Primordial fluctuations and non-Gaussianities in multi-field DBI inflation
David Langlois, Sebastien Renaux-Petel, Daniele A. Steer, Takahiro Tanaka
aa r X i v : . [ h e p - t h ] A p r Primordial fluctuations and non-Gaussianities in multi-field DBI inflation
David Langlois , S´ebastien Renaux-Petel , Dani`ele A. Steer , and Takahiro Tanaka APC, UMR 7164, 10 rue Alice Domon et L´eonie Duquet,75205 Paris Cedex 13, France CERN Physics Department, Theory Division, CH-1211 Geneva 23, Switzerland and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan. (Dated: November 3, 2018)We study Dirac-Born-Infeld (DBI) inflation models with multiple scalar fields. We show that theadiabatic and entropy modes propagate with a common effective sound speed and are thus amplifiedat the sound horizon crossing. In the small sound speed limit, we find that the amplitude of theentropy modes is much higher than that of the adiabatic modes. We show that this could stronglyaffect the observable curvature power spectrum as well as the amplitude of non-Gaussianities, al-though their shape remains as in the single-field DBI case.
The last decade has seen an accumulation of cosmo-logical data of increasing precision. Together with futureexperiments planned to measure the CMB fluctuationswith yet further accuracy, we may be able to piece to-gether more clues about early universe physics. In paral-lel with this observational effort, there has been tremen-dous progress in recent years in the construction of earlyuniverse models in the framework of high energy physicsand string theory.A particularly interesting class of models based onstring theory is known as DBI inflation [1, 2], associatedwith the motion of a D3-brane in a higher-dimensionalbackground spacetime. The characteristic of DBI infla-tion, and that which gives it its name, is that the actionis of the Dirac-Born-Infeld (DBI) type and thus containsnon-trivial kinetic terms. Most studies of DBI inflationmodels (or even of string based inflationary models) haveso far concentrated on a single-field description meaning,in the DBI case, that the inflaton corresponds to a radialcoordinate of the brane in the extra dimensions. Takinginto account the “angular” coordinates of the brane nat-urally leads to a multi-field description since each branecoordinate in the extra dimensions gives rise to a scalarfield from the effective four-dimensional point of view.This setup has started to be explored only very recently[3, 4].In this Letter, we show that the multi-field
DBI actioncontains some terms, higher order in space-time gradi-ents and vanishing in the homogeneous case, which havebeen overlooked. The inclusion of these terms leads todrastic consequences on the primordial fluctuations gen-erated in these types of models. The scalar-type pertur-bations in multi-field models can be divided into (instan-taneous) adiabatic modes, fluctuations along the trajec-tory in field space, and entropy modes which are orthog-onal to the former [5]. In contrast with previous expec-tations, we show that in DBI models, these two classesof modes propagate with the same speed, namely an ef-fective speed of sound c s smaller than the speed of light.As a consequence, the amplification of quantum fluctua-tions occurs at the sound horizon crossing for both typesof modes. Moreover, when c s ≪
1, this leads to an en-hancement of the amplitude of the entropy modes withrespect to that of the usual adiabatic modes. As primor- dial non-Gaussianities — potentially detectable in forth-coming experiments if strong enough — discriminate be-tween various models, we also study the impact of theentropy modes on non-Gaussianity in the DBI case.Our starting point is the DBI Lagrangian governingthe dynamics of a D3-brane, L DBI = − f q − det ( g µν + f G IJ ∂ µ φ I ∂ ν φ J ) , (1)where f = f ( φ I ) is a function of the scalar fields φ I ( I =1 , , . . . ), and G IJ ( φ K ) is a metric in field space. From ahigher-dimensional point of view, (1) is proportional tothe square root of the determinant of the induced metricon the brane, meaning that the φ I correspond to thebrane coordinates in the extra dimensions, f embodiesthe warp factor, and G IJ is (up to a scale factor) themetric in the extra dimensions. We also allow for thepresence of a potential and hence consider a full actionof the form S = Z d x √− g (cid:20) R P (cid:21) ,P = − f ( φ I ) (cid:16) √D − (cid:17) − V ( φ I ) , (2)where we have set 8 πG = 1. The determinant D =det( δ µν + f G IJ ∂ µ φ I ∂ ν φ J ) coming from Eq. (1) can berewritten as D = det( δ JI − f X JI )= 1 − f G IJ X IJ + 4 f X [ II X J ] J − f X [ II X JJ X K ] K + 16 f X [ II X JJ X KK X L ] L , (3)where we have defined X IJ ≡ − ∂ µ φ I ∂ µ φ J , X JI = G IK X KJ , (4)and where the brackets denote antisymmetrisation of thefield indices. In the single-field case, I = 1, the terms in f , f and f in (3) vanish. This is also true for mul-tiple homogeneous scalar fields for which X IJ = ˙ φ I ˙ φ J .However, for multiple inhomogeneous scalar fields, theseterms, which are higher order in gradients and have notbeen considered in previous works, do not vanish. Wenow show that they change drastically the behaviour ofperturbations.In order to study the dynamics of linear perturbationsabout a homogeneous cosmological solution, we expandthe initial action (2) to second order in the linear pertur-bations, including both metric and scalar field perturba-tions. This is a constrained system, and the number of(scalar) degrees of freedom is the same as the number ofscalar fields. It is convenient to express these degrees offreedom in terms of the scalar perturbations defined inthe flat gauge, usually denoted Q I . To obtain the second-order action, we follow the procedure outlined in [6] for aLagrangian of the form P ( X, φ J ), with X = G IJ X IJ : aswe have stressed above the multi-field DBI Lagrangian is not of this form, but despite that it can be rewritten as˜ P ( ˜ X, φ K ) = − f (cid:18)q − f ˜ X − (cid:19) − V , (5)where ˜ X = (1 − D ) / (2 f ). Although in the homogeneousbackground ˜ X and X coincide, their perturbed values dif-fer. Taking into account the corresponding extra terms,one can show [7] that the second-order action can bewritten in the compact form S (2) = 12 Z d t d x a h ˜ P , ˜ X ˜ G IJ D t Q I D t Q J − c s a ˜ P , ˜ X ˜ G IJ ∂ i Q I ∂ i Q J −M IJ Q I Q J + 2 ˜ P , ˜ XJ ˙ φ I Q J D t Q I i . (6)Here a is the scale factor; the effective (squared) massmatrix is M IJ = −D I D J ˜ P − ˜ P , ˜ X R IKLJ ˙ φ K ˙ φ L + X ˜ P , ˜ X H ( ˜ P , ˜ XJ ˙ φ I + ˜ P , ˜ XI ˙ φ J ) + ˜ X ˜ P , ˜ X H (1 − c s ) ˙ φ I ˙ φ J − a D t (cid:20) a H ˜ P , ˜ X (cid:18) c s (cid:19) ˙ φ I ˙ φ J (cid:21) , (7)and we have introduced covariant derivatives D I definedwith respect to the field space metric G IJ , as well as thetime covariant derivative D t Q I = ˙ Q I + Γ IJK ˙ φ J Q K whereΓ IJK is the Christoffel symbol constructed from G IJ and R IKLJ is the corresponding Riemann tensor. Finally, wehave defined the (background) matrix˜ G IJ = G IJ + 2 f X − f X e σI e σJ = ⊥ IJ + 1 c s e σI e σJ , (8)where e Iσ = ˙ φ I / √ X ( ˙ σ ≡ √ X is also used in the fol-lowing) is the unit vector pointing along the trajectory infield space, ⊥ IJ ≡ G IJ − e σI e σJ is the projector orthog-onal to the vector e Iσ , and c s ≡ ˜ P , ˜ X ˜ P , ˜ X + 2 ˜ X ˜ P , ˜ X ˜ X = 1 − f ˙ σ . (9) Let us stress that the only difference between action(6) and the corresponding expression in [6] is the termin spatial gradients, with coefficient c s ˜ P , ˜ X ˜ G IJ insteadof ˜ P ,X G IJ . This crucial difference shows that all per-turbations, both adiabatic and entropic, propagate withthe same speed of sound in multi-field DBI inflation, incontrast with [3, 4, 6] where they have different speeds.Finally, one should recall that the above expressions ap-ply to the DBI context where ˜ P is given in (5) so that˜ P , ˜ X = 1 /c s . For simplicity, let us now restrict our attention to twofields ( I = 1 , Q I = Q σ e Iσ + Q s e Is , where e Is , the unit vec-tor orthogonal to e Iσ , characterizes the entropy direction.(For N fields, the entropy modes would span an ( N − τ = R d t/a ( t ), to work in terms of the canonicallynormalized fields v σ ≡ ac s q ˜ P , ˜ X Q σ , v s ≡ a q ˜ P , ˜ X Q s . (10)Note that the adiabatic and entropy coefficients differbecause ˜ G IJ is anisotropic. The equations of motion for v σ and v s then follow from the action (6), reexpressed interms of the rescaled quantities (10). One finds v ′′ σ − ξv ′ s + (cid:18) c s k − z ′′ z (cid:19) v σ − ( zξ ) ′ z v s = 0 , (11) v ′′ s + ξv ′ σ + (cid:18) c s k − α ′′ α + a µ s (cid:19) v s − z ′ z ξv σ = 0 , (12)where ξ = a ˙ σ ˜ P , ˜ X c s [(1 + c s ) ˜ P ,s − c s ˙ σ ˜ P , ˜ Xs ] , (13) z = a ˙ σc s H q ˜ P , ˜ X , α = a q ˜ P , ˜ X , (14)and µ s follows from the mass matrix (7) (see [6, 7] fordetails). We will assume that the effect of the coupling ξ can be neglected when the scales of interest cross outthe sound horizon , so that the two degrees of freedom aredecoupled and the system can easily be quantized. In the slow-varying regime, where the time evolution of H , c s and ˙ σ is small with respect to that of the scale factor, onegets z ′′ /z ≃ /τ and α ′′ /α ≃ /τ . The solutions of (11)and (12) corresponding to the Minkowski-like vacuum onsmall scales are thus v σ k ≃ v s k ≃ √ kc s e − ikc s τ (cid:18) − ikc s τ (cid:19) , (15)when µ s /H is negligible for the entropic modes (if µ s /H is large the entropic modes are suppressed). Thepower spectra for v σ and v s after sound horizon crossingtherefore have the same amplitude P v = ( k / π ) | v k | .The power spectra for Q σ and Q s are thus P Q σ ≃ H π c s ˜ P , ˜ X , P Q s ≃ H π c s ˜ P , ˜ X , (16)evaluated at sound horizon crossing. One recognizes thefamiliar result of k-inflation for the adiabatic part [8, 9],while for small c s , the entropic modes are amplified withrespect to the adiabatic modes: Q s ≃ Q σ /c s .These results can be reexpressed in terms of the comov-ing curvature perturbation R = ( H/ ˙ σ ) Q σ with which itis useful to relate the perturbations during inflation tothe primordial fluctuations during the standard radiationand present era. We recover the usual single-field resultfor the power spectrum of R at sound horizon crossing: P R ∗ ≃ H π c s ˜ X ˜ P , ˜ X = H π ǫc s , (17)where ǫ = − ˙ H/H = ˜ P , ˜ X ˜ X/H (the subscript ∗ in-dicates that the corresponding quantity is evaluated atsound horizon crossing). It is then convenient to define anentropy perturbation S = c s H ˙ σ Q s such that P S ∗ ≃ P R ∗ .The power spectrum for the tensor modes is, as usual,governed by the transition at Hubble radius and its am-plitude, P T = (2 H /π ) k = aH , is much smaller than thecurvature amplitude for c s ≪ R = R ∗ + T RS S ∗ of the first-orderevolution equations for R and S which follow from (11),(12) in the slow-varying regime on large scales.This implies in particular that the final curvaturepower spectrum can be formally expressed as P R =(1 + T RS ) P R ∗ . Let us define the “transfer angle” Θ(Θ = 0 if there is no transfer and | Θ | = π/ T RS p T RS , (18)so that the curvature power spectrum at sound horizoncrossing and its observed value are related by P R ∗ = P R cos Θ. Finally the tensor to scalar ratio is given by r ≡ P T P R = 16 ǫc s cos Θ . (19)This expression combines the result of k-inflation, wherethe ratio is suppressed by the sound speed c s and of stan-dard multi-field inflation [5].We finally turn to primordial non-Gaussianities, whosedetection would provide an additional window on the very early universe. This aspect is especially importantfor DBI models since it is well known that (single-field)DBI inflation produces a (relatively) high level of non-Gaussianity for small c s [2]. How, therefore, do the en-tropic modes, whose amplitude is much larger than thatof the adiabatic fluctuations, affect the primordial non-Gaussianity? In the small c s limit, one can estimate thedominant contribution by extracting from the third-orderLagrangian the analogue of the terms giving the domi-nant contribution in the single-field case, but includingnow the entropy components. These terms are [7] S (main)(3) = Z d t d x a c s ˙ σ h ˙ Q σ + c s ˙ Q σ ˙ Q s i − a c s ˙ σ h ˙ Q σ ( ∇ Q σ ) − c s ˙ Q σ ( ∇ Q s ) + 2 c s ˙ Q s ∇ Q σ ∇ Q s i , where we have replaced f by 1 / ˙ σ since, for c s ≪ f ˙ σ ≃
1. Following the standard procedure [12, 13, 14]one can compute the 3-point functions involving adia-batic and entropy fields. The purely adiabatic 3-pointfunction is naturally the same as in single-field DBI[15, 16]. The new contribution is h Q σ ( k ) Q s ( k ) Q s ( k ) i = − (2 π ) δ ( X i k i ) H √ c s ǫc s Q 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) , (20)where K = P i k i .We now relate the correlation functions of the scalarfields to the 3-point function of the curvature pertur-bation R which is the observable quantity. It followsdirectly from above that R ≈ A σ Q σ ∗ + A s Q s ∗ A σ = (cid:18) H ˙ σ (cid:19) ∗ A s = T RS (cid:18) c s H ˙ σ (cid:19) ∗ . (21)Hence, for vectors k i whose norms have the same orderof magnitude (so that the slowly varying background pa-rameters are evaluated at about the same time) hR ( k ) R ( k ) R ( k ) i = ( A σ ) h Q σ ( k ) Q σ ( k ) Q σ ( k ) i + A σ ( A s ) [ h Q σ ( k ) Q s ( k ) Q s ( k ) i + perm . ]= ( A σ ) h Q σ ( k ) Q σ ( k ) Q σ ( k ) i (cid:0) T RS (cid:1) . (22)As we see, the above quantity depends on the sym-metrized version of the 3-point function (20), which hasexactly the same shape as in single-field DBI. Note thatthe enhancement of the mixed correlation h Q σ Q s Q s i bya factor of 1 /c s is compensated by the ratio between A σ and A s so that the adiabatic and mixed contributions in(22) are exactly of the same order. In principle, there areother contributions to the observable 3-point function, inparticular those coming from the 4-point function of thescalar fields, which can be reexpressed in terms of thepower spectrum via Wick’s theorem [17]. The amplitudeof this contribution will depend on the specific models.We implicitly ignore them in the following.The non-Gaussianity parameter f NL is defined by hR ( k ) R ( k ) R ( k ) i = − (2 π ) δ ( X i k i ) (cid:20) f NL ( P R ) (cid:21) P i k i Q i k i , (23)from which we obtain, for the equilateral configuration, f (3) NL = − c s
11 + T RS = − c s cos Θ . (24)One can easily understand this result. The curvaturepower spectrum is amplified by a factor of (1 + T RS ) dueto the feeding of curvature by entropy modes. Similarlythe 3-point correlation function for R resulting from the3-point correlation functions of the adiabatic and entropymodes is enhanced by the same factor (1 + T RS ). How-ever, since f NL is roughly the ratio of the 3-point func-tion with respect to the square of the power spectrum,one sees that f NL is now reduced by the factor (1 + T RS ).The so-called UV model of DBI inflation is under strongobservational pressure because it generates a high levelof non-Gaussianities that exceed the experimental bound[18, 19]. We stress that their reduction by multiple-fieldeffects may be very important for model-building.We end by revisiting the consistency condition relatingthe non-Gaussianity of the curvature perturbation, thetensor to scalar ratio r , and the tensor spectral index n T = − ǫ , given in [20] for single-field DBI. In our case,substituting f (3) NL ≃ −
13 1 c s cos Θ in (19), gives r + 8 n T = − r (cid:18)q − f (3) NL cos − Θ − (cid:19) , (25) As we can can see from (24) and (25), violation of thestandard inflation consistency relation (corresponding toa vanishing right-hand side in (25)) would be stronger inmulti-field DBI than in single-field DBI, and thus easierto detect. In the multi-field case the consistency condi-tion is only an inequality (unless Θ is observable whenthe entropy modes survive after inflation) from which wecan infer the transfer angle.To summarize, we have shown that both adiabatic andentropy modes propagate with the same speed of sound c s , in multi-field DBI models. Both modes are thus am-plified at the sound horizon crossing, with an enhance-ment of the entropy modes with respect to the adiabaticones in the small c s limit. The amplitude of the non-Gaussianities, which are important in DBI models, is alsostrongly affected by the entropy modes, although theirshape remains as in the single-field case. All these fea-tures are generic in any model governed by the multi-fieldDBI action. The model-specific quantity (depending onthe field metric, the warp factor and the potential) is thetransfer coefficient between the initial entropy modes andthe final curvature perturbation between the time whenthe fluctuations cross out the sound horizon and the endof inflation. Recent analyses [21, 22] in slightly differ-ent contexts show that this transfer can be very efficient,leading to a final curvature perturbation of entropic ori-gin (as in the curvaton scenario). More generally, ourresults show that multi-field effects, common in stringtheory motivated inflation models, deserve close atten-tion as the entropy modes produced could significantlyaffect the cosmological observable quantities. Acknowledgement . We are grateful for a CNRS-JSPSgrant. [1] E. Silverstein and D. Tong, Phys. Rev. D , 103505(2004)[2] M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D , 123505 (2004)[3] D. A. Easson, R. Gregory, D. F. Mota, G. Tasinato andI. Zavala, JCAP (2008) 010[4] M. x. Huang, G. Shiu and B. Underwood, Phys. Rev. D (2008) 023511[5] C. Gordon, D. Wands, B. A. Bassett and R. Maartens,Phys. Rev. D , 023506 (2001)[6] D. Langlois and S. Renaux-Petel, JCAP (2008) 017[7] D. Langlois, S. Renaux-Petel, D.A. Steer and T. Tanaka,in preparation, April 2008.[8] C. Armendariz-Picon, T. Damour and V. F. Mukhanov,Phys. Lett. B , 209 (1999)[9] J. Garriga and V. F. Mukhanov, Phys. Lett. B , 219(1999)[10] D. Langlois, Phys. Rev. D , 123512 (1999)[11] D. Wands, N. Bartolo, S. Matarrese and A. Riotto, Phys.Rev. D (2002) 043520 [12] J. M. Maldacena, JHEP , 013 (2003)[13] D. Seery and J. E. Lidsey, JCAP (2005) 003[14] D. Seery and J. E. Lidsey, JCAP (2005) 011[15] X. Chen, Phys. Rev. D (2005) 123518[16] X. Chen, M. x. Huang, S. Kachru and G. Shiu, JCAP (2007) 002[17] D. H. Lyth and Y. Rodriguez, Phys. Rev. Lett. (2005)121302[18] R. Bean, S. E. Shandera, S. H. Henry Tye and J. Xu,JCAP (2007) 004[19] H. V. Peiris, D. Baumann, B. Friedman and A. Cooray,Phys. Rev. D (2007) 103517[20] J. E. Lidsey and D. Seery, Phys. Rev. D , 043505(2007)[21] Z. Lalak, D. Langlois, S. Pokorski and K. Turzynski,JCAP0707