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November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla ModernPhysicsLetters A (cid:13)c WorldScientificPublishingCompany 2 1 0 2 v THE SUYAMA-YAMAGUCHI CONSISTENCY RELATION IN THE o N PRESENCE OF VECTOR FIELDS 2 2 JUANP.BELTRAŃALMEIDA ] Centro de Investigaciones, Universidad Antonio Narinõ, O Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia C [email protected] . h YEINZONRODRÍGUEZ p - Centro de Investigaciones, Universidad Antonio Narinõ, o Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia, r and t s Yukawa Institute for Theoretical Physics, KyotoUniversity, a Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto606-8502, Japan, [ and Escuela de F´ısica, Universidad Industrial de Santander, 2 Ciudad Universitaria, Bucaramanga 680002, Colombia v [email protected] 9 4 1 CE´SARA.VALENZUELA-TOLEDO 6 Departamento de F´ısica, Universidad del Valle, . Ciudad Universitaria Meléndez, Santiago de Cali 760032, Colombia 2 [email protected] 1 1 1 : Received22Nov2012 v Revised(DayMonthYear) i X r a Weconsiderinflationarymodelsinwhichvectorfieldsareresponsibleforpartoreventu- allyalloftheprimordialcurvatureperturbationζ.Suchmodelsarephenomenologically interesting since they naturally introduce anisotropies in the probability distribution functionoftheprimordialfluctuationsthatcanleaveameasurableimprintinthecosmic microwave background. Assuming that non-Gaussianity is generated due to the superhorizon evolution, we use the δN formalism to do a complete tree level calculation of thenon-GaussianityparametersfNL andτNL inthepresenceofvectorfields.Weisolate theisotropicpiecesofthenon-Gaussianityparameters,whichanywayhavecontributions fromthevector fields,andshowthattheyobeytheSuyama-Yamaguchi consistencyre- lation τNisLo > (56fNisLo)2. Other ways of defining the non-Gaussianity parameters, which couldbeobservationallyrelevant,arestatedandtherespectiveSuyama-Yamaguchi-like consistency relationsareobtained. Keywords: Non-gaussianity;statisticalanisotropy;vector fieldmodels. PACSNo.:98.80.Cq 1 November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla 2 J. P. Beltrán Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo 1. Introduction The study of non-Gaussianities in the primordial curvature perturbation ζ is a subject of major interest in modern cosmology because the evaluation of the non- Gaussianity(NG)parametersprovidecriteriatodiscriminateamongthemanymod- 1 elsproposedtoexplaintheoriginofthelarge-scalestructurethatweobservetoday . Particular attention has been paid to the study of a consistency relation between the f and τ parameters2,3, the Suyama-Yamaguchi (SY) consistency relation NL NL (τ > (6f )2 at least at tree level), because its violation would rule out many NL 5 NL of the popular models of inflation. So far, it has been shown that this consistency 2 relation works, in principle, on models that only include scalar fields under just a fewassumptions,butthereisnoanyconclusiveresultformodelsthatincludeother type of fields, for instance, vector fields (VF)a. Vector field models are particularly interestingbecausetheyaresuitablecandidatestoexplaintheapparentviolationof 4 statistical isotropy observed in recent analysis of data from the WMAP satellite , giventhat VF define inherently a preferreddirectionfor the expansion,for the distribution of primordial fluctuations, or for both5,6,7. Because of this reason, it is pertinent to study consistency relations in models that include scalar fields as well as vector fields as these not only include NG parameters but also the amount of 8 primordial statistical anisotropy g . ζ ThemainpurposeofthisletteristofindouthowthewellknownSYconsistency relation, valid for models involving only scalar fields under just a few assumptions, is modifiedwhen including VF. Tothis end,we do a complete treelevelcalculation 7 of the f and τ parameters based on the δN formalism , and concentrate on NL NL the isotropic pieces of these parameters since these are the ones that are currently constrained by observations. Under pretty the same few assumptions made in the scalar fields case, and although the VF do contribute to the isotropic pieces of the NG parameters, we prove that the SY consistency relation is still obeyed, at least at tree level. Finally, the modified SY consistency relations for other ways of defining the NG parameters, which could be observationally relevant when taking into account the level of statistical anisotropy, are obtained. 2. The curvature perturbation In the presence of VF, part or even all of the primordial curvature perturbation ζ can be generated by vector field perturbations. In such scenarios, the curvature perturbation can be calculated through the δN formalism as an expansion in the 7 perturbations of the fields a few e-folds after horizon exit : 1 1 ζ ≡δN =N δA + N δA δA + N δA δA δA +··· , (1) a a ab a b abc a b c 2 3! aVFnaturallygeneratescale-dependenceand,therefore,theproofsinRef.3donotapplytothose scenarioswhereVFcontributetothegenerationoftheprimordialcurvatureperturbation. November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 3 where N is the number of e-folds during inflation and N are the derivatives of ab···c N with respect to A . In the previous expression, and in order to avoid notational a clustering,we haveassumedthatthereis only onescalarandonevectorfield;thus, δA ≡δφ for a=0, and δA ≡δA , the latter being the spatial components of the a a i 6 vector field, for a=i=1,2,3 , where these field perturbations are calculated in a flat slicing (the threading must be the comoving one). The extension of the results found in this letter to the case of severalscalarand vectorfields is straightforward. 3. Calculation of the correlators 3.1. The Power Spectrum The power spectrum (PS) of the primordial curvature perturbation ζ is defined as: hζ(~k )ζ(~k )i= (2π)3δ(~k )P (~k ) (2) 1 2 12 ζ 1 2π2 = (2π)3δ(~k ) P (~k ), (3) 12 k3 ζ 1 whereP (~k)isthedimensionless power spectrum,and~k =~k +~k .Thestatistical ζ 12 1 2 anisotropyinζ,duetotheexistenceofapreferreddirectionnˆ,canbeparametrized 9 by a function g as follows : ζ P (~k)=Piso(k) 1+g (kˆ·nˆ)2 , (4) ζ ζ ζ (cid:16) (cid:17) where Piso(k) denotes the isotropic part of the PS and the vectors kˆ and nˆ are of ζ unit norm. The first assumption we will make is that the NG in the field perturbations is negligible, i.e., the NG is generated due to the superhorizon evolution. The second assumption is to consider that the field perturbationsb are statistically isotropic, which is equivalent to the fact that there is no correlation between scalar and 6 vectorfieldsandthattheexpansionisisotropic .Thismaybereasonableasagood 7 approximationinseveralcases ,althoughitmustbeprovenineachparticularcase, and is essential in the development of this letter because, otherwise, there would not exist any isotropic contributions to the NG parameters which are what really is constrained by observations. We shall adopt the notation hδA (~k )δA (~k )i=(2π)3δ(~k )Π (~k ), (5) a 1 b 2 12 ab 1 with the components hδA (~k )δA (~k )i =(2π)3δ(~k )P (k ), (6) 0 1 0 2 12 δφ 1 hδA (~k )δA (~k )i =(2π)3δ(~k )Π (~k ), (7) i 1 j 2 12 ij 1 and zero otherwise. In the latter expression: Π (~k)=Πeven(~k)P (k)+Πodd(~k)P (k)+Πlong(~k)P (k), (8) ij ij + ij − ij long bWerefer,inthecaseofVF,tothescalarperturbationsthatmultiplytherespectivepolarization vectors inapolarizationmodeexpansion. November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla 4 J. P. Beltrán Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo whereΠeven(~k)=δ −kˆ kˆ , Πodd(~k)=iǫ kˆ ,andΠlong(~k)=kˆ kˆ ,P (k)being ij ij i j ij ijk k ij i j long the longitudinal powerspectrum andP (k) and P (k) being the parity conserving + − and violating power spectra respectively6,7. The fact that none of P (k), P (k), δφ + P (k), and P (k) depend on the direction of the wavevector is a consequence of − long the second assumption. At tree level, the PS is6,7: P (~k)=N N Π (~k). (9) ζ a b ab For concreteness, we shall restrict to the case of vector field perturbations which preserve parity, then, we just keep the P (k) and P (k) parts of the vector field + long power spectra. We also suppose that all the δA perturbations evolve in the same i wayafter horizoncrossingsothat they havethe same spectralindex in their power spectra.Insuchconditions,thelongitudinalandtheparityconservingpowerspectra arerelatedbyP (k)=rP (k)whererisascaleindependentnumber.Withthese long + considerations in mind, the vector power spectra are: Π (~k)= δ +(r−1)kˆ kˆ P (k). (10) ij ij i j + h i Our next steps are to evaluate the PS by employing the δN expansion and to separate the isotropic and anisotropic pieces obtaining in this way: 2 P (~k)=N N P (k)+(r−1) N kˆ P (k)=Piso(k) 1+g (nˆ kˆ )2 , (11) ζ a b ab i i + ζ ζ i i (cid:16) (cid:17) (cid:16) (cid:17) where Piso(k)= N N P (k), (12) ζ a b ab 2 Paniso(k)= (r−1) N kˆ P (k), (13) ζ i i + (cid:16) (cid:17) nˆ =N /(N N )1/2 and the relation between the (r−1) factor and the anisotropy i i j j parameter g in the power spectrum is given by: ζ N N P i i + g =(r−1) . (14) ζ N N P a b ab In the latter expression, the dimensionless power spectra P assume the values ab P =P and P =δ P . 00 δφ ij ij + 3.2. The Bispectrum. The bispectrum (BS) of the primordial curvature perturbation is defined as: 3 h ζ(~k )i =(2π)3δ(~k )B (~k ,~k ,~k ) (15) i 123 ζ 1 2 3 iY=1 4π4 =(2π)3δ(~k ) B (~k ,~k ,~k ). (16) 123 k3k3 ζ 1 2 3 1 2 November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 5 Using the δN expansion we get the tree level BS: B =N N N Π (~k )Π (~k )+2perm. . (17) ζ a b cd ac 1 bd 2 h i Expanding the above by employing the expression in (10) and separating the tree level BS into the isotropic and the anisotropic pieces we get Biso(k , k , k )=N N N P P (k k )−3, (18) ζ 1 2 3 a b cd ac bd l m lX<m Baniso(~k ,~k ,~k )=(r−1)N N N P P (k k )−3(kˆ kˆ +kˆ kˆ ) ζ 1 2 3 a i bj ab + l m (l)i (l)j (m)i (m)j lX<m +(r−1)2N N N P2 (kk )−3kˆ kˆ kˆ kˆ . (19) i k jn + l m (l)i (l)j (m)k (m)n lX<m 3.3. The Trispectrum Finally,weevaluatethe trispectrum(TS)ofthe primordialcurvatureperturbation: 4 h ζ(~k )i=(2π)3δ(~k )T (~k ,~k ,~k ,~k ) (20) i 1234 ζ 1 2 3 4 iY=1 8π6 =(2π)3δ(~k ) T (~k ,~k ,~k ,~k ). 1234 k3k3k3 ζ 1 2 3 4 1 2 23 (21) Once again, we use the δN expansion to get the tree level contribution to the TS, which results in T =TτNL +TgNL (22) ζ ζ ζ =N N N N Π (~k )Π (~k )Π (~k )+11perm. a b cd ef ac 1 be 2 df 13 h i +N N N N Π (~k )Π (~k )Π (~k )+3perm. . a b c def ad 1 be 2 cf 3 h i (23) As we are interested in the τ amplitude, we only need the term TτNL given that NL ζ the term TgNL enters in the definition of the g parameter. The decomposition of ζ NL November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla 6 J. P. Beltrán Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo TτNL in isotropic and anisotropic pieces is given by ζ Tζiso(τNL) = NaNbNcdNefPacPbePdf (klkmkls)−3, (24) l<m,s6=m,l X Tζaniso(τNL) = (r−1)NiNbNjdNefPbePdfP+ (klkmkls)−3(kˆ(l)ikˆ(l)j+kˆ(m)ikˆ(m)j) l<mX,s6=m,l + (r−1)NbNdNeiNfjPbePdfP+ (klkmkls)−3kˆ(ls)ikˆ(ls)j l<mX,s6=m,l + (r−1)2NiNkNjdNlfPdfP+2 (klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)n l<mX,s6=m,l + (r−1)2NiNdNjkNfnPdfP+2 (klkmkls)−3kˆ(ls)kkˆ(ls)n(kˆ(m)ikˆ(m)j+kˆ(l)ikˆ(l)j) l<mX,s6=m,l + (r−1)3NiNkNjmNhnP+3 (klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)hkˆ(ls)mkˆ(ls)n. (25) l<mX,s6=m,l 4. The NG parameters fNL and τNL In this work we consider the definition of the non-gaussianity parameters f and NL τ as follows NL 6 B = f (P (~k )P (~k )+2perm.), (26) ζ NL ζ 1 ζ 2 5 T =τ (P (~k )P (~k )P (~k )+11perm.). (27) ζ NL ζ 1 ζ 2 ζ 14 Noticethat,inthedefinitions above,wehaveusedthefullPSincludingbothscalar and vector field perturbations which, accordingly,include isotropic and anisotropic contributions. Nevertheless, it is usual to find in the literature a definition which 1 considers only the isotropic piece of the PS , so the NG parameters are given by B = 6f′ (Piso(~k )Piso(~k )+2perm.) and T = τ′ (Piso(~k )Piso(~k )Piso(~k )+ ζ 5 NL ζ 1 ζ 2 ζ NL ζ 1 ζ 2 ζ 14 11perm.). We can relate both definitions using the expression in (11) for the PS: 1 1 f = f′ = f′ , (28) NL 1+χ1 NL 1+ 2i=1mi(gζ)i NL 1 P 1 τ = τ′ = τ′ , (29) NL 1+χ2 NL 1+ 3i=1li(gζ)i NL P wherethe parametersm andl areobtainedfromthe expansionofthe products of i i the PS in (11) which are in the definitions of the NG parameters. They depend on November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 7 the derivatives of N and the momenta~k . Explicitly: i Pζ(~k1)Pζ(~k2)+2perm.=(1+χ1) Pζiso(k1)Pζiso(k2)+2perm. = 1+g nˆ nˆ l<m(klkm)−3(kˆ(hl)ikˆ(l)j+kˆ(m)ikˆ(m)j) i " ζ i j P l<m(klkm)−3 +gζ2nînˆjnˆknˆn Pl<m(klkm)Pl<−m3kˆ((kl)likkˆm(l))−jkˆ3(m)kkˆ(m)n#(NaNbPab)2lX<m(klkm)−3, P (30) Pζ(~k1)Pζ(~k2)Pζ(~k13)+11perm.=(1+χ2) Pζiso(k1)Pζiso(k2)Pζiso(k13)+11perm. l<m,s6=m,l(klkmkls)−3(kˆ(hl)ikˆ(l)j +kˆ(m)ikˆ(m)j+kˆ(ls)ikˆ(ls)j) i = 1+g nˆ nˆ " ζ i j P l<m,s6=m,l(klkmkls)−3 +gζ2nînˆjnˆknˆn Pl<m,s6=m,l(klkmkPls)−3hkˆ(l)ikˆ(l)l<jkˆm(m,s6=)kmkˆ(,lm(k)nlk+mkkˆl(sls))−ik3ˆ(ls)j(kˆ(m)kkˆ(m)n+kˆ(l)kkˆ(l)n)i +gζ3nînˆjnˆknˆhnˆtnˆn Pl<m,s6=m,l(klkmlk<lms),−s6=3mkˆP(,ll)(ikkˆl(kl)mjkˆk(lms))−kk3ˆ(m)hkˆ(ls)tkˆ(ls)n# ×(NaNbPab)3 (klkmkls)−3P. (31) l<mX,s6=m,l Certainly, both definitions of the NG parameters are approximately equal if the correctionsarisingfromtheexistenceofsomestatisticalanisotropieswerenegligible. Otherwise, such corrections would be very important from the observational point of view. 4.1. The fNL parameter For the f parameter we can use the results (12), (13), (18), and (19) to sepa- NL rate the isotropic and the anisotropic contributions as follows: f =fI +fII = NL NL NL Biso/(P P + 2p.)+ Baniso/(P P + 2p.). The labels I and II are related to the ζ ζ ζ ζ ζ ζ isotropicandthe anisotropicpieces ofthe bispectrum in (18) and(19) respectively. Using this decomposition, we can compare the fI and fII contributions by eval- NL NL uating its ratio fII 2 NL ≡ξ ≡ f (g )l fI 1 l ζ NL Xl=1 N N N N N P P (k k )−3(kˆ kˆ +kˆ kˆ ) =g a bj i c d ab cd l<m l m (l)i (l)j (m)i (m)j ζN N N N N P P P (k k )−3 a b cd k k ac bd l<m l m (N N P )2N N N (k kP)−3kˆ kˆ kˆ kˆ +g2 a b ab i k jn l<m l m (l)i (l)j (m)k (m)n . (32) ζ(N N )2N N N P P P (k k )−3 i i a b cd ac bd l<m l m P In the latter expression, we have used the expression in (14) to write the series expansion in terms of the level of statistical anisotropy g . The coefficients f can ζ l November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla 8 J. P. Beltrán Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo be read off directly from the latter expression. Using the ratio fII /fI , we can NL NL write the total f as: f = (1+ξ )fI = (1+ξ )/(1+χ )fiso, where fiso is NL NL 1 NL 1 1 NL NL the piece of f that depends only on the isotropic BS in (18) and the isotropic NL PS in (12). A very appealing feature that we can see from the previous analysis is that the ratio of the terms related to the anisotropic and the isotropic parts of the NG parameter f depends on the parameters of the model; for instance, it NL depends on the value of the ratio r, the level of statistical anisotropy g , and the ζ shape of the triangle defined by the ~k momenta. Moreover, interestingly enough, i for some values of g and for some configurations of the momenta, the anisotropic ζ contribution could be of the same order of magnitude as the isotropic piece and cannot be neglected. 4.2. The τNL parameter. For the τ parameter, we apply the same decomposition related to the isotropic NL andanisotropicpiecesthatwedidbeforewiththef parameter:τ =τI +τII = NL NL NL NL Tiso/(P P P +11p.)+Taniso/(P P P +11p.). We follow the same steps that we ζ ζ ζ ζ ζ ζ ζ ζ did in the case of the f parameter and write the ratio of both contributions as a NL series in powers of the g parameter: ζ τII 3 τNIL ≡ξ2≡ τl(gζ)l NL Xl=1 =g (NaNbPab)NiNcNjdNefPcePdf l<m,s6=m,l(klkmkls)−3(kˆ(l)ikˆ(l)j+kˆ(m)ikˆ(m)j) ζ(NiNi)NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3 +g (NaNbPab)NbNdNeiNfjPbePdf l<m,s6=m,l(Pklkmkls)−3kˆ(ls)ikˆ(ls)j ζ(NiNi)NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3 +g2 (NaNbPab)2NiNkNjdNnfPdf l<Pm,s6=m,l(klkmkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)n ζ(NiNi)2NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3 +g2 (NaNbPab)2NiNdNjkNfnPdf l<m,s6=mP,l(klkmkls)−3kˆ(ls)kkˆ(ls)n(kˆ(m)ikˆ(m)j+kˆ(l)ikˆ(l)j) ζ(NiNi)2NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3 +g3 (NaNbPab)3NiNkNjmNhn l<m,s6=m,l(klkmPkls)−3kˆ(l)ikˆ(l)jkˆ(m)kkˆ(m)hkˆ(ls)mkˆ(ls)n . ζ(NiNi)3NaNbNcdNefPacPbePdf P l<m,s6=m,l(klkmkls)−3 P (33) The coefficients τ can be read off directly from the expression in (33). Then, the l τ parameter canbe written as: τ =(1+ξ )τI =(1+ξ )/(1+χ )τiso, where NL NL 2 NL 2 2 NL τiso isthepartofτ thatdependsonlyontheisotropicTSin(24)andtheisotropic NL NL PS in (12). The ratio again depends on the g parameter and on the configuration ζ described by the~k momenta. i 5. A consistency relation for fNL and τNL Searching for relations between the NG parameters, we start exploring their isotropic pieces. Here we give an explicit proof that the isotropic pieces of the November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla The Suyama-Yamaguchi consistency relation inthe presence of vectorfields 9 NG parameters, in presence of scalar and vector field perturbations, obey the SY 2 consistency relation . Using the expressions in (12), (18), and (24) we find that: 6 N N N P P f(iso) = a b cd ac bd , (34) 5 NL (N N P )2 a b ab N N N N P P P (iso) a b cd ef ac be df τ = . (35) NL (N N P )3 a b ab Now, we can write the scale invariant power spectra P as: ab 1 0 P =D P = P , (36) ab ab δφ (cid:18)0sδ (cid:19) δφ ij ab wheres=P /P isthevectortoscalarPSratio.DefiningthevectorsK =D N + δφ a ab b and T =N K , we evaluate the ratio A =τiso/(6fiso)2: a ab b NL NL 5 NL (N K )(D T T ) (D N N )(D T T ) A = a a ab a b = ab a b ab a b >1, NL (K T )2 (D N T )2 a a ab a b (37) which implies: 6 τiso >( fiso)2. (38) NL 5 NL The latter expression is a direct consequence of the Cauchy-Schwartz inequality for the vectors N and T with the positive definite metric D , and reflects the a a ab fact that the SY consistency relation, at least at tree level, applies for the isotropic pieces of the NG parameters even in the presence of vector fields. Now,wecantrytoestablisharelationforthecompleteNGparametersf and NL τ .ThefactthattheisotropicNGparametersobeytheSYinequalityimpliesthat NL the total NG parametersobey a modified consistency relation.The modificationto this relation will depend on several aspects, for instance, on the level of statistical anisotropy, the momenta configuration, etc. Employing the SY inequality for the isotropic parameters, we deduce that the total NG parameters obey the relation: (1+ξ )(1+χ )2 6 2 τ > 2 1 f . (39) NL (1+χ )(1+ξ )2 (cid:18)5 NL(cid:19) 2 1 On the other hand, the relation obeyed by the NG parameters f′ and τ′ is the NL NL following: 2 (1+ξ ) 6 τ′ > 2 f′ . (40) NL (1+ξ )2 (cid:18)5 NL(cid:19) 1 All of these relations represent a constraint on the NG and statistical anisotropy parameters for models of inflation that involve vector fields. November 26, 2012 1:14 WSPC/INSTRUCTION FILE ws-mpla 10 J. P. Beltrán Almeida, Y. Rodr´ıguez, and C. A. Valenzuela-Toledo 6. Conclusions In this paper,we have studied under which conditions new kinds ofthe well known Sayama-Yamaguchiconsistencyrelationareallowedwhenvectorfieldsareincluded in the inflationary dynamics. In particular we have shown that the isotropic pieces ofthenon-gaussianityparametersobeytheSuyama-Yamaguchiconsistencyrelation τiso >(6fiso)2 when its calculation is performed at tree level. NL 5 NL We have also derived a modified consistency relation for the complete non- gaussianity parameters f and τ and for a set of new parameters f′ and τ′ NL NL NL NL that could be observationally relevant; these relations are given by the Eqs. (39) and (40). Since the latter depend on the level of statistical anisotropy and the configuration of the wavevectors, the naive relations τ > (6f )2 and τ′ > NL 5 NL NL (6f′ )2 could eventually be violated in some particular inflationary model that 5 NL includes vector fields. On this basis, we expect that in a near future the derived relationscouldprovideacriteriatodiscriminateamongthedifferentmodelsforthe generationoftheprimordialcurvatureperturbationwhenvectorfieldsareinvolved. Acknowledgments J.P.B.A. and Y.R. are supported by Fundación para la Promoción de la Investi- gaciónylaTecnolog´ıadelBancodelaRepu´blica(COLOMBIA)grantnumber3025 CT-2012-02. J.P.B.A. is supported by VCTI (UAN) grant number 2010251. Y.R. was a JSPS postdoctoral fellow (P11323) and, in addition, is supported by DIEF de Ciencias (UIS) grant number 5177. C.A.V.-T. is supported by Vicerrector´ıa de Investigaciones (UNIVALLE) grant number 7858. References 1. E. Komatsu et. al., arXiv:0902.4759 [astro-ph.CO]; E. Komatsu, Class. Quantum Grav. 27, 124010 (2010); A. P. S. Yadav and B. D. Wandelt, Adv. Astron. 2010, 565248 (2010). 2. T.SuyamaandM.Yamaguchi,Phys. Rev.D77,023505 (2008);T.Suyama,T.Taka- hashi, M. Yamaguchi, and S. Yokoyama, JCAP 1012, 030 (2010); N. S. 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