δ A ⊥ ≡ δ A − δ A || (3.39) This is so even for the case in which f ∝ a − (and m ∝ a ) if the vector field remains light untilthe end of inflation [13]. – 20 –he equations become: (cid:34) m f + (cid:32) H + ˙ ff (cid:33) ∂ t + ∂ t + (cid:18) ka (cid:19) (cid:35) δ A ⊥ = 0 (3.40) (cid:34) m f + (cid:32) H + ˙ ff (cid:33) ∂ t + (cid:32) H + 2 ˙ mm − ˙ ff (cid:33) (cid:0) ka (cid:1) ∂ t (cid:0) ka (cid:1) + m f + ∂ t + (cid:18) ka (cid:19) (cid:35) δ A || = 0 , (3.41)which are identical to those obtained in Ref. [12].At this point we define the spatial components of the canonically normalised,physical (as opposed to comoving) vector field as follows: W ≡ (cid:112) f A /a . (3.42)Moving to Fourier space, we expand the perturbations of the physical vector field, δ W , as δ W ( t, x ) = (cid:90) d k (2 π ) / δ W ( t, k ) e i k · x . (3.43)The field equations for the spatial vector perturbations (3.40) then become (cid:34) ∂ t + 3 H∂ t + M + (cid:18) ka (cid:19) (cid:35) δ W ⊥ = 0 , (3.44)and (cid:40) ∂ t + (cid:34) H + 2 H ( ka ) (cid:0) ka (cid:1) + M (cid:35) ∂ t + M + (cid:18) ka (cid:19) (cid:41) δ W (cid:107) = 0 , (3.45)which are the same as those which were found in Ref. [13]. Thus in what follows weapply the results in [13] to our present set up.
4. Statistical Anisotropy
We have now all the necessary ingredients to study possible cosmological implicationsof the D-brane set up with varying dilaton discussed in the previous section. We startby reviewing the relevant results in [13], which we then apply to our scenario.Historically, statistical anisotropy due to the contribution of vector field pertur-bations to the curvature perturbation was first studied in the context of the inho-mogeneous end of inflation mechanism [25] (see also Ref. [26]). However, the firstcomprehensive mechanism independent study of the effect of vector field perturba-tions onto ζ and the resulting statistical anisotropy in the spectrum and bispectrum– 21 –s presented in Ref. [9]. In the present work, we focus on the vector curvaton mecha-nism, which has the advantage of not being restrictive on the inflation model as wellas keeping the inflaton and vector curvaton sectors independent (they can correspondto two different branes for example) . Gravitational particle production of a vector field proceeds analogously to that of ascalar field: vacuum fluctuations are exponentially stretched by the expansion andapproach classicality on superhorizon scales, imprinting their spectrum on the homo-geneous background. As a realisation of Hawking radiation in a de Sitter background,it can be shown that this process gives rise to the appearance of real particles, whichare then interpreted as having been created by the changing gravitational field [5].In our scenario studying particle production for the vector boson field amounts tosolving Eqs. (3.44) and (3.45) for the mode functions of the vector field perturbations.This has been done in Ref. [13], where the following power spectra were obtained: P L,R = (cid:18) H π (cid:19) . (4.1)for the transverse components and P || = 9 (cid:18) HM (cid:19) (cid:18) H π (cid:19) . (4.2)for the longitudinal component. In our present case M is constant and is given byEq. (3.13). It is evident that both power spectra above are scale-invariant. Com-paring the results for the power spectra of the transverse and longitudinal modefunctions, we see that, for M (cid:28) H , P || (cid:29) P L,R . (4.3)The above results are central to the claim that a vector field modulated by an evolvingscalar field can contribute measurable statistical anisotropy to the curvature pertur-bation. As we discussed in the previous section, we consider the dilaton as the fieldwhich plays the role of a modulating field. We assume that the curvature perturbation ζ receives contributions from both thescalar inflaton field as well as the vector field. In terms of its isotropic and anisotropiccontributions, the power spectrum of ζ may be written as [29, 9] P ζ ( k ) = P iso ζ ( k )[1 + g ( ˆ N A · ˆ k ) ] , (4.4) Indirectly, statistical anisotropy in ζ can also be generated by considering a mild anisotropisa-tion of the inflationary expansion, due to the presence of a vector boson field condensate. In thiscase, it is the perturbations of the inflaton scalar field which are rendered statistically anisotropic[27] (for a recent review see Ref. [28]). – 22 –here P iso ζ ( k ) is the dominant isotropic contribution, ˆ k ≡ k /k and ˆ N A ≡ N A /N A are unit vectors in the directions k and N A respectively, and g quantifies the degreeof statistical anisotropy in the spectrum. The components of N A are given by N iA ≡ ∂N/∂W i , where W i are the spatial components of the physical vector field(cf. Eq. (3.42)), N A ≡ | N A | and N gives the number of the remaining e-foldingsof inflation. The degree of statistical anisotropy in the spectrum may be defined interms of the various power spectra that contribute to P ζ as follows [9] g ≡ N A P || − P L,R P iso ζ . (4.5)It is clear that, for the case at hand in which P || (cid:29) P L,R , the degree of statisticalanisotropy can be non-negligible. The CMB data provide no more than a weak upperbound on g , which allows statistical anisotropy as much as 30% [30]. The forthcomingobservations of the Planck satellite will reduce this bound down to 2% if statisticalanisotropy is not observed [31]. This means that, for statistical anisotropy to beobservable in the near future, g must lie in the range0 . ≤ g ≤ . . (4.6)Thus, the spectrum is predominantly isotropic but not to a large degree. To avoidgenerating an amount of statistical anisotropy that is inconsistent with the obser-vational bounds, we then require that the dominant contribution to the curvatureperturbation comes from the scalar inflaton.The vector curvaton contribution to the curvature perturbation is [9, 14] ζ A = 23 ˆΩ A W i δW i W , (4.7)where ˆΩ A ≡ A − Ω A (cid:39) Ω A , with Ω A ≡ ρ A /ρ being the density parameter of the vectorfield, where ρ A and ρ are the densities of the vector field and of the Universe respec-tively. Therefore, in this case, we should have Ω A (cid:28) ζ only serves to imprint statistical anisotropy at a measurable level.Let us attempt to quantify this contribution in the case of our D-brane vectorcurvaton scenario. In Ref. [9] (see also Ref. [14]) it was shown that, for the vectorcurvaton, when Ω A (cid:28) N A (cid:39)
12 Ω A W , (4.8)where W = | W | is the modulus of the physical vector field. In view of the above,Eq. (4.5) can be recast as √ g ∼ Ω A ζ δWW , (4.9)– 23 –here we have used that P iso ζ ≈ ζ , since the curvature perturbation is predominantlyisotropic. We also used that δW ∼ (cid:112) P (cid:107) ∼ H ∗ /M , (4.10)since P (cid:107) (cid:29) P L,R and we have employed Eq. (4.2), with H ∗ denoting the Hubble scaleduring inflation.The contribution of the vector field to ζ is finalised at the time of decay of thevector curvaton, since until then it is evolving with time. Thus, we need to evaluatethe above at the time of the vector curvaton decay, which we denote by ‘dec’. InRef. [13] it was shown thatΩ dec A ∼ Ω end A (cid:18) ΓΓ A (cid:19) / min (cid:26) , MH ∗ (cid:27) / min (cid:26) , M Γ (cid:27) − / , (4.11)where Γ and Γ A denote the decay rates of the inflaton and the vector curvaton fieldsrespectively, and ‘end’ denotes the end of inflation.During inflation, as long as the dilaton is varying and f ∝ a and m ∝ a , thevector curvaton remains frozen with W = W ∗ = (cid:112) f A /a ∝ A (cid:39) constant , (4.12)where we used Eq. (3.42). However, the dilaton is not expected to roll throughoutthe remaining 50 or so e-foldings of inflation, after the cosmological scales exit thehorizon. Instead, being a spectator field, it will most probably stop rolling N x e-foldings before the end of inflation. After this the modulation of f and m ceases andwe have f →
1. While the mass of the physical vector curvaton M = m/ √ f remainsconstant, this is not so for W . Indeed, taking f = 1, it is easy to see that A remainsfrozen. Thus, in view of Eq. (3.42), we find W ∝ /a . (4.13)It is clear that the same is also true of the vector field perturbation, i.e. δW ∝ /a .Putting the above together we can estimate the value of Ω end A as follows:Ω end A ∼ e − N x (cid:18) MH ∗ (cid:19) (cid:18) W ∗ M P (cid:19) , (4.14)where we used the Friedman equation ρ = 3( H ∗ M P ) and also that, while the dila-ton is varying, the density of the frozen vector curvaton remains constant with ρ A ∼ M W ∗ [13].Combining the above with Eqs. (4.9), (4.10) and (4.11) we obtain √ g ∼ ζ − e − N x H ∗ W ∗ M P (cid:18) MH ∗ (cid:19) / min (cid:26) , M Γ (cid:27) − / (cid:18) ΓΓ A (cid:19) / , (4.15)where ‘*’ denotes the epoch when the cosmological scales leave the horizon and weconsidered that M (cid:28) H ∗ for particle production of the vector curvaton to take place.– 24 – .3 The bispectrum Now, let us consider the bispectrum. As is well known, the bispectrum of the curva-ture perturbation is a measure of the non-Gaussianity in ζ since it is exactly zero forGaussian curvature perturbation. This non-Gaussianity is quantified by the so-callednon-linearity parameter f NL , which connects the bispectrum with the power spectra.When we have a contribution of a vector field to ζ , f NL can be statistically anisotropic[9, 32]. In the case of the vector curvaton with a varying kinetic function and mass, asis the case in our scenario, it was shown in Ref. [13] that non-Gaussianity is predom-inantly anisotropic. This means that, if non-Gaussianity is indeed observed withouta strong angular modulation on the microwave sky, scenarios of the present type willbe excluded from explaining the dominant contribution to the non-Gaussian signal.The value of f NL depends on the configuration of the three momentum vectorswhich are used to define the bispectrum. In Ref. [13] it was demonstrated thatstatistical anisotropy is strongest in the so-called equilateral configuration, wherethe three momentum vectors are of equal magnitude. In this case,65 f eqNL = 2 g Ω dec A (cid:18) M H ∗ (cid:19) (cid:34) (cid:18) H ∗ M (cid:19) ˆ W ⊥ (cid:35) , (4.16)where ˆ W ⊥ is the modulus of the projection of the unit vector ˆ W ≡ W /W onto theplane determined by the three momentum vectors which define the bispectrum. Fromthe above, we see that the amplitude of the modulated f eqNL is [13] (cid:107) f eqNL (cid:107) = 524 g Ω dec A , (4.17)while the degree of statistical anisotropy in non-Gaussianity is G ≡ (cid:18) H ∗ M (cid:19) (cid:29) , (4.18)which demonstrates that non-Gaussianity is predominantly anisotropic.Combining Eqs. (4.15) and (4.17) we can eliminate the dependence on W ∗ andobtain (cid:107) f eqNL (cid:107) ∼ gζ − e − N x (cid:18) H ∗ M P (cid:19) (cid:18) MH ∗ (cid:19) / min (cid:26) , M Γ (cid:27) − / (cid:18) ΓΓ A (cid:19) / . (4.19)To proceed further we note that H ∗ M P < δWW < , (4.20)where the upper bound is to ensure that our perturbative approach remains validand the lower bound is due to the requirement that Ω A < i.e. the vector field– 25 –oes not dominate the Universe at any stage. Indeed, since the ratio δW/W remainsconstant throughout the evolution of the vector field, we find δWW ≈ δWW (cid:12)(cid:12)(cid:12)(cid:12) ∗ ∼ H ∗ M W ∗ ∼ H ∗ M P √ Ω A ∗ , (4.21)where Ω A ∗ = ( ρ A /ρ ) ∗ . Employing Eq. (4.20), Eq. (4.15) gives gζ e N x (cid:18) M P H ∗ (cid:19) Γ A Γ < (cid:18) MH ∗ (cid:19) / min (cid:26) , M Γ (cid:27) − / < gζ e N x (cid:18) M P H ∗ (cid:19) Γ A Γ . (4.22)Using this, Eq. (4.19) gives524 g / ζ H ∗ M P < (cid:107) f eqNL (cid:107) < g / ζ . (4.23)From Eq. (4.6) the above suggests that12 ≤ (cid:107) f eqNL (cid:107) max ≤ , (4.24)where ζ = 4 . × − is the observed curvature perturbation. Thus, we see that, if thegenerated statistical anisotropy in the spectrum is observable then non-Gaussianityhas also a good chance of being observable, especially since the upper bound in theabove is already excessive and violates the observational constraints, − < f eq NL < To explore the possible observational consequences of the type of scenarios we arediscussing, we need to consider the evolution of the Universe after the end of inflation.Once the expansion rate has dropped sufficiently such that H ( t ) becomes of order M ,the vector field condensate begins quasi-harmonic coherent oscillations and producescurvaton quanta. It has been shown in Refs. [4, 13] that the energy density andaverage pressure of the field during the oscillations scale as ρ A ∝ a − and p ≈ i . e . the field behaves as pressureless isotropic matter. After inflation f = 1 and m = M . Therefore the action for the vector field which is minimallycoupled to gravity becomes S = M P (cid:90) d x √− g (cid:18) R − M P F µν F µν − M M P A µ A µ (cid:19) . (4.25)We can make a lower bound estimate on the decay rate of the curvaton field quantabased on gravitational decay, for which the decay rate is given byΓ A ∼ M M P . (4.26)– 26 –he decay products of the vector curvaton are much lighter degrees of freedom whichare, therefore, relativistic.The physical mass of the vector field in terms of the Planck mass is given inEq. (3.13), which we quote here: M = (2 π ) √ π V M P . (4.27)This should be compared to the inflationary Hubble scale H = 1 M P (cid:114) V , (4.28)where V / is the energy scale of inflation. Therefore, in order to obtain M (cid:28) H werequire V / (cid:29) (2 π ) √ π V M P . (4.29)We can estimate the scale of inflation for slow roll and DBI scalar inflation in orderto obtain a bound on the compact volume. • In a slow roll inflationary scenario, the CMB observations suggest [33] V / = 0 . (cid:15) / M P , (4.30)where (cid:15) is the slow-roll parameter. Assuming that (cid:15) is not tiny ( e.g. taking (cid:15) > − )we obtain V / ∼ GeV. Thus, for M (cid:28) H to be fulfilled, we require that thesize of the dimensionless volume V is around 10 − or larger, in which casethe physical mass of the vector field is M < GeV. For such a physical mass,Eq. (4.26) suggests Γ A < GeV. The temperature at the time of the vector curva-ton decay is T dec ∼ √ M P Γ A . It is easy to see that the decay occurs before Big BangNucleosynthesis if M >
10 TeV. • If inflation is instead driven by a DBI scenario, an equivalent expression to(4.30) can be found (see for example [34]) to be V / = 0 .
03 ˜ h − / M P , (4.31)where ˜ h = h M P is a dimensionless warp factor. For a GUT inflationary scale, whichis consistent with DBI inflation, we require ˜ h / ∼ , which can be achieved nearthe tip of the throat, where the warp factor is larger. Using this, we again obtaina limit on the 6D compact volume of the same order as above, and thus analogousresults follow. Moreover, if inflation happens in a DBI fashion, a second source forlarge non-Gaussianites of equilateral shape is generated [20].What about the inflaton decay, which reheats the Universe (since the vectorcurvaton is always subdominant)? In the case where the vector field brane is static,– 27 –nflation is driven by some other sector. Possibly, this is another D-brane undergoingmotion along the warped throat that may be either of the slow roll or DBI variety.The end of inflation takes place when the inflaton brane approaches the tip of thethroat, where there is an IR cutoff which allows the brane to reach the origin atfinite time and oscillate around the minimum of the potential [20, 35, 36]. In thiscase, the inflaton decay rate Γ depends on the couplings of the inflaton to standardmodel particles, to which it decays. A more dramatic end of inflation can take placeif the inflaton brane meets and annihilates with the antibrane located at the end ofthe tip. This is a complex process which occurs via a cascade that begins with agas of closed strings, followed by Kaluza-Klein modes and eventually standard modelparticles [37].The above possibilities may also arise in the case that our vector curvaton fieldis living on a moving brane, which can indeed be the inflaton brane, as will besubsequently discussed. However, in this case, prompt reheating by annihilation isnot possible because we need the vector field to survive after the end of inflation inorder to play the role of the curvaton. We are now ready to illustrate through a few examples how the D-brane curvaton toymodel we presented in section III.B can lead to observable statistical anisotropy inthe spectrum and bispectrum of the curvature perturbation. In all cases we considerinflation at the scale of grand unification, with H ∗ ∼ GeV, which is favoured byobservations [cf. Eq. (4.30)].Even though there is no compelling reason why the value of the physical vectorfield cannot be super-Planckian (since Ω A < , we make the conservative choice W ∗ (cid:46) M P in the examples below. As mentioned above, the dilaton is not expected tocontinue to roll for the remaining 50 or so e-folds after the cosmological scales exit thehorizon. The cosmological scales span about 10 e-folds. Therefore, for the signal tobe generated and span all the cosmological scales, one would strictly require that thedilaton rolls appropriately for up to 10 e-folds. However, if inflation is continuous, N x is expected to be small, and the dilaton will need to roll for a substantial numberof e-folds to allow for observable statistical anisotropy. On the other hand, if there isan early phase of inflation that lasts for up to 10 e-folds after which the vector fielddecays and imprints the spectrum, then the modulation period needs only to last forthis early phase, for once the spectrum is imprinted it can no longer be diluted byfurther bouts of inflation. While there are several motivations in string theory forconsidering successive periods of inflation, such as bouncing branes or inflation from This is in contrast to scalar fields, where super-Planckian values are expected to blow-up non-renormalisable terms in the scalar potential and render the perturbative approach invalid. Note, however, that it is possible for statistical anisotropy not to span all of the cosmologicalscales. For example, it might be there only for very large scales, comparable to the present horizon. – 28 – subset of the appropriately light moduli, in what follows we will take the simplestcase and assume that inflation is continuous, and will briefly comment at the end oncases where inflation is not continuous.
Let us consider first the case of prompt reheating, where we simply assume thatall of the initial vacuum energy of the branes is converted into radiation [38]. Thiswould lead to almost instantaneous reheating (prompt reheating) with a reheatingtemperature T reh ∼ V / , which implies Γ ∼ H ∗ .Then, using also Eq. (4.26), we can recast Eq. (4.22) as e − N x (cid:112) gζ H ∗ M P < M < e − N x (cid:112) gζ H ∗ . (4.32)Using that 10 TeV < M < H ∗ , the above givesln (cid:34) ( gζ ) − / (cid:114) H ∗ M P (cid:35) < ∼ N x < ∼ ln (cid:34) ( gζ ) − / (cid:114) H ∗
10 TeV (cid:35) . (4.33)Employing Eq. (4.6) and considering H ∗ ∼ GeV, we obtain1 < ∼ N x < ∼ . (4.34)This range is further truncated if we postulate W ∗ < ∼ M P as mentioned. Now, withprompt reheating Eq. (4.15) becomes g ∼ ζ − e − N x (cid:18) W ∗ M P (cid:19) , (4.35)which is independent of the value of M . If we choose W ∗ ∼ M P then g ∼ . N x ≈
6. With these values Eq. (4.19) gives (cid:107) f eqNL (cid:107) ∼ . × (cid:18) H ∗ M (cid:19) . (4.36)Thus, we can obtain observable non-Gaussianity ( (cid:107) f eqNL (cid:107) (cid:38)
1) for 10 GeV ≤ M ≤ GeV.
Let us consider another example, where we now assume Γ ∼ M . Then, following thesame process as above, we arrive atln (cid:34) ( gζ ) − / (cid:114) H ∗ M P (cid:35) < ∼ N x < ∼ ln (cid:34) ( gζ ) − / (cid:18) H ∗
10 TeV (cid:19) / (cid:35) . (4.37)– 29 –mploying Eq. (4.6) and considering H ∗ ∼ GeV, we obtain1 < ∼ N x < ∼ . (4.38)This range is further truncated if we postulate W ∗ < ∼ M P .With W ∗ ∼ M P , Eq. (4.15) becomes g ∼ ζ − e − N x (cid:18) MH ∗ (cid:19) / . (4.39)Combining this with Eq. (4.19) we obtain (cid:107) f eqNL (cid:107) ∼ ζ − e − N x (cid:18) MM P (cid:19) . (4.40)Using N x ≈
4, we find that (cid:107) f eqNL (cid:107) ∼
100 can be attained if M ∼ − M P ∼ GeV.Using this value, Eq. (4.39) gives g ∼ .
02. These are just sample “large” values, theycan become smaller if M is reduced. As a last example, we assume now that the inflaton decays through gravitationalcouplings. If we further take the inflaton mass to be of order H ∗ (this is natural insupergravity [39]), then we have Γ ∼ H ∗ M P . (4.41)For inflation at the scale of grand unification we find Γ ∼ GeV.Now, let us consider that the dilaton rolls throughout inflation, i.e. N x = 0.Taking W ∗ ∼ M P , Eq. (4.15) becomes MH ∗ ∼ ( gζ ) (cid:18) M P H ∗ (cid:19) , (4.42)where we have used Eq. (4.26) and we have assumed M >
Γ. Employing Eq. (4.6)and considering H ∗ ∼ GeV, the above gives10 − H ∗ (cid:46) M (cid:46) − H ∗ . (4.43)for an observable signal. Let us take M ∼ − H ∗ ∼ GeV > Γ, which meansΓ A ∼ − GeV, according to Eq. (4.26). Then, Eq. (4.15) yields g ∼ .
1, whileEq. (4.19) gives (cid:107) f eqNL (cid:107) ∼ . Again, these values can be reduced for a smaller M .From the above examples we see that it is possible to generate observable sta-tistical anisotropy in the spectrum and bispectrum of the curvature perturbation.If we allow for super-Planckian W ∗ we can increase N x but no more than N x (cid:39) e.g. thermal inflation, which can contribute about 20e-foldings or so [40]). In particular, for super-Planckian W ∗ and for a total periodof inflation that lasts N (cid:38)
60 e-foldings, of which 20 e-foldings may be generatedsubsequently by thermal inflation for example, we see that the dilaton must evolvefor about 20 e-foldings. Further improvement can be achieved if one considers severalbouts of inflation, e.g. as in [36]. The reheating temperature T reh ∼ √ M P Γ in theabove examples is rather large and would result in an overproduction of gravitinos,if the latter were stable. This problem is also overcome by adding a late period ofinflation since the entropy release can dilute the gravitinos.
5. Moving brane
We now consider the brane whose world volume hosts our vector curvaton to bethe D3-brane which is driving inflation. In open string D-brane models of inflation,inflationary trajectories can arise from motion in the radial direction of a warpedthroat, in which brane motion may be slow or relativistic, leading to slow-roll or DBIinflation respectively . In addition to the radial direction, one may also consider thebrane to have non-trivial motion in any of the five angular directions of the throat,giving an inherently multifield scenario. As shown in Ref. [36, 42] for the case of radialmotion plus one angular field, motion in the angular directions experiences strongHubble damping such that the behaviour of the brane very soon tends towards theconventional single field scenario. This situation was recently confirmed in Ref. [43]in which motion in all six directions in the throat is considered. Thus we can assumethat for most of the inflationary period, the motion of the brane in the throat iseffectively along a single direction. Nevertheless, it is useful to comment on themultifield case since these scenarios overcome the essential problems with single fieldDBI inflation, which, for example, are related to the lack of consistency of predictedbounds on the scalar-to-tensor ratio [44]. In what follows, we briefly outline the twopossibilities, i.e. slow-roll inflation and DBI inflation, where we consider both singlefield and multifield DBI scenarios. All the results from the previous sections canthen just follow straightforwardly.To take into account the possible effects of relocating our vector curvaton toa moving brane, we consider the same scenario as is discussed in Sec. 3.3 ( i.e. acanonical vector field modulated by the dilaton field such that e − φ ∝ a ), but with aposition field for general brane motion in the radial direction. We therefore rewrite As previously mentioned, in a realistic scenario where the St¨uckelberg mass mechanism is used,one would need to consider branes of higher dimensionality. For D-brane inflation in a warpedthroat as we consider here, one may consider the inflaton brane to be wrapped D5-brane as inRef. [41]. – 31 –he action in Eq. (2.12) as S = (cid:90) d x √− g (cid:26) M P R − M P ∂ µ φ ∂ µ φ − V ( φ ) − h − (cid:2) he − φ F αβ F αβ + h∂ α ϕ∂ α ϕ +3 h e − φ (cid:0) ∂ α ϕ∂ α ϕ F βγ F βγ − ∂ α ϕ F αβ ∂ γ ϕ F γβ (cid:1)(cid:3) / − V ( ϕ ) + h − − m A µ A µ (cid:27) . (5.1)We consider cosmological scenarios such that all background fields are functions oftime only, in which case there is a cancelation of the terms mixed in ϕ and A µ atbackground level. Nonetheless, in principle this action could still lead to mixed termsin the perturbations, which can be functions of space as well as time. It turns outhowever, that the mixed terms do not appear in the equations of motion for theperturbations of both ϕ and A µ , as is clear from examining the complete equationsof motion which are given below and considering the possible perturbations of thevarious terms.The equations of motion for ϕ and A µ calculated from the action in Eq. (5.1)are given respectively by, h (cid:48) h (cid:16) − √ Σ (cid:17) + V (cid:48) ( ϕ ) + h (cid:48) h e − φ F αβ F αβ + ∂ α ϕ∂ α ϕ + 6 he − φ [( ∂ α ϕ ) F − ∂ α ϕ F αβ ∂ γ ϕ F γβ ]2 √ Σ= ∂ µ √− g (cid:26) √− g [2 ∂ µ ϕ + 3 he − φ (2 F αβ F αβ ∂ µ ϕ − ∂ α ϕ F αβ F µβ )] √ Σ (cid:27) (5.2) m A ν = ∂ µ √− g (cid:26) √− g ( e − φ F µν + 3 he − φ [2( ∂ α ϕ ) F µν − ∂ α ϕ F αν ∂ µ ϕ + 2 ∂ α ϕ F αµ ∂ ν ϕ )] √ Σ (cid:27) , (5.3)where Σ = 1 + h ( ∂ α ϕ ) + he − φ F + 3 h e − φ [( ∂ α ϕ ) F − ∂ α ϕ F αβ ∂ γ ϕ F γβ ] . Neglecting the mixed terms at both background and perturbation level, and con-sidering the derivatives acting on A µ to be small while keeping those acting on theposition field to be general, we may expand √ Σ, and, keeping only up to quadraticorder in F , the equations of motion then become V (cid:48) ( ϕ ) + h (cid:48) h (cid:104) − (cid:112) h ( ∂ α ϕ ) (cid:105) + h (cid:48) h (cid:0) e − φ F αβ F αβ + ∂ α ϕ∂ α ϕ (cid:1)(cid:112) h ( ∂ α ϕ ) = ∂ µ √− g (cid:34) √− g∂ µ ϕ (cid:112) h ( ∂ α ϕ ) (cid:35) , for the inflaton field, and m A ν = ∂ µ √− g (cid:34) √− ge − φ F µν (cid:112) h ( ∂ α ϕ ) (cid:35) , (5.4)– 32 –or the vector field.An important feature of (5.4) to note is the new form of the gauge kinetic function f = e − φ √ h ∂ α ϕ∂ α ϕ = γ ϕ e − φ . (5.5)Employing the metric (2.1) into these equations, we find that the equations of motionfor the background fields ϕ ( t ) and A µ ( t ) are given respectively by¨ ϕ − h (cid:48) h + 32 h (cid:48) h ˙ ϕ + 3 H ˙ ϕ γ ϕ − h (cid:48) h e − φ (cid:32) ˙ A γ ϕ a (cid:33) + (cid:18) V (cid:48) ( ϕ ) + h (cid:48) h (cid:19) γ ϕ = 0 (5.6)and A t = 0 , (5.7) ¨ A + ˙ A (cid:32) H + ˙ ff (cid:33) + m f A = 0 . (5.8)In the absence of the vector field, Eq. (5.6) reduces to the standard equation ofmotion for the DBI inflaton (see Ref. [20]), where we note that now, since the scalarfield is homogenised by inflation, γ ϕ = 1 (cid:112) − h ˙ ϕ . (5.9)Eq. (5.9) gives the Lorentz factor for brane motion in the internal space, and is adirect generalisation of the Lorentz factor for a relativistic point particle. In an AdSthroat, the warp factor is simply given by h = λϕ (5.10)where λ = g Y M is the ’t Hooft coupling, and we require λ (cid:29) γ ϕ remains real at all times. Given the form of the warp factor in Eq. (5.10), we seethat warping becomes significant as ϕ →
0, therefore at small ϕ the velocity of therelativistic brane is forced to decrease. Indeed, in Ref. [20] it is shown that at latetimes ϕ ( t ) → t , (5.11)which implies that for a pure AdS throat, the brane takes an infinite time to crossthe horizon. A realistic throat may be approximated as AdS in the regions of interestbut has a finite cut-off at the IR end, therefore the brane may cross the horizon infinite time. The fact that the brane is forced to slow down as it moves towards thehorizon leads to inflationary trajectories in this region.– 33 –n an AdS geometry, γ ϕ may indeed become arbitrarily large at late times, andthis leads to a suppression of all but the first three terms in Eq. (5.6). Therefore, wesee that in this case the vector term has a negligible impact on the dynamics of theinflaton, along with the potential term and friction term, as soon as the brane startsto approach the speed limit ˙ ϕ = ϕ / √ λ . The same is true for the potential andfriction terms in the case of standard DBI inflation in an AdS throat (see Ref. [20]).In a Klebanov-Strassler throat, the behaviour of γ ϕ is such that its maximumvalue is reached almost immediately as the brane moves from the UV end of thethroat, dropping for subsequent times. At late times, when the brane is moving inthe IR region of the throat, the value of γ ϕ is roughly constant, remaining withina single order of magnitude. Ultimately γ ϕ →
1, as the brane stops. This meansthat the vector, friction and potential terms in Eq. (5.6) are no longer suppressed forlater times. In this case the vector A µ can have an influence on the dynamics of theinflaton. In the standard DBI scenarios in Klebanov-Strassler throats, i.e. withoutthe vector contribution, the presence of non-negligible potential and friction terms inthe dynamics of the inflaton does not change the result: inflation still takes place inthe throat. For our case, the presence of the vector field may contribute an effectiveterm in the potential for the inflaton, along the lines of what has been demonstratedin Ref. [45]. For the time being we focus on the simpler AdS case, such that the vectorterm is subdominant in the dynamics of the inflaton, and we can treat the systemas undergoing standard DBI inflation in an AdS background. However, further workis currently in progress that will assess the impact of the vector backreaction in aKlebanov-Strassler throat.Let us now consider conventional slow-roll, which is possible if the potentialadmits a particularly flat section. When the brane is slowly rolling along a flatsection of its potential in the throat, the derivatives of both the vector as well as theposition field are small and we can expand the √ Σ factor in Eqs. (5.2). Keeping onlythose terms that are up to quadratic order in the derivatives of both of the fields, werecover the standard Klein-Gordon equation for a minimally coupled scalar field,¨ ϕ + 3 H ˙ ϕ + V (cid:48) ( ϕ ) = 0 . (5.12)We can now implement the vector curvaton scenario in this set-up as follows.Assuming that the vector field and the dilaton give a subdominant contribution tothe energy density during inflation, the energy density and pressure calculated fromthe action in Eq. (5.1) are given by ρ = 1 h ( γ ϕ −
1) +
V , (5.13) p = 1 h (cid:18) − γ ϕ (cid:19) − V . (5.14)– 34 –e consider the brane to be moving relativistically, therefore γ ϕ is large. As dis-cussed above, in the limit of strong warping the velocity of the brane is forced todecrease, hence the energy density becomes dominated by the potential. For large γ ϕ and strong warping, the pressure is clearly also dominated by the potential. Thisillustrates how inflation can arise in a DBI scenario.Taking into account the new form of the gauge kinetic function, we see that if γ ϕ (cid:54) = 1 the scaling necessary for statistical anisotropy could in principle be spoilt bynew powers of the scale factor that are introduced as a result of the inflaton. Asshown in Ref. [20], for DBI inflation in an AdS geometry with a warp factor as inEq. (5.10), the scale factor a ( t ) → a t /(cid:15) at late times, where (cid:15) is a generalisation ofthe slow-roll parameter and is given by (cid:15) = 2 M P γ ϕ (cid:18) H (cid:48) H (cid:19) , (5.15)such that ¨ a/a = H (1 − (cid:15) ) and one obtains de Sitter expansion for (cid:15) →
0. For abackground expanding in this way, the vector field will undergo gravitational par-ticle production as outlined in Sec. 3.3 and obtain a scale invariant spectrum ofsuperhorizon perturbations as long as we still have f ∝ a and as long as the vectorfield remains light (the vector mass m does not depend on the inflaton and thereforethe condition m ∝ a is not impacted, however the physical mass M = m/ √ f isimpacted).It is further shown in Ref. [20] that γ ϕ ∝ t at late times, i.e. γ ϕ ∝ a (cid:15) whichmeans that our gauge kinetic function is now f = e − φ γ ϕ ∝ a (cid:15) . This couldcontribute a small degree of scale dependance to the power spectrum of vector per-turbations, however clearly the scaling f ∝ a still holds. Furthermore, as shownin Ref. [13], when f ∝ a (cid:15) ) , the spectral tilt for the transverse and longitudinalcomponents of the vector field are different with the corresponding spectral indexesbeing n L,R − − (cid:15) and n (cid:107) − (cid:15) , (5.16) i.e. the transverse spectrum is slightly red and the longitudinal is slightly blue.The physical mass of the vector field is now given by M = m/ √ f ∝ / √ γ ϕ ∝ a − (cid:15) , which means that M now experiences a slight evolution during inflation. Inparticular, when γ ϕ (cid:29) M only serves to make the condition M (cid:28) H easier to fulfill.In addition to the suppression of the vector term by γ ϕ , we saw before that for aphysical vector mass M (cid:28) H during inflation and a gauge kinetic function f ∝ a ,the equation of motion for A µ , given in Eq. (3.12), implies that the vector field freezesat constant amplitude, such that ˙ A µ (which appears in Eq.(5.6)) is expected to bevery small during this time. The same behaviour occurred during inflation for thenon-canonical vector field, as can be seen in Fig. 1. Note also, that the vector field– 35 –s coupled to the inflaton through the Lorentz factor in Eq. (5.9), which features thederivatives of ϕ , which are expected to be small during slow-roll inflation. In Refs. [21, 46] the potential for a D3-brane has been explicitly calculated tak-ing into account all corrections from fluxes and bulk objects, and the results showthat it is possible, albeit with fine-tuning, to obtain a flat region in which a slow-rollphase could occur. Similarly, an explicitly calculated potential is studied in Ref. [47],in which it is shown that a sufficiently long period of inflation as well as a correctspectrum of perturbations can be achieved from the combination of a slow-roll andDBI phase, where slow-roll is obtained by fine-tuning the potential in the region closeto the tip. In such pictures in which the dominant contribution to the curvature per-turbation can be successfully generated by the inflaton, modulated vector curvatonof the type considered in the present work could add the new feature of measurablestatistical anisotropy.Let us now comment on the multifield case. In the simplest situation one mayconsider a generalization to a two field model in which, in addition to its motionin the radial direction, the brane moves in one of the five angular directions of thewarped throat. Such a scenario is considered in Ref. [36, 42] and we briefly discussthis picture here to illustrate the multifield generalisation of our work, keeping inmind that the dominant behaviour of the brane is always well approximated by asingle field scenario at the times of interest to us.In general for a multifield scenario, several of the terms contained in the deter-minant in Eq. (2.8) computed for the DBI action in Eq. (2.4) may no longer vanishafter the antisymmetrisation. However, these terms become subdominant as soon asthe brane tries to move radially only, and therefore we do not need to consider themto explain the essentials of the multifield picture. The important point is that thevector field and the scalar fields are decoupled for late times at both background andperturbation level, as we saw for the single field case. Considering only the dominantterms in Eq. (2.8), the transition to a multifield scenario will impact the form of γ ϕ ,which now becomes, γ ϕ = 1 (cid:112) − h ˙ ϕ i ˙ ϕ j g ij , (5.17)where g ij is the metric on the internal space.The energy density and pressure calculated from this action are analogous toEqs. (5.13) and (5.14), but where γ ϕ is now given by (5.17). Once the brane starts tosettle onto a radial trajectory, we recover single field DBI inflation and the familiarform of γ ϕ given in (5.9). All of the results outlined before for the single field caseare then applicable here. This is in contrast to Ref. [45], where the kinetic function of the vector field is modulated bythe inflaton field itself and not by its derivatives. – 36 – . Conclusions
In this paper we have discussed the possibility to embed the vector curvaton scenarioin string theory where the vector field which lives on a D3-brane plays the role of thevector curvaton. We focused on the simplest case which may outline the mechanismwithin string theory, and which may be generalised to realistic explicit cases, orused as a means to inform the search for such cases. We have investigated how thisscenario can affect the observed curvature perturbation ζ in the universe. We havefirst considered the case in which the vector curvaton brane is stationary and inflationoccurs in some other sector, for example via warped D ¯D inflation, or via the motionof a different D3-brane. For suitable values of the parameters, such a vector curvatoncan generate observable statistical anisotropy in the spectrum and bispectrum of ζ provided that the dilaton field, which is a spectator field during inflation, varies withthe scale factor as e φ ∝ a − when the cosmological scales exit the horizon. If thisis the case, both the transverse and the longitudinal components of the vector fieldobtain a scale-invariant superhorizon spectrum of perturbations. However, particleproduction is anisotropic, which means that the vector curvaton cannot generate ζ by itself but it can give rise to observable statistical anisotropy. In view of theforthcoming data from the Planck satellite, this is a finding that will be testable inthe very near future. Indeed, in the class of models we have discussed, statisticalanisotropy in the bispectrum is predominant. This means that non-Gaussianity hasto have a strong angular modulation on the microwave sky, which may or may notbe found by Planck. Moreover, these models can still produce observable statisticalanisotropy in the spectrum even if its contribution to the bispectrum is negligible(and vice-versa, see also Ref. [48]).We also showed that these results are robust when we allow for the possibilitythat the same brane which hosts the vector curvaton, is also responsible for drivingcosmological inflation, which can be of either the slow roll or the DBI variety. All theresults obtained for the stationary brane case follow and we again can obtain mea-surable statistical anisotropy both in the spectrum and the bispectrum. Moreover,in the case of DBI inflation, the constraints on the vector mass can be considerablyimproved. Furthermore, since DBI inflation also contributes to the generation oflarge non-Gaussianities of the equilateral type, in this case there would be two dif-ferent sources for large Gaussian deviations. Thus we have seen that the presence ofseveral light fields in string theory models of cosmology can provide us with a uniquesource for distinctive features, which can help us to distinguish such models frompure field theory models. Certainly these possibilities deserve further investigation,and in the present work we have only begun to explore the prospects for sources ofstringy statistical anisotropies in the curvature perturbation.As we have shown, our results are based on the assumption that the dilaton rollsas e − φ ∝ a while the cosmological scales exit the horizon, and then for a further– 37 –ase-dependent amount of time in order for statistical anisotropy to be observable.Nevertheless, for the simplest cases in which inflation is continuous, we require thatthe minimum of the dilaton potential is located at an extremely weak coupling, g s (cid:28) Acknowledgments
We would like to thank M. Blaszczyk, T. Koivisto, A. Maharana, D. Mayorga-Pe˜na,P. Oehlmann, F. Quevedo, F. Ruehle, M. Schmitz and G. Tasinato for helpful discus-sions. K.D. wishes to thank the University of Crete for the hospitality. K.D is sup-ported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physicsunder STFC grant ST/J000418/1. D.W. and I.Z. are supported by the DFG clus-ter of excellence Origin and Structure of the Universe, the SFB-Tansregio TR33“The Dark Universe” (Deutsche Forschungsgemeinschaft) and the European Union7th network program “Unification in the LHC era” (PITN-GA-2009-237920). D.W.would also like to thank BIGS-PA for support.– 38 – . Mass generation mechanism for U (1) field Consider a Lagrangian of the following form [49]: L = − e φ H µνρ H µνρ − e − φ g F µν F µν + c (cid:15) µνρσ B µν F ρσ , (A.1)where F µν = ∂ µ A ν − ∂ µ A ν , H µνρ = ∂ µ B νρ + ∂ ρ B µν + ∂ ν B ρµ , ˜ g and c are arbitraryconstants and we have included a dilaton-like coupling to the fields. The Lagrangian(A.1) describes a massless two-form field B µν , with one degree of freedom, couplingto a massless gauge field A µ , with 2 degrees of freedom, plus the dilaton field. Thefact that the two-form field has one degree of freedom is evident by the fact thatin four dimensions, it transforms under the little group SO (2). We will see that inits dual form, this Lagrangian describes a gauge field A µ with 3 degrees of freedom, i.e. the single degree of freedom carried by the two-form field is “eaten” by the gaugefield to provide a mass, and the two-form field no longer appears.To arrive at the dual form, we make an intermediate step which involves rewritingEq. (A.1) by integrating the coupling term by parts, and then imposing the constraint H = dB by way of a Lagrange multiplier field η . Integration by parts changes theform of the coupling term from (cid:15) µνρσ B µν F ρσ to (cid:15) µνρσ H µνρ A σ , and so eliminates B µν from the Lagrangian. To retain the same information as was present in the originalform, we need to impose the constraint H = dB , however, as we have alreadyeliminated B µν , we formulate the constraint in terms of the new field H µνρ as dH = 0(which is of course true in the case that H = dB ). The Lagrangian in Eq. (A.1) canbe thus rewritten as: L = − e φ H µνρ H µνρ − e − φ g F µν F µν − c (cid:15) µνρσ H µνρ A σ − c η (cid:15) µνρσ ∂ µ H νρσ . (A.2)Now, integrating by parts the last term in Eq. (A.2), L = − e φ H µνλ H µνλ − e − φ g F µν F µν − c (cid:15) µνλβ H µνλ ( A β + ∂ β η ) (A.3)and solving for H , we find: H µνλ = − c e − φ (cid:15) µνλβ ( A β + ∂ β η ) . Inserting this back into Eq. (A.3), we find L = − e − φ g F µν F µν − c e − φ A σ + ∂ σ η ) . (A.4)Normalising the kinetic term, we see that the gauge field A µ has acquired a mass m = ˜ g c e − φ . Notice the the scalar η can be gauged away via a gauge transformation– 39 –f A → A + ∂ Λ, thus we are left with only the mass term. By absorbing the scalarfield η into the gauge field, we have explicitly chosen a gauge.In our set-up discussed in the main text, this mechanism is realised via thecoupling of the gauge field F to the RR two form C (see Eq. (2.9)). The kineticterm for C descends from the 4D components of the RR field strength, F = dC .This arises from the ten dimensional type IIB action. The relevant piece in theEinstein frame, is given by12 κ (cid:90) d x √− g (cid:18) − e φ F µνλ F µνλ (cid:19) , (A.5)where 2 κ = (2 π ) ( α (cid:48) ) [22]. After dimensional reduction to four dimensions thisbecomes: M P (cid:90) d x √− g (cid:18) − e φ F µνλ F µνλ (cid:19) , (A.6)where we have used that 12 κ (cid:90) d x h √− g = V κ = M P . (A.7)where h is the warped factor. Using this and (2.9), we find the appropriate La-grangian for the mass generation in our set-up, L mass = − e φ (cid:18) M P (cid:19) F µνρ F µνρ − T (2 πα (cid:48) ) e − φ F µν F µν + T (2 πα (cid:48) )4 (cid:15) µνρσ C µν F ρσ . (A.8)Rescaling the 2-form as, C = √ M P (cid:101) C , the Lagrangian takes the form of Eq. (A.1) L mass = − e φ (cid:101) F µνρ (cid:101) F µνρ − e − φ g F µν F µν + c (cid:15) µνρσ (cid:101) C µν F ρσ , (A.9)where ˜ g = 1 T (2 πα (cid:48) ) , c = T (2 πα (cid:48) ) √ M P . (A.10)Thus the dilaton dependent mass for the vector field is given by m = ˜ g c e − φ = 2 T e − φ M P = (2 π ) M s e − φ V , (A.11)where M s = α (cid:48)− / is the string scale, V = V /(cid:96) s is the dimensionless six dimensionalvolume and we have used that T = (2 π ) − ( α (cid:48) ) − , M P = 2(2 π ) − V M s . (A.12) Note that in our case, C = C . – 40 –oing back to the WZ action for the D3-brane in Eq. (2.9), using C = √− g h − dx ∧ dx ∧ dx ∧ dx and the mass term discussed above, we have S WZ = q (cid:90) d x √− g (cid:18) h − − m A µ A µ + C (cid:15) µνλβ F µν F λβ (cid:19) , (A.13)where here ˜ g A µ = A µ is the canonical normalised gauge field (and correspondingly F µν its field strength). Further, (cid:15) µναβ is the Levi-Civita tensor, such that (cid:15) = √− g . References [1] E. 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