Combined local and equilateral non-Gaussianities from multifield DBI inflation
CCombined local and equilateral non-Gaussianities from multifield DBI inflation
S´ebastien Renaux-Petel ∗ APC (UMR 7164, CNRS, Universit´e Paris 7),10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France (Dated: November 6, 2018)We study multifield aspects of Dirac-Born-Infeld (DBI) inflation. More specifically, weconsider an inflationary phase driven by the radial motion of a D-brane in a conical throatand determine how the D-brane fluctuations in the angular directions can be converted intocurvature perturbations when the tachyonic instability arises at the end of inflation. Thesimultaneous presence of multiple fields and non-standard kinetic terms gives both local andequilateral shapes for non-Gaussianities in the bispectrum. We also study the trispectrum,pointing out that it acquires a particular momentum dependent component whose amplitudeis given by f locNL f eqNL . We show that this relation is valid in every multifield DBI model,in particular for any brane trajectory, and thus constitutes an interesting observationalsignature of such scenarios. Contents
I. Introduction II. Generating the curvature perturbation at the end of brane inflation δN formulae 10 III. Power spectrum and primordial non-Gaussianities from the bispectrum ∗ [email protected] a r X i v : . [ h e p - t h ] O c t D. Implication of the results 18
IV. Primordial non-Gaussianities from the trispectrum
V. Conclusions Appendix 1 - δN formulae beyond leading order Appendix 2 - Higher order loop contributions References I. INTRODUCTION
Current measurements of the cosmic microwave background (CMB) anisotropies, such as thoseobtained by the WMAP satellite, already provide us a with a wealth of valuable information aboutthe very early universe. Furthermore, with the successful launch of the Planck satellite and theincreasing precision of large scale structure surveys, one can hope to get yet more precise infor-mation in the near future. In this context non-Gaussianity [1] is particularly exciting since it hasthe ability to discriminate between models which are otherwise degenerate at the linear level: forinstance, a detection of so-called local non-Gaussianity would rule out all single field scenarios ofinflation in a model independent way [2]. Amongst such single field models, those with standardkinetic terms, in which the inflaton slowly rolls down its potential, come with an unobservably lowlevel of non-Gaussianity and hence are still consistent with present-day observations [3]. Theoret-ically though, their embedding in high-energy physics theories is hampered by the eta-problem,namely that Planck-suppressed corrections often lead to potentials that are too steep to supportslow-roll inflation. In the context of string theory, this problem was demonstrated to be particu-larly acute in slow-roll inflationary models based on the dynamics of D-branes moving in higherdimensional spaces [4, 5]. Nonetheless, this set-up precisely motivates an interesting way to bypassthe eta-problem that has attracted a lot of attention: since the inflaton field in brane inflation isgoverned by a Dirac-Born-Infeld (DBI) action characterized by non-canonical kinetic terms, thereexists an upper bound on the inflaton velocity that allows one to achieve an inflationary phase withotherwise too steep potentials [6]. Besides alleviating the eta-problem, the non standard kineticterms also enhance the self-interactions of the inflaton, resulting in significant non-Gaussianitiesof equilateral type [7]. Indeed, the simplest single-field DBI models are already under strain fromobservations [8, 9].However, as is central to the discussion of this paper, DBI inflation is naturally a multifieldscenario, since the position of the brane in each compact extra dimension gives rise to a scalarfield from the effective four-dimensional point of view [10, 11, 12, 13]. In multiple field models,the scalar perturbations can be decomposed into (instantaneous) adiabatic and entropy modes byprojecting, respectively, parallel and perpendicular to the background trajectory in field space [14].If the entropy fields are light enough to be quantum mechanically excited during inflation, theydevelop super-Hubble fluctuations that can be transferred to the adiabatic mode on large scales.This effect, as well as the nonlinearities imprinted on the fields at the epoch of horizon crossing,was taken into account in [15] where it was shown that the shape of equilateral non-Gaussianitiesis the same as in the single-field case while their amplitude is reduced by the entropy to curvaturetransfer, which therefore eases the confrontation with the data. This paper, as well as subsequentones on multifield DBI inflation [16, 17, 18, 19, 20, 21, 22], focused mainly on equilateral non-Gaussianities. More generally, multiple-field inflationary models are known to produce possiblylarge non-Gaussianities of another shape, namely local non-Gaussianities, that arise due to thenonlinear classical evolution of perturbations on superhorizon scales. The formalism developed inthese papers also remained very general and no particular model was presented for the entropictransfer on large scales. We address both these questions here, using a mechanism outlined byLyth and Riotto [23] to convert entropic into adiabatic perturbations at the end of brane inflation.In this scenario, inflation is still driven by a single inflaton scalar field, namely the D3-branesolely moves along the standard radial direction of the throat in the context of warped conicalcompactifications. When the mobile D3-brane and an anti D3-brane sitting at the tip of the throatcome within a string length, an open string mode stretched between them becomes tachyonic,triggering their annihilation and the end of inflation [24]. As the brane-antibrane distance is six-dimensional , it acquires some dependence upon the fluctuations of the light fields parametrizingthe angular position of the brane. Hence the value of the inflaton at which the instability signalsis modulated and the duration of inflation varies from one super-Hubble region to another. In thisway initially entropic perturbations are converted into the curvature perturbation. The relevanceof this mechanism in the slow-roll ’Delicate Universe’ scenario [4] has been recently investigated[25]. Here however we would like to combine it with the DBI inflationary regime. This was lookedat by Leblond and Shandera [26] regarding the power spectrum. In the present paper, we extendtheir analysis by taking into account the enhancement of the angular fluctuations by the low speedof sound [15], as well as by investigating the non-Gaussian properties of the curvature perturbation.It should be noted that while no explicit model of DBI inflation in a string theory framework hasbeen constructed yet that both satisfy observational constraints and are theoretically self-consistent[105] – mainly because of the existence of a geometrical limit for the size of the throat in Planckianunits [27, 28][106] – it was so far assumed that the fluctuations of the primary inflaton createthe curvature perturbation. One can therefore hope to embed consistently the DBI inflationaryscenario in string theory with other mechanisms to generate the density perturbations, such as theone considered in this paper.In our model, the curvature perturbation is nonlinearly related to the entropy perturbations,therefore the conversion process gives rise to local non-Gaussianities besides the standard equilat-eral ones generated at horizon crossing. To the best of our knowledge it is the first time that definitepredictions are made for an inflationary scenario where both significant local and equilateral non-Gaussianities can arise, characterized in the bispectrum respectively by the parameters f locNL and f eqNL . While the logical possibility that a linear combination of different shapes of the bispectrumcan arise is an acknowledged fact, it should be stressed that what we consider is not merely ajuxtaposition of an inflaton with non standard kinetic terms generating f eqNL and another lightscalar generating f locNL . In multifield DBI inflation, light scalar fields other than the inflaton and with derivative interactions naturally contribute to both types of non-Gaussianities. We show thatthis nontrivial combination leaves a distinct imprint on the primordial trispectrum , which acquiresa particular momentum dependent component whose amplitude is given by the product f locNL f eqNL .This relation is structural and is valid independently of the details of the inflationary scenario, i.e. for any brane trajectory and any process by which entropic perturbations feed the adiabatic ones.Hence it constitutes an interesting observational signature of multifield DBI inflation.The layout of this paper is the following. In section II, we describe our set-up and recall resultsconcerning the amplification of quantum fluctuations in multifield DBI inflation. We explain themechanism by which entropic perturbations are converted into the curvature perturbation at theend of brane inflation. Using the δN formalism, we also derive the relevant formulae for quantifyingthis effect. In section III, we calculate the power spectrum of the primordial curvature perturbation.We show, in particular, that the entropic transfer is more efficient in the DBI than in the slow-roll regime. We also calculate the primordial bispectrum which acquires a linear combination ofboth the local and equilateral shapes of non-Gaussianities. We then combine our results for thespectrum and bispectrum to derive constraints on the model. In section IV we turn to the studyof the primordial trispectrum. We calculate the local trispectrum parameters τ NL and g NL anddiscuss the purely quantum contribution coming from the field trispectra. Then we point out thepresence of a particular component of the trispectrum whose amplitude is given by the product f locNL f eqNL . We finish by plotting the corresponding shape of the trispectrum in different limits, andthis turns out to have characteristic features. We summarize our main results in the last section. II. GENERATING THE CURVATURE PERTURBATION AT THE END OF BRANEINFLATIONA. The set-up and the amplification of quantum fluctuations
Our setting is that of a flux compactification of type IIB string theory to four dimensions [29],resulting in a warped geometry in which the six-dimensional Calabi-Yau manifold contains one ormore throats. The ten-dimensional metric inside a throat has the generic form ds = h ( y K ) g µν dx µ dx ν + h − ( y K ) G IJ ( y K ) dy I dy J , (1)where g µν is the metric of the four-dimensional, non-compact, spacetime and we have factoredout the so-called warp-factor h ( y I ) from the metric G IJ in the six compact extra dimensions. Inthe following, we assume that the geometry presents a special radial direction, in agreement withknown solutions of the supergravity equations [30], so that the warp factor is a function of a radialcoordinate ρ only, decreasing along the throat down to its tip at ρ = 0 [107]. In this framework,we consider the following internal metric G IJ ( y K ) dy I dy J = dρ + b ( ρ ) g (5) mn dψ m dψ n , (2)where we refer to b ( ρ ) as the radius of the throat and ψ m ( m = 5,6,7,8,9) denote its angularcoordinates. Since we aim to present a general mechanism, we do not specify a precise form for h ( ρ ) and b ( ρ ) and only require that they approach constant values h tip and b tip near the tip, whichis the situation encountered for instance in the Klebanov-Strassler throat [30].We take then a probe D3-brane, of tension T = m s (2 π ) g s , (3)where m s is the string mass and g s the string coupling, filling the four-dimensional spacetime, andpoint like in the six extra dimensions. The D3-brane has coordinates y I ( b ) and falls down to the tipof the throat where a static D φ I = (cid:112) T y I ( b ) → φ = (cid:112) T ρ ( b ) , θ m = (cid:112) T ψ m ( b ) , (4)and rescaled warp factor f = (cid:0) T h (cid:1) − , (5)the D3-brane low-energy dynamics is captured by the Lagrangian P = − f ( φ ) − (cid:18)(cid:113) det( δ µν + f G IJ ∂ µ φ I ∂ ν φ J ) − (cid:19) − V ( φ I ) , (6)where V ( φ I ) is the field interaction potential. Note that in general, there are also contributionsfrom the presence of various p -forms in the bulk as well as the gauge field confined on the brane.In [18], these fields were shown to have no observable effects on scalar cosmological perturbationsat least to second-order so we have omitted them here for simplicity.The explicit calculation of the potential V in (6) is extremely difficult and requires a detailedknowledge of the compactification scheme (see e.g [31] and references therein). In general though,we know that bulk as well as moduli stabilizing effects break the isometries of the throat, stabiliz-ing some of the angular coordinates of the branes. However, there typically remain approximateresidual isometries of the potential, as shown explicitly in [25] for the most-advanced brane in-flation model [5]. For simplicity, we consider only one such isometry direction ψ , entering thefive-dimensional metric g (5) mn (2) through g (5) mn dψ m dψ n = dψ + . . . (7)and we assume that the four other brane angular degrees of freedom are frozen in their minima oftheir effective potential at the position of the antibrane along these directions. Therefore we takethe potential V = V ( φ ) to depend only on the radial position of the brane, which itself moves alongthe radial direction only – ˙ φ (cid:54) = 0 , ˙ θ m = 0. As opposed to single-field inflation, the perturbationsalong the isometric, i.e. flat direction ψ , can be quantum mechanically excited during inflation. Inthat case, the angular D − D θ ≡ √ T ∆ ψ varies from one Hubble patch to the others.Although this does not modify the dynamics during inflation, this will turn out to be crucial atthe end of inflation, as we will see below.We now recall the relevant results of [16] regarding the amplification of quantum fluctuationsin multifield brane inflation, in particular the amplitude of the inflaton and of the angular per-turbations at horizon crossing. For that purpose, it is convenient, after going to conformal time τ = (cid:82) d t/a ( t ), where a ( t ) is the cosmological scale factor, to work in terms of the canonicallynormalized fields given by v σ = ac / s Q φ , v s = a √ c s b ( φ ) Q θ , (8)where Q I denotes the perturbations of the field φ I in the flat gauge and c s ≡ (cid:113) − f ( φ ) ˙ φ (9)is the propagation speed of scalar perturbations (see Eq. (10) below), or speed of sound. Clearly,from Eq. (9), there is an upper bound on the inflaton velocity | ˙ φ | ≤ √ f ( φ ) . When c s ≈
1, onecan expand the square-root in the Lagrangian (6) to quadratic order in the fields. Then the actionbecomes canonical and one recovers the slow-roll regime. However, when the brane almost saturatesits speed limit – c s (cid:28) v σ and v s at linear order take the simple form[16, 32] v (cid:48)(cid:48) σ + (cid:18) c s k − z (cid:48)(cid:48) z (cid:19) v σ = 0 , v (cid:48)(cid:48) s + (cid:18) c s k − z (cid:48)(cid:48) s z s + a µ s (cid:19) v s = 0 , (10)where we have introduced the two background-dependent functions z ( τ ) = a √ (cid:15)/c s , z s ( τ ) = a/ √ c s , with (cid:15) ≡ − ˙ HH the inflationary deceleration parameter. The effective entropic mass squared µ s is given by µ s ≡ c s b (cid:48) ( φ ) b ( φ ) V (cid:48) ( φ ) − (1 − c s ) f b (cid:48) ( φ ) b ( φ ) f (cid:48) ( φ ) − ˙ φ b (cid:48)(cid:48) ( φ ) b ( φ ) , (11)where b ( φ ) is the radius of the throat evaluated at the brane position. In the following we assumethat the time evolution of (cid:15) and c s is very slow with respect to that of the scale factor [108], asquantified by the slow-varying parameters η ≡ ˙ (cid:15)H(cid:15) (cid:28) , (12) s ≡ ˙ c s Hc s (cid:28) , (13)so that z (cid:48)(cid:48) /z (cid:39) z (cid:48)(cid:48) s /z s (cid:39) /τ . The amplification of the vacuum fluctuations at horizon crossingis possible only for very light degrees of freedom. Although this is automatically verified for theadiabatic perturbation v σ because of our assumption z (cid:48)(cid:48) /z (cid:39) /τ , this is not necessary true for v s ; if µ s is larger than H , this amplification is suppressed and there is no production of entropymodes. Note that the effective entropic mass squared (11) is non zero despite the angular directionbeing exactly isometric. In particular, because the potential and radius of the throat typicallyincrease and f typically decreases with φ , the first two terms in (11) are positive. Below we assumethat | µ s | /H (cid:28) v σ k (cid:39) v s k (cid:39) √ kc s e − ikc s τ (cid:18) − ikc s τ (cid:19) . (14)As a consequence, the power spectra for v σ and v s after sound horizon crossing have the sameamplitude. However, in terms of the initial field perturbations, one finds, using (8), P Q φ ∗ (cid:39) (cid:18) H ∗ π (cid:19) , P b ∗ Q θ ∗ (cid:39) (cid:18) H ∗ πc s ∗ (cid:19) (15)(the subscript ∗ indicates that the corresponding quantity is evaluated at sound horizon crossing kc s = aH ). Therefore, for small c s ∗ , the entropic modes are amplified with respect to the adiabaticmodes. As we will discuss in subsection II C, the standard formulae of the δN formalism areexpressed in terms of fields whose perturbations have the canonical amplitude H ∗ / π at horizoncrossing. We therefore define the ’canonical’ entropy field asΞ ≡ b ∗ c s ∗ θ . (16) B. The conversion process
We now address the question of how initially entropic perturbations can be converted into theadiabatic modes. We assume that inflation does not end not by the breakdown of the slow-rollconditions but rather persists all along down the throat. Then, when the D3-brane comes withina string length of the anti D3-brane, a tachyonic instability arises which ends inflation. UsingEqs. (1), (2) and (7), this happens when1 h (cid:0) (∆ ρ ) + b (∆ ψ ) (cid:1) = l s , (17)where ∆ ρ and ∆ ψ are the radial and angular D − D l s = m − s is the string length.In terms of the rescaled fields, the tachyon surface, represented in Figs. (1) and (2), is given by anti D3-braneD3-brane Tachyon surface FIG. 1: A simplified picture of the geometry at the tip of the throat, with one angular direction θ only.The radial inflationary trajectory is represented by the blue line. [109] φ + b θ = T l s h . (18)The main point is that the end value of the inflaton acquires a spatial dependence through thefluctuations of the light angular field θ (see Fig. (2)). Consequently, the duration of inflation variesfrom one super-Hubble region to another and this can be interpreted as a curvature perturbation,as we will quantify in the next subsection. To simplify the notation, we define φ c ≡ (cid:112) T l s h tip = m s h tip (2 π ) / g / s (19)and the angle − π/ < α < π/ φ e is given by φ e = cos( α ) φ c . (20)(here and in the following, the subscript e indicates the end of infation). For example, the D − D α = 0.Let us now make some remarks regarding the validity of our scenario: clearly the D ψ verifies ∆ ψ ≤ l s h tip b tip . (21)0 Tachyon surface BackgroundtrajectoryPerturbed trajectory
FIG. 2: The tachyon surface, at which inflation ends, in the φ − θ plane. The end value of the inflaton isshifted from φ e to φ e + δφ e due to the angular fluctuation Q θ , hence the duration of inflation varies fromone super-Hubble region to another. For the supergravity approximation to be valid, the radius at the tip b tip must be large in localstring units h tip l s (it is ∼ √ g s M in the KS throat with M (cid:29) C. δN formulae In order to quantify the curvature perturbation generated by the entropic fluctuations, it isconvenient to use the δN formalism [43, 44, 45, 46, 47], in which a key role is played by the local1integrated expansion, or local number of e-folds, between some initial and final hypersurfaces N ( x ) = (cid:90) fi H ( t, x ) dt . (22)In this formalism, the curvature perturbation on uniform energy density hypersurfaces, which wedenote ζ , is identified as the perturbation in the local number of e-folds of expansion from aninitially flat hypersurface to a final uniform energy density hypersurface ζ = δN ≡ N ( x ) − ¯ N , (23)where ¯ N is the number of e-folds in the homogeneous background spacetime. In the long-wavelengthlimit, according to the separate universe picture [48], the integrated expansion can be calculatedfrom solutions to the unperturbed Friedmann equation, with initial conditions specified by the perturbed scalar fields ζ = N ( ϕ A ∗ ) − ¯ N , where ϕ A ∗ = ϕ A ∗ + Q A ∗ is the sum of the homogeneous valuesplus fluctuations of the scalar fields on the initial spatially-flat slice, which we take to be soon afterhorizon crossing during inflation. Taylor-expanding this relation in terms of the field fluctuationsleads to the formal expression ζ = N A Q A + 12 N AB Q A Q B + 16 N ABC Q A Q B Q C + . . . (24)Once N ( ϕ A ∗ ) is known, one can work out the coefficients N A , N AB , N ABC in (24) and determinethe curvature perturbation.In our model, since the background dynamics is solely determined by the radial scalar field φ ,the local number of e-folds, evaluated right after the end of inflation, simply reads N ( φ ∗ ( x ) , Ξ ∗ ( x )) = (cid:90) φ e (Ξ ∗ ) φ ∗ (cid:18) H ˙ φ (cid:19) dφ , (25)where the canonical entropy field Ξ was defined in (16) and, using Eqs. (18) and (19), the endvalue of the inflaton reads φ e (Ξ ∗ ) = (cid:115) φ c − (cid:18) βc s ∗ Ξ ∗ (cid:19) , (26)with β ≡ b tip b ∗ (27)being the ratio between the radius at the tip of the throat and at sound horizon crossing. Herewe assumed that no significant expansion is generated after the field reaches the tachyon surface,that is, the sudden end approximation [49]. If not negligible, the extra curvature perturbation2generated can be taken into account similarly to [39], though even at this stage, ζ may not havesettled down to its final value. For example, the further evolution of the tachyon can give othercontributions to ζ [50, 51]. This interesting aspect is outside the scope of this paper but it shouldbe borne in mind that it is present in general.From Eq. (25), one obtains ζ = ζ ∗ + ζ e (28)where ζ ∗ = − (cid:90) ¯ φ ∗ + Q φ ∗ ¯ φ ∗ (cid:18) H ˙ φ (cid:19) dφ = − (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∗ Q φ ∗ −
12 dd φ (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∗ Q φ ∗ −
13! d d φ (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∗ Q φ ∗ + . . . (29)and ζ e = (cid:90) φ e (¯Ξ+ Q Ξ ∗ )¯ φ e (¯Ξ) (cid:18) H ˙ φ (cid:19) dφ = (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e δφ e + 12 dd φ (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e δφ e + 13! d d φ (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e δφ e + . . . (30)are the contributions to the curvature perturbation from respectively, the epoch of horizon crossingand the end of inflation. Loosely speaking, ζ e is associated to the time delay generated by thefluctuation δφ e in Fig. (2), where we use the notation δφ e = φ (cid:48) e Q Ξ ∗ + 12 φ (2) e ( Q Ξ ∗ ) + 13! φ (3) e ( Q Ξ ∗ ) + . . . (31)and the derivatives of φ e (Ξ ∗ ) (26), given in Appendix 1, are evaluated on the background. Notethat in order to trust the perturbative expansion, the angular fluctuations must be small comparedto the background separation, namely βH ∗ /c s ∗ (cid:28) φ c .Adding (29) and (30) gives an expression of ζ of the form (24), where A, B = σ, s with Q σ ≡− Q φ ∗ and Q s ≡ Q Ξ ∗ , which are normalized to share the canonical amplitude H ∗ / π . To leadingorder in the slow-varying parameters η and s (12)-(13) and their time derivatives, only the firstterms remain in the expansions (29) and (30) so that the non-zero coefficients in (24) are then N σ = − √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12) ∗ N s = − φ (cid:48) e √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12) e (32) N ss = − φ (2) e √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e N sss = − φ (3) e √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (33)where we have used H/ ˙ φ = − (2 (cid:15)c s ) − / /M pl [16]. For completeness, we include in Appendix 1the full expansion of ζ , not restricting to leading order in the slow-varying parameters. Noticealso that they cannot be neglected in the computation of the scalar spectral index n s and runningnon-Gaussian parameter n NG as they give then the leading order result (see below).3 III. POWER SPECTRUM AND PRIMORDIAL NON-GAUSSIANITIES FROM THEBISPECTRUM
According to Eq. (24), the statistical properties of the curvature perturbation ζ are inheritedfrom those of the field fluctuations Q A at horizon crossing. For clarity, we first recall the results,determined in [16], for the field two-point and three-point functions in two-field DBI inflation,before calculating the power spectrum and bispectrum of ζ in the following subsections. Theinvestigation of the trispectrum will be the subject of section IV. We use the notations of [52]. A. Statistical properties of the field perturbations at horizon crossing
In Fourier space the power spectrum of the scalar field perturbations is defined by (cid:104) Q A k Q B k (cid:48) (cid:105) = C AB ( k )(2 π ) δ ( k + k (cid:48) ) . (34)To leading order in the field perturbations C AB ( k ) = H ∗ k δ AB , (35)where δ AB is the Kronecker delta-function. Notice that the cross-correlation between adiabatic andentropy modes is zero for the straight line background trajectory considered here as the couplingbetween them exactly vanishes in that case (see Eq. (10) as well as [16] for details).The bispectrum of the field perturbations is defined by (cid:104) Q A k Q B k Q C k (cid:105) ≡ B ABC ( k , k , k )(2 π ) δ ( k + k + k ) . (36)In slow-roll inflation, where the self interactions of the fields are suppressed by the flatness of thepotential, the bispectrum of the fields is small, both for single [53] and multifield inflation [54]. Onthe contrary, self-interactions are enhanced in models with non-standard kinetic terms [55], and inDBI inflation in the low sound speed limit in particular, with the result [16] B ABC ( k , k , k ) = H ∗ √ (cid:15) ∗ c s ∗ c s ∗ M pl b ABC ( k , k , k ) . (37)The fully adiabatic momentum dependent factor b ABC ( k , k , k ) is given by b σσσ ( k , k , k ) = 1 (cid:81) i k i K (cid:2) k k k − k ( k · k )(2 k k − k K + 2 K ) + perm . (cid:3) (38)where K = k + k + k and the ‘perm.’ indicate two other terms with the same structure as thelast term but permutations of indices 1, 2 and 3. Note also that it depends only on the norm of4the three wave-vectors as for instance, k · k = ( k − k − k ) due to momentum conservation.This is the standard result from single field DBI inflation [56]. In the relativistic limit, there existsonly one other non-zero three-point correlation function at leading order, namely b σss ( k , k , k ) = 1 (cid:81) i k i K (cid:2) k k k + k ( k · k )(2 k k − k K + 2 K ) − k ( k · k )(2 k k − k K + 2 K ) − k ( k · k )(2 k k − k K + 2 K ) (cid:3) . (39) B. Power spectrum, scalar spectral index and tensor to scalar ratio
The power spectrum of the curvature perturbation is defined as (cid:104) ζ k ζ k (cid:48) (cid:105) = P ζ ( k )(2 π ) δ ( k + k (cid:48) ) . (40)The corresponding variance per logarithmic interval in k -space is given, to leading order in thefield perturbations, by P ζ ( k ) ≡ k π P ζ ( k ) = (cid:18) H ∗ π (cid:19) (cid:0) N σ + N s (cid:1) , (41)where we have used Eqs. (24) and (35). As in [16, 57], it is convenient to introduce the transferfunction T σs , such that N s = T σs N σ . The curvature power-spectrum then takes the form P ζ = 18 π (cid:15) ∗ c s ∗ H ∗ M (cid:0) T σs (cid:1) , (42)where T σs quantifies the contribution of the entropy modes to the final curvature perturbation.This vanishes in single-field DBI inflation while in our case, from Eq. (32), T σs = (cid:15) ∗ (cid:15) e c se c s ∗ tan ( α ) β . (43)Let us comment on the ranges of the various parameters entering Eq. (43). First, since the radiusof the throat decreases from the UV to the IR end, β ≡ b tip /b ∗ is bounded by one and this boundis saturated when the last 60 efolds of inflation take place at the tip of the throat, where b ( φ )becomes a constant [58]. Second, the entropic transfer depends on the angular D − D α (20). When tan( α ) = 0, the angular fluctuations give no time-delay to theend of inflation at linear order, as is clear from Fig. (2), and hence the transfer function vanishes.In that case, the first effect appears through higher order loop corrections (see Appendix 2). Inthe following, we have in mind that tan( α ) = O (1) although we keep formulae general. Finally,the result (43) indicates that in slow-roll inflation, where the speed of sound is one, the entropiccontribution to the curvature power spectrum can be significant, compared to the inflaton one,5only if the deceleration parameter (cid:15) is smaller at the end of inflation than at horizon crossing.In DBI inflation, however, the transfer function is amplified by the inverse of the product of thesound speed at horizon crossing and at the end of inflation, hence it is more efficient. Note thatthe enhancement of entropic perturbations by the inverse of the sound speed (15) was crucial inderiving this result.It is straightforward to calculate the scalar spectral index and tensor to scalar ratio from thepower spectrum. We find n s − ≡ d ln P ζ d ln k = − (cid:15) ∗ − η ∗ − s ∗ + T σs T σs (cid:32) η ∗ − s ∗ − b ∗ H ∗ b ∗ (cid:33) , (44)and [16] r = 16 (cid:15) ∗ c s ∗
11 + T σs ≈
16 1tan ( α ) 1 β (cid:15) e c se c s ∗ , T σs (cid:29) C. Primordial bispectrum
1. General definitions
The bispectrum of the curvature perturbation is defined as (cid:104) ζ k ζ k ζ k (cid:105) ≡ B ζ ( k , k , k )(2 π ) δ ( k + k + k ) , (46)where, from Eqs. (24), (34) and (36) and to leading order [59] B ζ ( k , k , k ) = N A N B N C B ABC ( k , k , k )+ N A N BC N D (cid:2) C AC ( k ) C BD ( k ) + C AC ( k ) C BD ( k ) + C AC ( k ) C BD ( k ) (cid:3) . (47)Observational quantities are usually expressed in terms of the dimensionless non-linearity parameter f NL – generally momentum-dependent – defined by B ζ ( k , k , k ) = 65 f NL [ P ζ ( k ) P ζ ( k ) + perm . ] . (48)Hence there are two contributions to f NL : the first, coming from the first term in Eq. (47), isrelated to the three-point functions of the fields at horizon crossing, and we will denote it as f (3) NL : f (3) NL = 56 N A N B N C B ABC ( k , k , k )( P ζ ( k ) P ζ ( k ) + perm . ) . (49)6The second, coming from the second group of terms in Eq. (47), comes from the leading ordernonlinear relation between the curvature perturbation and the field perturbations, and we willdenote it as f locNL [47]: f locNL = 56 N A N B N AB ( N C N C ) , (50)the total f NL being the sum of the two f NL = f (3) NL + f locNL .
2. Equilateral and local bispectra
Let us first discuss f (3) NL : if inflation is of slow-roll type when the observables modes cross thehorizon – c s ∗ ≈ f (3) NL is negligibly small. We therefore concentrate on the relativistic regime c s ∗ (cid:28)
1, in which case (37), (38), (39) and (41) give [15] f (3) NL = − c s ∗ (1 + T σs ) (cid:18) b σσσ ( k , k , k ) (cid:81) i k i (cid:80) i k i (cid:19) . (51)Here we used the symmetry property b σss ( k , k , k ) + b sσs ( k , k , k ) + b ssσ ( k , k , k ) = b σσσ ( k , k , k ), which implies that the shape dependence of f (3) NL is the same as in single-fieldDBI, as can be understood in a geometrical way [21]. In the equilateral limit k = k = k , f eqNL = − c s ∗
11 + T σs , (52)where T σs is given in (43) in our model. Hence the entropic to curvature transfer in multifieldDBI inflation diminishes the amount of equilateral non-Gaussianities with respect to the singlefield case. This comes from the fact that the transfer not only enhances the bispectrum of ζ butit also enhances its power spectrum by the same amount. Since f eqNL is roughly the ratio of thethree-point function with respect to the square of the power spectrum, f eqNL is effectively reduced.We now turn to the local shape of the bispectrum: from the definition (50), we obtain [111] f locNL = 56 N s N ss ( N σ + N s ) . (53)When there is a large entropic transfer – N s (cid:29) N σ – (the case where the entropic transfer is smallis discussed in Appendix 2) Eq. (53) reduces to the single field entropic result f locNL = 56 N ss N s = − (cid:114) (cid:15) e c se φ (2) e M pl φ (cid:48) e , T σs (cid:29) , (54)where we have used Eqs. (32)-(33) in the last equality. Up to the factor √ c se which diminishes theeffect, this expression is identical to the slow-roll result [23] in the same limit. We have however a7concrete model at hand, for which one obtains f locNL = 53 1sin ( α ) cos( α ) (cid:114) (cid:15) e c se M pl φ c , T σs (cid:29) . (55)This shows that f locNL is always positive, in agreement with the discussion in [60], and can be sig-nificant, given that the inflaton field φ is highly sub-Planckian in brane inflation [27] and thereforethat M pl φ c (cid:29)
1. As we will see in subsection III D, it can even saturate the existing observationalbound and put constraints on the model.Let us stress that the bispectrum in our scenario displays a combination of two different shapes[112]. First, the classical nonlinear relation between the curvature perturbation and the lightentropic scalar field gives rise to local non-Gaussianities, that peak for squeezed triangles ( k (cid:28) k ≈ k ), and this is quantified by the parameter f locNL . This type of non-Gaussianities constantlyarises when the curvature perturbation is generated at the end of inflation through light fields otherthan the inflaton [49, 61, 62, 63, 64, 65, 66, 67, 68, 69]. Second, derivative interactions producequantum correlations at the epoch of horizon crossing between modes of comparable wavelengths.The associated non-Gaussian signal peaks for equilateral triangles in momentum space ( k ∼ k ∼ k ) and this is quantified by the parameter f eqNL . Note that as the local and equilateral signals havefairly orthogonal distributions in momentum space [70], observational bounds on each of them canbe used when they are both present, each one being almost blind to the other [113].
3. Running non-Gaussianities
Besides its amplitude and shape, the scale dependence of the primordial bispectrum is anotherprobe of the early universe physics, and recently it has been shown that combining CMB and largescale structure observations give interesting constraints on the running of non-Gaussianities [71, 72].In our scenario, while f locNL (55) is scale-independent, it is clear that f eqNL (52) is scale-dependent.This can be quantified by the running non-Gaussian parameter, defined as n NG ≡ ∂ ln | f eqNL ( k ) | ∂ ln k = − s ∗ − T σs T σs (cid:32) η ∗ − s ∗ − b ∗ H ∗ b ∗ (cid:33) , (56)where we have used Eqs. (43) and (52) in the second equality.As the speed of sound generally decreases with time in models in which the brane goes fromthe UV to the IR end of the throat, s ≡ ˙ c s /Hc s < n NG is positive if the entropy modesdo not feed the curvature perturbation. On the contrary, using the relation η = 2 (cid:15) − s − ˙ fHf valid8in the DBI regime (see [18] for details), one obtains, for a large transfer, n NG = − (cid:15) ∗ + 2 ˙ b ∗ H ∗ b ∗ + ˙ f ∗ H ∗ f ∗ , T σs (cid:29) . (57)Hence, if observable modes cross the horizon at the tip of the throat where b and f become constant,the running non-Gaussian index rather becomes negative. D. Implication of the results
In the recent paper [73], it was noticed that in some models, brane inflation ends by tachyonicinstability in the relativistic DBI regime, even if inflation is of slow-roll type when the observablemodes cross the horizon. In this subsection, we concentrate on this limit, namely c se (cid:28)
1, andadditionally assume that the curvature perturbation is mostly of entropic origin T σs (cid:29)
1. Notethat the deceleration parameter (cid:15) has been used until now only as a small parameter, quantifyinghow much the inflationary expansion is close to de-Sitter. However, in the DBI regime, it can berelated to the brane tension and the warp factor. Indeed, since (cid:15) ≡ − ˙ HH = ˙ φ c s H M , (58)when c s = 1 − f ˙ φ (cid:28)
1, this gives, with Eq. (5), (cid:15)c s = T h H M . (59)Therefore, from (43) and (59), our hypotheses imply the condition2 H e M T h (cid:15) ∗ c s ∗ tan ( α ) β (cid:29) f locNL given in (55), together with the definitions of φ c in Eq. (19) and T in Eq. (3), weobtain f locNL = 56 1sin ( α ) cos( α ) h tip m s H e . (61)To avoid stringy corrections, one requires that at least h tip m s H e (cid:38) P ζ = 2 πg s tan ( α ) (cid:18) βc s ∗ (cid:19) ( H ∗ H e ) ( m s h tip ) . (62)9Then, Eqs. (61) and (62) together lead to f locNL = 56 2 / (2 πg s ) / sin / (2 α ) 1 P ζ / (cid:18) βc s ∗ (cid:19) / (cid:18) H ∗ H e (cid:19) / . (63)For instance, taking g s = 0 . α = π/ P ζ = 2 . . − [3] gives f locNL (cid:39) (cid:18) βc s ∗ (cid:19) / (cid:18) H ∗ H e (cid:19) / . (64)Let us recall that the only hypotheses that we used to derive (64) are that the curvature pertur-bation is of entropic origin and that the end of infation takes place in the relativistic regime. It isin particular valid for any value of c s ∗ between zero and one. Furthermore, since the Hubble scaledecreases during inflation, namely H ∗ /H e >
1, then we find the lower bound f locNL ≥ β / . (65)Therefore, the WMAP5 observational constraint − < f locNL <
111 (95% CL) [3] implies the upperbound β ≡ b tip /b ∗ (cid:46) . f eqNL (cid:39) − . c s ∗ T σs = − . ( α ) (cid:15) e c se (cid:15) ∗ c s ∗ β . (66)With Eq. (59), this gives f eqNL (cid:39) − . ( α ) (cid:18) H ∗ H e (cid:19) β (cid:18) h tip h ∗ (cid:19) . (67)One needs a precise model to actually determine the amplitude of f eqNL . Let us simply commentthat if one tunes β ≡ b tip b ∗ to a small value to low down f locNL (64) in the observational range, thisenhances f eqNL . However, in that case, observable modes cross the horizon far from the tip of thethroat, and one expects a huge hierarchy (cid:16) h tip h ∗ (cid:17) (cid:28)
1, which tends to put f eqNL within the currentobservational bounds − < f eqNL <
253 (95% CL) [3].
IV. PRIMORDIAL NON-GAUSSIANITIES FROM THE TRISPECTRUM
As the next generation of experiments will be able to probe refined details of the statistics ofdensity fluctuations [75, 76, 77], the study of the primordial trispectrum is becoming increasinglyimportant. In DBI inflation, another motivation comes from the fact that f (3) NL acquires the samemomentum dependence in single- and multiple-field models. Therefore, we cannot observationally0differentiate between them with the bispectrum alone. As the degeneracy between models tendsto be broken as we go to higher-order correlation functions, the investigation of the trispectrum inmultifield DBI inflation is thus very natural. A. General definitions
The primordial trispectrum is defined as the connected part of the four-point correlation functionof the curvature perturbation in Fourier space (cid:104) ζ k ζ k ζ k ζ k (cid:105) c ≡ T ζ ( k , k , k , k )(2 π ) δ ( k + k + k + k ) . (68)We need twelve real numbers to specify a set of four three-dimensional momenta. However, momen-tum conservation, k + k + k + k = 0, eliminates three of them and invariance under rotationalsymmetry eliminate three others. We are thus left with six independent parameters that specifyinequivalent configurations of the tetrahedron formed by the four k vectors: we will use the set { k , k , k , k , k , k } where k = | k + k | and k = | k + k | . The others k ij = | k i + k j | canthen be reexpressed in terms of them as follows: k = k = (cid:113) k + k + k + k − k − k ,k = k , k = k . (69)Note that there are geometrical limitations on the parameter space, for instance in the form oftriangle inequalities (see [78] for details).From the general δN expansion (24), the primordial trispectrum can be evaluated and one finds[52] T ζ ( k , k , k , k ) = N A N B N C N D T ABCD ( k , k , k , k )+ N AB N C N D N E (cid:2) C AC ( k ) B BDE ( k , k , k ) + 11 perms (cid:3) + N AB N CD N E N F (cid:2) C BD ( k ) C AE ( k ) C CF ( k ) + 11 perms (cid:3) + N ABC N D N E N F (cid:2) C AD ( k ) C BE ( k ) C CF ( k ) + 3 perms (cid:3) (70)(where we have omitted special configurations of the wavevectors, for instance when any k vectoror the sum of any two k vectors is zero). Here the T ABCD are the connected four-point correlationfunctions of the field perturbations at the epoch of horizon crossing, defined by (cid:104) Q A k Q B k Q C k Q D k (cid:105) c ≡ T ABCD ( k , k , k , k )(2 π ) δ ( (cid:88) i k i ) . τ NL ≡ N AB N AC N B N C ( N D N D ) (71)and g NL ≡ N ABC N A N B N C ( N D N D ) (72)simplifies the expression of (70) to T ζ ( k , k , k , k ) = N A N B N C N D T ABCD ( k , k , k , k )+ N AB N C N D N E (cid:2) C AC ( k ) B BDE ( k , k , k ) + 11 perms (cid:3) + τ NL [ P ζ ( k ) P ζ ( k ) P ζ ( k ) + 11 perms]+ 5425 g NL [ P ζ ( k ) P ζ ( k ) P ζ ( k ) + 3 perms] . (73)In order to compare the various momentum dependences entering (73), we use the form factor T defined by [78] T ζ ( k , k , k , k ) = (2 π ) P ζ (cid:32) (cid:89) i =1 k i (cid:33) T ( k , k , k , k , k , k ) , (74)whilst the amplitude of the trispectrum signal is quantified by the estimator t NL for each shapecomponent 1 k T ( k , k , k , k , k , k ) component RT −−−→ limit t NL . (75)Here the regular tetrahedron (RT) limit means k = k = k = k = k = k ≡ k [22, 78].In the DBI regime of brane inflation, the quantum non-Gaussianities of the fields are large –as set by their bispectrum B ABC and connected part of their trispectrum T ABCD – and the firsttwo lines in Eq. (73) a priori gives sizeable contributions to the primordial trispectrum. Moreover,the last two lines in Eq. (73) could be significant as we will see later. Their respective amplitudeare determined by the two parameters τ NL and g NL , which are similar to f locNL for the bispectrumin that they describe the nonlinearities generated classically outside the horizon. In that sense wewill occasionally refer to them as the local trispectrum parameters. They contribute to the formfactor as T ⊃ τ NL T loc + g NL T loc , (76)2where the two local shapes are T loc = 950 (cid:18) k k k + 11 perms (cid:19) , (77) T loc = 27100 (cid:88) i =1 k i , (78)and the size of the trispectrum for each is t loc NL = 1 . τ NL , t loc NL = 1 . g NL . (79)We study the local and intrinsically quantum contributions to the trispectrum in the nextsubsection while the discussion of the second term in the right hand side of Eq. (73) is deferred tosubsection IV C. B. Local and intrinsically quantum trispectra in our model
We begin by considering the standard local parameters τ NL and g NL . From their definitionsEqs. (71) and (72), they are given by τ NL = N s N ss ( N σ + N s ) , g NL = 2554 N s N sss ( N σ + N s ) . (80)Notice that using Eqs. (104)-(106) and (32)-(33), the entropic derivatives satisfy N s N sss = 3 sin ( α ) N ss . (81)We therefore obtain g NL = 2518 sin ( α ) τ NL . (82)As τ NL and g NL are in principle observationally distinguishable [79], this gives us the possibilityto deduce the angle α from the observations. Given that g NL < τ NL , it should be stressedalso that one can not obtain a large g NL without having a large τ NL , contrary to the cases of thecurvaton [80], ekpyrotic models [81] or non slow-roll multifield inflation [82]. Specializing to thelimit in which the curvature perturbation is mostly of entropic origin, one obtains from Eq. (80)the single-field entropic result τ NL = N ss N s = (cid:18) f locNL (cid:19) , T σs (cid:29) f locNL is given in this limit by (55) (and the relation (82) still holds).3The intrinsic trispectra of the fields also leave their imprint on the primordial trispectrum,through the linear relation between the curvature perturbation and the field perturbations at hori-zon crossing (see the first term in Eq. (73)). The calculation of the field trispectra in multifieldmodels with non-standard kinetic terms, and in multifield DBI inflation in particular, begun onlyvery recently [19, 21, 22]. For example, the authors of [21, 22] considered the trispectra from theintrinsic fourth-order contact interaction. However, it was pointed out that there are other impor-tant contributions coming from interactions at a distance [19, 78, 83, 84]. As the full calculationof the field bispectra in multifield DBI inflation has not yet been completed, below we simply com-ment on the partial results coming from the contact interaction. The important result is that thedegeneracy between single and multiple-field models is broken at this level, i.e. with our notationsone obtains T ζ ( k , k , k , k ) ⊃ N σ (cid:18) T σσσσ ( k , k , k , k ) + T σs (cid:16) T σσss ( k , k , k , k ) + 5 perms (cid:17) + T σs T ssss ( k , k , k , k ) (cid:19) = N σ (1 + T σs ) (cid:18) T σσσσ ( k , k , k , k ) + T σs T ssss ( k , k , k , k ) (cid:19) (84)where the last line follows from a particular relation between the mixed term T σσss and the adia-batic and entropic trispectra, which have the same order of magnitude ∼ N σ H ∗ /c s ∗ but differentmomentum dependence. Therefore the trispectrum can a priori discriminate between single- andmultifield DBI inflation. In particular, one can hope that looking at the trispectrum signal indifferent limits of the tetrahedron’s parameter space might enable a measurement of T σs . As forthe overall amplitude of these terms though, with P ζ = (cid:0) H ∗ π (cid:1) N σ (1 + T σs ) (41), this schematicallygives a contribution to t NL of the form t NL ∼ P ζ N σ H ∗ c s ∗ N σ (1 + T σs )(1 + a T σs ) ∼ c s ∗ (1 + T σs ) 1 + a T σs (1 + T σs ) (85)where a is a O (1) numerical coefficient. Similarly to the case of the bispectrum, notice that the fieldtrispectra having the same amplitude implies that the transfer function T σs solely determines themultiple-field modification to the single-field result t NL ∼ c s ∗ [85]. Therefore, even if one does notsucceed extracting T σs from measurements of the trispectrum alone – for example because T σs (cid:29) f eqNL ≈ − . c s ∗ (1+ T σs ) todetermine both c s ∗ and T σs . This illustrates how the study of higher-order correlation functionscan be used to extract information one can not obtain with the power spectrum alone.4 C. An observational signature of multifield DBI inflation
Going back to the general expression Eq. (73), one sees that the intrinsic bispectrum of thefields also contribute to the primordial trispectrum of the curvature perturbation, i.e. T ζ ( k , k , k , k ) ⊃ N AB N C N D N E (cid:2) C AC ( k ) B BDE ( k , k , k ) + 11 perms (cid:3) . (86)Remarkably, this term is solely determined by quantities that already appeared in lower-ordercorrelation functions. In particular, while N AB determine the amplitude of f locNL (50), the intrinsicbispectra of the fields B ABC set the magnitude of f (3) NL (49). Therefore, this contribution can benon negligible only if significant classical nonlinearities and quantum non-Gaussianities are presentin the same model, which explains why it has been neglected before and why we refer to it as T loc , eq ζ in the following. Note that since the shape of the field three-point functions does nottake a universal form, one cannot extract a number that characterizes the amplitude of T loc , eq ζ independently of a model, contrary to τ NL (71), which also depends only on lower-order terms.In our scenario, T loc , eq ζ reads T loc , eq ζ = (cid:2) N σσ N σ C σσ ( k ) B σσσ ( k , k , k ) + N σσ N σ N s C σσ ( k ) B σss ( k , k , k )+ N ss N σ N s C ss ( k ) ( B ssσ ( k , k , k ) + B sσs ( k , k , k )) + 11 perms (cid:3) . (87)Notice that the first two terms, being proportional to N σσ , are negligible to leading order in theslow-varying parameters and we are left with the second line alone at this order. We have includedthe first line only to stress that the trispectrum breaks the degeneracy between the two shapes b σσσ and b σss entering the bispectrum, defined respectively in Eqs. (38) and (39). Recall indeedthat we used the symmetry property b σss ( k , k , k ) + b sσs ( k , k , k ) + b ssσ ( k , k , k ) = b σσσ ( k , k , k ) (88)to show that the multifield effects do not modify the shape of f (3) NL compared to the single-fieldcase. Here, one sees that because N σσ (cid:54) = N ss in general, one can not use this identity and theprimordial trispectrum truly depends on the shape b σss , not on its symmetrized version b σσσ .Let us now concentrate on the leading-order term, namely the second line in Eq. (87), and writeits contribution to the form factor as T ⊃ s NL T loc, eq , (89)where T loc, eq ≡ k k k ) [ b ssσ ( k , k , k ) + b sσs ( k , k , k )] + 11 perms . (90)5Here, s NL is a dimensionless non-linearity parameter that set the amplitude of T loc, eq , similarly to τ NL and g NL for T loc and T loc respectively (see Eq. (76)), and the numerical factor in the definitionof T loc, eq is for convenience only. The overal amplitude of this contribution to the trispectrum is t loc, eqNL = 0 . s NL . (91)By noting that (37) can be rewritten in the form B ABC ( k , k , k ) = − N σ H ∗ c s ∗ b ABC ( k , k , k ) , (92)from (87), one formally obtains s NL = − N ss N σ c s ∗ T σs (1 + T σs ) , (93)which is exactly the product of f locNL (53) and f eqNL (52) s NL = f locNL f eqNL . (94)It is worth emphasizing that, in order to derive the result (94), we solely used the δN formalismtogether with the general results for the field bispectra [16]. Hence it is clear that it is valid forevery DBI model, not restricting to a radial trajectory nor to a specific scenario for the entropy tocurvature transfer. In particular, we considered the two-field case for simplicity of presentation butthe proof generalizes straightforwardly to a higher number of light fields. The consistency relation(94) is thus structural to multifield DBI inflation. It represents a distinct observational imprint onthe trispectrum of the presence of light scalar fields, other than the inflaton, and with non standardkinetic terms, hence the appearance of both the local and equilateral non-linearity parameters. D. Shapes of trispectra
To test the consistency relation (94), one must observationally disentangle the contributionproportional to s NL from other components of the trispectrum. The study of its six-dimensionalparameter space is challenging and in the following, we follow the discussion in [78] and simplyplot the form factor T loc, eq in various limits to reduce the number of variables, and compare it to T loc and T loc . The shape functions are left white when the momenta do not form a tetrahedron.We consider the following cases (see [78] for details):1. Equilateral limit: k = k = k = k . In Fig. 3, we plot T loc, eq , T loc and T loc as functionsof k /k and k /k . One observes that T loc, eq and T loc blow up at all boundaries, cor-responding to k /k , k /k and k /k →
0. However, the signal in T loc, eq is much more6important, as is clear from the extremely large cutoff for the z-axis in this figure. On thecontrary, T loc is constant in this region of parameter space.2. Specialized planar limit: we take k = k = k , and the tetrahedron to be a planar quad-rangle with [78] k = (cid:20) k + k k k (cid:18) k k + (cid:113) (4 k − k )(4 k − k ) (cid:19)(cid:21) / . (95)We plot the shape functions as functions of k /k and k /k in Fig. 4. At the k → k limit,we have k →
0, so that T loc, eq and T loc blow up. Again, the signal in T loc, eq is muchmore important and we had to use an extremely large cutoff for the z-axis for the sake ofpresentation. The most distinctive difference comes from the k → k → T loc and T loc take finite (non-zero) values whereas T loc, eq blows up. Moreover, itssign alternates non-trivially over the parameter space. Interestingly, this feature was notpresent in any other shape studied in [78].3. Near the double-squeezed limit: we consider the case where k = k = k and the tetrahe-dron is a planar quadrangle with [78] k = (cid:113) k (cid:0) − k + k + k (cid:1) − k s k s + k k + k k + k k − k k − k + k k √ k , (96)where k s and k s are defined as k s ≡ (cid:112) ( k k + k · k )( k k − k · k ) ,k s ≡ (cid:112) ( k k + k · k )( k k − k · k ) . (97)We plot T loc, eq , T loc and T loc as functions of k /k and k /k in Fig. 5. Similarly tothe specialized planar limit, notice that the sign of T loc, eq varies non trivially over thedomain. Several differences between the shapes are visible. 1) In the squeezed limit, at( k /k = 1 , k /k = 1) where k →
0, and in the double-squeezed limit, k = k → T loc, eq blows up while the local shapes are finite. 2) In the folded limit ( k /k = 1 , k /k = 0),both T loc, eq and T loc blow up and T loc takes a finite value. 3) In the other folded limit,( k /k = 1 , k /k = 2), all shapes remain finite. Close to it, note that the bump in T loc actually remains finite while T loc, eq truly blows up.To summarize, although we did not make an exhaustive study of the shape factor T loc, eq , we haveseen that it presents characteristic features. In particular, in the last two envisaged situations where7 FIG. 3: In this group of figures, we consider the equilateral limit k = k = k = k , and plot T loc, eq , T loc and T loc , respectively, as functions of k /k and k /k . Note that T loc, eq and T loc blow up when k (cid:28) k and k (cid:28) k , as well as in the other boundary, corresponding to k (cid:28) k . FIG. 4: In this group of figures, we consider the specialized planar limit with k = k = k , and plot T loc, eq , T loc and T loc , respectively, as functions of k /k and k /k . Along the diagonal k → k , T loc, eq and T loc blow up because in this limit, k →
0. At the k → k → T loc, eq blow upwhile the local shapes remain finite. Notice that the sign of T loc, eq varies non trivially over the parameterspace. FIG. 5: In this group of figures, we look at the shapes near the double squeezed limit: we consider thecase where k = k = k and the tetrahedron is a planar quadrangle. We plot T loc, eq , T loc and T loc ,respectively, as functions of k /k and k /k . Notice that the sign of T loc, eq varies non trivially over thedomain. In the squeezed limit, at ( k /k = 1 , k /k = 1) where k →
0, and in the double-squeezed limit, k = k → T loc, eq blows up while the local shapes are finite (see the main text for additional comments). V. CONCLUSIONS
In this paper, we have analyzed a scenario to convert entropic into curvature perturbations inthe context of multifield brane inflation, and multifield DBI inflation in particular. Considering astandard single-field driven inflationary phase ending by tachyonic instability, we have shown howthe perturbations of light fields, parametrizing the angular position of the brane in the throat, cangenerate curvature perturbation by spatially modulating the duration of inflation. The entropictransfer generated by this mechanism is more efficient in the DBI than in the slow-roll regime,being enhanced by the inverse of the sound speed at horizon crossing and at the end of inflation.We have investigated the non-Gaussianities created in such models. Single-field DBI inflationis known to produce a substantial amount of non-Gaussianities of equilateral type and previousworks on multifield DBI inflation showed that the entropic perturbations reduce the amplitude f eqNL of these non-Gaussianities. However, as soon as light scalar fields other than the inflaton arepresent, non-Gaussianities of another shape, namely local non-Gaussianities, can be present. Inour model, they arise because of the nonlinear relation between the curvature perturbation andthe angular ones. This local contribution f locNL to the non-linearity parameter f NL can be large andeven saturate the observational bounds for some specific scenarios.We have also calculated τ NL and g NL , which are trispectrum parameters similar to f locNL for thebispectrum, in that they describe the nonlinearities generated classically outside the horizon. Wehave shown that g NL is related to τ NL by a simple parameter describing the angular position ofthe mobile brane, implying g NL < τ NL . When the curvature perturbation is mostly of entropicorigin, τ NL = (cid:0) f locNL (cid:1) , easily reaching the expected Planck sensitivity | τ NL | ∼
560 [75] as soon as f locNL (cid:38)
25. The trispectrum also contains a part that is directly related to the trispectra of the fieldperturbations. One can in principle extract the entropic transfer function from its complicatedmomentum dependence, as well as determine the sound speed when the observable modes crossthe horizon by combining it with measurements of f eqNL .Finally, due to the presence of light scalar fields, other than the inflaton, and with non standardkinetic terms, the primordial trispectrum acquires a particular momentum dependent component1whose amplitude (94) is given by the product f locNL f eqNL . Moreover, this consistency relation is validin every multifield DBI model, whatever the brane’s trajectory or the mechanism to convert entropicperturbations into the curvature perturbation. It thus represents an interesting observationalsignature of multifield DBI inflation. We have represented the corresponding form factor in differentlimits and it turned out to display important differences with the shapes associated to τ NL and g NL ,such as its sign varying over the tetrahedron’s parameter space. Overall, should future experimentscome to detect both local and equilateral non-Gaussianities in the primordial bispectrum, thetrispectrum may well help to to confirm or exclude this kind of scenarios. Note added : On the day this work was submitted, the paper [86] appeared in the arXiv, whichcompletes the calculation of the trispectrum coming from the four-point functions of the fieldperturbations in multifield DBI inflation.
Acknowledgements
We would like to thank Eiichiro Komatsu, David Langlois, Liam McAllister, Fransesco Nitti,Sarah Shandera and Daniele Steer for valuable discussions related to the topic of this paper, andparticularly David Langlois and Daniele Steer for their careful reading of the manuscript.
Appendix 1 - δN formulae beyond leading order Using the expressions of ζ ∗ (29) and ζ e (30) together with the definitions of η (12) and s (13), one obtains the coefficients of the δN expansion without restricting to leading order in theseslowly-varying parameters and their time derivatives: N σ = − √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12) ∗ (98) N s = − φ (cid:48) e √ (cid:15)c s M pl (cid:12)(cid:12)(cid:12)(cid:12) e (99) N σσ = η + s (cid:15)c s M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ (100) N ss = − (cid:32) φ (2) e √ (cid:15)c s M pl + η + s (cid:15)c s φ (cid:48) e M (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (101)2 N σσσ = 12 / ( (cid:15)c s ) / M (cid:18) ˙ ηH + ˙ sH − ( η + s ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ (102) N sss = − (cid:32) φ (3) e √ (cid:15)c s M pl + 3 ( η + s )4 (cid:15)c s φ (cid:48) e φ (2) e M + φ (cid:48) e / ( (cid:15)c s ) / M (cid:18) ˙ ηH + ˙ sH − ( η + s ) (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (103)The derivatives of φ e (Ξ ∗ ) (26) are evaluated at the background value ¯Ξ = b ∗ c s ∗ ¯ θ and theirexplicit expressions read φ (cid:48) e = − tan( α ) βc s ∗ , (104) φ (2) e = − ( α ) (cid:18) βc s ∗ (cid:19) φ c , (105) φ (3) e = − α )cos ( α ) (cid:18) βc s ∗ (cid:19) φ c . (106) Appendix 2 - Higher order loop contributions
Given we encountered large entropic derivatives in the δN formalism, one must ensure that weare in a simple perturbative regime in which one can safely neglect higher order loop corrections(see [87, 88] for discussions on the viability of a perturbative expansion and [89] for a diagrammaticapproach to loop effects). The integral over the loop momenta give rise to logarithmic infrareddivergences ln( kL ) where L is the large scale cut off. In the following we assume that one can takeln( kL ) ∼ δN expansion (24) to second order forsimplicity, and neglecting for the moment the non-Gaussianities of the fields at horizon crossing,the dominant loop corrections to the power spectrum, bispectrum and trispectrum are [91, 92] P loopζ P treeζ = N AB N AB ( N C N C ) P ζ , (107) f NL = 56 N AB N BC N AC ( N D N D ) P ζ , (108) τ NL = N AB N BC N CD N AD ( N E N E ) P ζ . (109)Below we discuss the two limiting cases of a large and small entropic transfer. • Large entropic transfer. P loopζ P treeζ = (cid:18) N ss N s (cid:19) P ζ = (cid:18) f locNL (cid:19) P ζ , (110) f NL = 56 (cid:18) N ss N s (cid:19) P ζ = 3625 (cid:16) f locNL (cid:17) P ζ , (111) τ NL = (cid:18) N ss N s (cid:19) P ζ = (cid:18) f locNL (cid:19) P ζ . (112)Therefore, given that P ζ ∼ − and that observations already show that the level of non-Gaussianities is relatively small, f locNL = O (100), this demonstrates that loop corrections canbe neglected, and clearly the argument is valid for every scenario in which one Gaussian fieldonly is responsible for the curvature perturbation [93].In general, the field non-Gaussianities lead to additional loop contributions, for instanceto f NL . This was shown to be completely negligible for slow-roll models [94] but in DBIinflation where intrinsic non-Gaussianities are large, one has to determine if this remainstrue. In particular, there is a contribution to the primordial trispectrum B ζ (46) from termsof the form N A N BC N DE (cid:104) Q A ( k )( Q B (cid:63) Q C )( k )( Q D (cid:63) Q E )( k ) (cid:105) , (113)where the symbol (cid:63) denotes a convolution product. The most dangerous terms in (113)involve the entropic derivatives and, as to leading order there is no purely entropic threepoint function, one is led to consider N σ N ss (cid:104) Q σ ( k )( Q s (cid:63) Q s )( k )( Q s (cid:63) Q s )( k ) (cid:105) . (114)From the definition of f NL (48) as well as the result (37) for the three point functions of thefields, its contribution is of order f NL ⊃ (cid:18) N σ N ss H ∗ H ∗ √ (cid:15) ∗ c s ∗ c s ∗ M pl (cid:19) (cid:14)(cid:18) H ∗ N σ (cid:0) T σs (cid:1) (cid:19) . (115)For a large entropic transfer, one therefore obtains f NL ⊃ f eqNL ( f locNL ) P ζ , (116)where we have used (52) and (53). Again, the observational bounds f eqNL , f locNL = O (100) aswell as the normalization P ζ ∼ − show that such a loop-correction is negligible and onecan verify that the same conclusion applies to other higher order corrections.4 • Small entropic transfer
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