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Stable solutions of inflation driven by vector fields

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Inflationary tensor fossils in large-scale structure Emanuela Dimastrogiovanni, Matteo Fasiello, Donghui Jeong et al. JCAP03(2017)058 hysics P n known to be le and , ity ic d,c t ar ience and Technology, niversity of Tokyo, evade all of the mentioned rse (WPI), nce, 10.1088/1475-7516/2017/03/058 d explicitly show its stability. e standard slow-roll one and ca Theory with an extension ain the conditions required for oretical Physics, smouth, dient instabilities. In this work, doi: nary scenarios, reviving potential ations. Identifying the remedy to strop Ryo Namba A b,c [email protected] , [email protected] , osmology and and osmology Shinji Mukohyama, C a e,f 1612.09581 cosmological perturbation theory, inflation, modified grav Many models of inflation driven by vector fields alone have bee [email protected] rnal of rnal ou An IOP and SISSA journal An IOP and 2017 IOP Publishing Ltd and Sissa Medialab srl [email protected] Institute for Advanced Study, The Hong Kong University of Sc National Astronomy Observatories, Chinese Academy of Scie Clear Water Bay, Kowloon, HongCenter Kong for Gravitational Physics,Kyoto Yukawa Institute University, for The 606-8502, Kyoto, Japan Kavli Institute for theThe Physics University and of Mathematics Tokyo of InstitutesKashiwa, the for Chiba Unive Advanced 277-8583, Study, Japan TheDepartment U of Physics, McGill QC,Montr´eal, University, H3A 2T8, Canada Beijing 100012, People’s Republic ofInstitute China of Cosmology andPortsmouth Gravitation, PO1 University 3FX, of U.K. Port E-mail: b c e d a f c ArXiv ePrint: plagued by pathological behaviors,we namely ghost seek and/or a gra instabilities. new class We ofto build vector-driven three our inflationary vector analysis models fieldsquasi on to that de-Sitter the realize solutions isotropic Generalizedthose to expansion. Pro for be their We an stability obt the at existing attractor unstable the analogous models, levelThis to we significantly of broadens provide th linearized our a perturb knowledgephenomenological simple on interests example vector for inflatio an this class of models. Keywords: Received January 13, 2017 Revised March 3, 2017 Accepted March 13, 2017 Published March 29, 2017 Abstract. Stable solutions of inflationvector driven fields by Razieh Emami, J Ying-li Zhang

JCAP03(2017)058 9 ], 1 4 6 8 e- ty 11 13 14 15 17 18 19 17 21 25 The – 2 18 ], Ekpyrotic 4 , 3 t require the slow roll, is more appropriate term of a scalar field. ]. 9 ], and ground-based [ r multiple scalar fields. l as providing the initial seeds diation and for the large-scale 17 ], matter bounce [ solve the horizon, flatness and 2 – e, and the use of scalar fields is ], introduction of potential terms easure and/or constrain inflation- 15 tions driven by higher-spin fields, oming experiments are expected to n often employed breaking of gauge 42 . A small subset of them consists of pre- – t with a spatially isotropic expansion. 39 precision cosmology ]. 38 Since the standard U(1) gauge field only – ], and Galilean Genesis [ 8 29 3 , 7 – 1 – ]), balloon-borne [ 14 -form fields in [ p 4 is conformally coupled to gravity and thus its energy densi / 2 F − A number of past, ongoing and upcoming CMB experiments, spac 1 ], where the inflationary solution is supported by the kinetic ] (and references therein), string gas cosmology [ ] for a good collection of models. There are examples that do no 28 1 27 , ] (latest results by Planck [ 26 ], cosmology in gravity Hoˇrava-Lifshitz [ 13 6 – , 5 -inflation [ 10 k Most of the realizations of inflation rely on slow roll of one o Some alternative mechanisms to inflation have been proposed See, e.g., [ There are examples of inflation by 5.2 A simple successful example 4.4 Scalar4.5 sector Attractor4.6 condition against anisotropic Stability expansion conditions 5.1 Existing (unstable) examples 4.1 General4.2 procedure Tensor4.3 sector Vector sector 1 2 3 quickly decays away, the attempts ofinvariance, vector-driven such inflatio as non-minimal coupling to gravity [ than has ever been. 6 Conclusion 1 Introduction Inflation has since itsunwanted relics birth problems been in the afor Hot successful Big fluctuations Bang paradigm in cosmology the to as cosmicstructure re wel microwave background formation. (CMB) ra 5 Illustrative examples 3 Inflationary background 4 Perturbations Contents 1 Introduction 2 Generalized Proca theory with O(3) invariance based [ with the kinetic action scenario [ such as as well as those forary the large-scale parameters. structure The (LSS), observational aim precisionsbe to of yet m at the unprecedented forthc levels, and the nomenclature slow roll is tothe ensure simplest a realization in prolonged field periodOn theories that the of is other inflationary consisten hand, stag especially possibilities to vector obtain fields, inflationary have solu been sought for. big-bang scenario [ JCAP03(2017)058 . , ]. ], ν φ ) 67 c 80 ( , – ] and A 5 66 with a ]. µ 78 96 ) or field, ]). b ( 48 φ , the dan- – 85 A ) ] a 46 that respects ν ( A gical signatures per se µ [ ]. Unfortunately, pled to a pseudo- is the dual of the ∂ nflation. Another 92 ˜ F abc ǫ he three spatial direc- all of the vector-only- ]. ile the dominant energy 86 ], magnetogenesis [ y a scalar field ]. However, these models pe of coupling but without ], was a novel vector-driven 77 nd due to the local property is allowed/motivated. (The , where 45 assive Gauge-flation [ en spontaneously but in this – ound to suffer instabilities on ˜ 91 nt with the data. F vely studied in the context of oken SU(2) gauge symmetry is ls of inflation solely driven by ation around horizon crossing, observational upper bound, for 71 , is abandoned, the SU(2) specific to be pathological [ ost/gradient instabilities or ob- nd they involve the SU(2) struc- In the case of a pseudo-scalar ional constraints: essentially the al) attractor [ ld, i.e. a Higgs field.) Therefore, φF izations in the propagation between the 90 ]. Recently, the variants of these 4 . ] (for further review, see [ 95 etogenesis have been considered in [ – ds on the produced amplitude of mag- 84 93 , that would otherwise roll down its poten- 83 φ ogical birefringence [ ], which can be interpreted as a limit of 89 , – 2 – ], in which the U(1) gauge invariance is preserved 88 49 term, and this coupling leads the vector to an attrac- 2 ], while it has been shown that the CMB constraints on F ) 62 heavy and integrated out [ ], helical gravitational waves [ – φ ( φ 70 2 50 ], and it has been proposed that the mass terms can enlarge the I , and therefore this is not a vector-driven inflation 97 − 69 φ . It results in a copious, exponential production of the vect F ], and primordial black holes [ in an essential way through the term of the form 82 ]. This is the only coupling between a scalar and a vector field abc , ǫ 65 ]. In addition, this model turns out to have rich phenomenolo ], and fixing the norm by Lagrange multiplier [ – 81 ] promoted the Abelian vector to a non-Abelian gauge field cou 68 44 63 87 , SO(3), one can orient the vev of the three vector fields along t as in the Abelian case, dubbing the model as Chromo-natural i 43 φ ∼ = ) [ As described above, among the plethora of the models, almost Ref. [ A different model was proposed in [ 2 Some derivative coupling operators and their effects on magn The same type of coupling can invoke a rotation of photon polar A 4 5 ( model called Gauge-flation proposed in [ driven ones areservationally either unfavored. theoretically So inconsistentvector far, due fields the are to only the gh Gauge-flation knownture and stable constants its mode massive variant, a such as non-Gaussianity [ scalar gerous longitudinal mode isthe absent vector in vacuum expectation thisprimordial values model. magnetogenesis (vev) [ The has same been ty extensi However, as mentioned above, theobservationally Gauge-flation disfavored. with the Once unbr the SU(2)structure gauge is symmetry notSU(2) necessary specific and structure a iscase more kept the general if system class the inevitably of gauge involves models an symmetry additional is scalar brok fie V symmetries admit a coupling to a gauge field in the form and its backreaction can slow down the motion of tial quickly [ model of inflation. TheseSU(2) models employ an SU(2) gauge field, a models, with massive SU(2) fields, were considered, dubbed M Chromo-natural model with parameter space so that the model predictions are in agreeme contain a longitudinal modeinflationary of backgrounds, the and vector therefore field all that have has turned been out while f the kinetic term of the vector (gauge) field is modified b gauge invariance and parity, without invoking derivatives baryogenesis [ however, neither of thesetensor models modes survives experience against aand observat tachyonic the growth for tensor-to-scalar a ratioany limited is acceptable dur always values beyond of the the level scalar of spectralHiggsed Chromo-natural index [ [ higher-order correlation statistics place stringent boun netic fields [ field-strength tensor tor phase andcomponent prevents is it the from one decaying of away by expansion. Wh flat potential, through a last scattering surface and the present time, called cosmol tions, and then an isotropic expansion is manifestly an (loc JCAP03(2017)058 d 0, ≈ l free ]). In ) , where a i 105 δ , A/a ) ( t t ( p the theory ∂ 104 A = ent with this work, current purpose. ]. i ) opic configuration, e stability of these a i ( thus its longitudinal 134 – d expansion was ana- A h nstrained by the Planck 130 tains derivative terms of e. almost constant Hubble ons of the action give rise stant, not allowing a quasi ponent, and for this reason, e start from a highly general models of vector-only driven zed Proca theory nor contains the s theory has been introduced heory is constructed in such a ur knowledge on vector-driven fore, Generalized Proca theory gainst both ghost and gradient oll attractor. tive instabilities, also known as s are proposed in [ onents take a vev, a vector field, lobal O(3) symmetry in the field relation functions become statis- ents a homogeneous vev pointing e conditions under which all the r, namely ion of the universe and vanishing in space. An expansion driven by el window that satisfies all of the vector fields without the mentioned . To this end, we introduce 3 vector ere the background is quasi de-Sitter in which vector fields couple non-minimally ations. e considered in [ of Generalized Proca to multiple vectors ap- order. ] was studied intensively in the context of This setup, together with the conditions for 49 8 – 3 – and anisotropic solutions in Generalized Proca are explore 7 ], The theory is quite general, as it is characterized by severa 129 6 – ]. ]. Our deformation of the original theory is mutually consist 109 ]. In fact, the model in [ 103 137 45 ) is the label of the three vectors. Moreover, in order to set u a 0, that are given by a slowly varying physical vector vev, i.e. ≈ ]; however, this model is neither within the class of Generali ˙ ], the application of this theory to the late-time accelerate H ] and consists of a vector field without gauge invariance, and ], we deform the original theory so as to minimize the anisotr 106 is the scale factor. Our interest in this work is to analyze th 108 , a ]. However, since the amount of anisotropy is stringently co 102 136 – Some deformation of the original theory is necessary for our We focus on quasi de-Sitter solutions for the background, i. Taking the lessons from the preceding unsuccessful cases, w 107 Recently an inflationary solution is investigated in a model Vector curvaton scenarios in a resembling circumstance wer During our preparation of this manuscript, generalization 135 98 6 7 8 under the concerned symmetry. in [ parameter where background solutions. Specifically, we investigate the mod our aim of thisinflation, paper to is, search a as stable a inflationarySU(2) first solution specific driven step by towards structure. ainflationary wider This models. class would of significantly broaden o consistent with this background configuration,space we impose and a retrieve g the terms that respect it. data [ while preserving the no-Ostrogradsky-ghost construction fields and correspondingly give eachin of each their of spatial compon the spatial directions perpendicular to each othe terms, identified in our work, to cure ghosts at the linearized statistical anisotropy [ a background attractor, indeed ensures an isotropic expans class of vector-fieldin models: [ Generalized Proca theory. Thi functions (five, in the mostthis general work, case we — investigate furtherperturbations this extension around general inflationary classinstabilities. backgrounds and We are identify also restrict stable th ourwith a attention to an the attractor case wh behavior analogous to the standardIn slow-r [ lyzed. There thethe vector field value is of given thede-Sitter a solution. Hubble vev parameter On in the (and itsas other the its temporal hand, name when vev) com stands, its is inevitably spatialsuch exactly points comp a to con field a therefore specifictically has direction a anisotropic privileged [ direction, and the cor statistical anisotropy, at least in the linearized perturb mode propagates asthe a vector physical field in degreeway the of that action it freedom. withoutonly takes gauge to the While invariance, the the most it equations t is general con up by form to construction with second-order guaranteedOstrogradsky which derivatives. to ghosts the [ be There variati free from higher-deriva to gravity [ the superscript ( peared in the literature [ JCAP03(2017)058 ) il- are 2.4 (2.1) ρ ν )–( F 2.2 µρ F ν ric is excited = 1, take the A µ c . We summarize A = and then provide a 4.5 . p . We finally conclude ~ ld with a timelike vev M 5.1 5.2 = 0 is a trivial solution of the i ] and thus we need to resort ) a 0 ( of vector fields can be stable ralized Proca theory to have 48 hand, a single spacelike vector thout the structure specific to , then proceed the analysis for on to the so-called generalized lely by vector fields, to identify expansion. A , respectively, and provide the – and then provide a (relatively) . multaneously. This is the first h us introduce three vector fields c in section rbations against both ghost and lutions to be an analogue of the he present paper. (See ( 46 4.1 d models. It has been known that ackground isotropic configuration 4.4 s at the linearized order, analogous erms of the form ion of the triplet vector fields is not the olved, and those in the general case e inevitably introduces anisotropy. = 0 s[ t the vector vev does not run away ] for clarification. s in section d in section i 93 ) is devoted for the stability analyses of and , we introduce the theory we consider a , and we separately examine the stability 0 ( 2 4 9 88 A h 4.3 , , 10 4.2 a i illustrates some examples: we consider existing δ – 4 – and refs. [ ) 5 t 4 ( ] as the straightforward extension of the Horndeski A 137 = , i ) a , we seek for quasi de-Sitter background solutions and in ( i 102 3 . Section – A h ] to a vector-tensor one (thus free from Ostrogradsky instab 98 4.6 = 0 instead, but then they can see that 142 i 6 – ) a ( 0 138 A h and therefore that no anisotropic part of the background met a i δ . Throughout the paper we use the natural units with 6 i∝ ) ]). As already stated in the introduction, a single vector fie a ( i with vacuum expectation values (vev), A 103 h 3) This paper is organized as follows. In section , 2 This analogy is only an approximate one, since the decomposit One can start from , + ++) metric signature, and denote the reduced Planck mass by 9 10 (1 µ − dynamically. The perturbations are decoupledto into 3 the sector standard tensor/vector/scalar decomposition, standard one under spatial rotation. See section crucial for the absencewithout those of terms within ghosts, the class and of model no considered inflationary in t models ( following conditions: (i)standard the slow-roll one attractor, for (ii)and the the geometry, quasi one and de-Sitter to (iii)gradient preserve so the instabilities. the ones b The for condition thefrom (ii) stabilities is of to pertu ensure tha background. We thus set this from the beginning. in section to more dedicated models.Proca In theory, this introduced regard, in we [ turn our attenti ities [ (therefore unstable) models and identify the instabilitie scalar-tensor theory [ below.) The conditions forare gradient stability not are illuminating. quite inv simple We therefore example consider that some canexample examples, of indeed inflationary satisfy model allthe in SU(2) the which gauge conditions vector symmetry, fields si drive alone, quasi wi de Sitter inflationary particular provide the attractor condition. Section tensor, vector and scalarcondition sectors to ensure in the sections all background the attractor conditions to in be section isotropi successful model with all the stability conditions satisfie 2 Generalized Proca theoryOur with aims are O(3) to search invariance necessary for building a blocks model and of to inflationsimple broaden that implementations the is of driven class this so of are allowe plagued by instabilitie does not allow for afield graceful intrinsically exit carries from inflation. aIn On order privileged the to direction other realize andA inflation henc with isotropic expansion, we th conditions for each sector. Our analysis clarifies that the t and describe the necessary deformation of the original Gene isotropic expansion. In section perturbations: we outline the general procedure in section JCAP03(2017)058 6 , and ], or X/ = 0) (2.4) (2.2) (2.3) ly the µν Y 88 4 ) − , a ( G 2 X ( ˜ ) indices, / F 2 ) a p a X ss of simple ( µν 2 M . Notice the F m = 3 =1 + 4 term for simplic- a X /∂X 4 G Y ) and demonstrate 6 1 4 L = , ∂G 2.4 2 and ρ ] and gauge-flation [ , ≡− ν ≡ G ) 87 i X b ( µ 2 d the ) ,X F a 4 m n, we impose a global O(3) , Z s to contract the ( ( G find a stable model of inflation µρ A olutions. More importantly, we + nly driven inflation, i.e. Gauge- tor fields. Note that the case of ) µν flation [ ν SU(2)) structure constants. This a , ] (with the terms proportional to ) SU(2)) invariant scalar combina- ( Y a ∇ ( ) ν ≃ metry is naturally realized since SU(2) is F 42 4 = ) ) ≃ – F b a ) and thus significantly broaden a class L ν ( 2 ( 2, and a 40 ( ν A / µν A G + ) ) ) c µ F a a 2 ( ( µ ( ρσ L A 3 A , and that the superscript on the vectors is =1 F ( µ a 3 X , but they can be re-expressed in terms of −∇ , ) g ρ b 2 =1 4 1 , ν ( ν 3 − 1 ˜ ) µνρσ X Requiring the form of Generalized Proca ac- F ] to be zero for simplicity. Since they vanish from the A a,b a – 5 – ǫ √ ) ( ]. Recovering them can be done straightforwardly, and the W µρ a 99 x ( ≡ ≡− ˜ A µν 4 F = ν 107 ν F 2 d 11 , and that of A ∇ µν µ abc µ m Z term [ ) ǫ ) A a a 4 , , W ( represents the SO(3) ( = ) is the action we study in this work, and we derive the ( L ˜ ρ ρ F , A S ν ν on yet another SO(3) ( 2.2 µ abc ) , ) ) ) , they do not contribute to the background dynamics and affect on ǫ b 3 a 2 ( ∇ a ( π ( µ h etc., we take the minimal choice as in ( µ G F F in the ∂ the terms ,W A ν 2 3 ν =1 , Y 2 2 ) µρ µρ c a X → b ) ) G µ ( b b µ ) ( ,W ( , where ,X A a 1 A −∇ ) 4 ( ν F F ) a ) ) ) c ( ν A G a b ( a ) ( ν ( ν ( ν A a A + ( µ ] with three vectors and the symmetry and restricting to a cla A A µ A µ = 0 is the standard free U(1) gauge theory, that of ) ) ) ) µ A a a b R b 4 ( µ ( µ ( ( ) 137 ∇ 3 A A G =1 A A , ) a X X,Y,Z,W X ) a = ( ( a ( µ =1 =1 2 1 ( 2 4 3 3 µν ) 102 A X X and 13 F a a,b a,b G G , – ( µν 12 Y dropped by hand). Eq. ( abc ≡ ≡ = = ≡− F 99 ǫ 1 3 2 4 In the case of SU(2) gauge fields, such as in Chromo-natural in Here we set parameters One can also include in = X and therefore are redundant. ,X L L 11 13 12 4 2 W W tion with specific combination is includedflation in and known its models massive of variant.without vector-o In relying the on present the paper combination we shall different contractions in the definitions of Moreover, in ordersymmetry to in preserve the this internal background field configuratio space. merely a label of the three vector fields. Also we have exclude W the original Proca theory with mass action in the scalar limit that it equips sufficientexclude room the to provide dependence stable of inflationary s G stability conditions around the inflationary background. any larger group thathomomorphic has to SO(3). SU(2) as its subgroup, this type of sym procedure of our subsequent calculations will be the same. tion [ actions, we have of allowed models ofG inflationary models driven solely by vec where such as ity. Although there are in principle many other possible way vector (and tensor in our calculations) sectors [ the non-minimally coupled model considered in e.g. [ JCAP03(2017)058 ) , B B 2 2.1  , (3.9) (3.8) (3.2) (3.3) (3.1) (3.7) (3.4) (3.5) (3.6) ,X 2 ¯ G HB + + 14 ,  ˙ , B as  2 ˙  HB = 0 A B 2 4 2 a  + 2 B eedom in the back- A 2 ˙
B 3 B , and their vev ( 3  H )  − ,X a a A 2 µ ( ,W 2 ,Y ¯ + 3 =  A G 2 flat Friedmann-Lemaˆıtre- ¯ t G ook for one with quasi de 3 ˙ ¯ G ∂ H + and the homogeneous vector ¯ + 2 W to gravity. 2 ) H a A ave  2 a  k ( = ˙ rivative with respect to physical e background action reads a a , B ,W A + 2 2 2 j j 3 HB ,X ¯ ¯ ∂ ,X W G 4 ) 4 o dx a ¯ 2 correctly reproduces the equations + ¯ i G i ( . G / = ˙ + 3 2 A p B ) as a solution only if the vector vev dx 1 a t A 1 HB  + 2 . ∂ ¯ ij M W  δ 3.8 + 12  + ≡ ,W 2 ) . . and “physical” vector vev ijk 2 2 = t ˙ 2 2 ǫ B ( ¯ 0 B 0 , = 0 ˙ G B a a 4 2
H  2 ∝ a 4  3 G ≈ is a tetrad component of the vectors and thus does not ≈  2 2 , B ¯ H 2 – 6 – G ˙ Z ˙ + = 0 ,W B B H ˙ H − 6 2 a a B . 2 − ¯ a Z ¯ and
G 2 − dt ,Y 3 ≡ ˙ + 3 2 B 2 − + X , ¯ ,W ˙ G  ¯ 2 H 2 G 2 H  2 2 = ¯  2 m ˙ B G ) consists of three vector fields a ,W A 2 3 
2 2 3 + + 2 3 + 3 ¯ 2.2 ds G 2 ˙ B dta ,W Y = B 2 ). The corresponding equations of motion are, respectively ,W ,X t ¯ ¯ 2 + 3 = Y Z G 4 ( ¯ 1 ¯ a G 2 HB G V + , G ,W ,X 2 2 = 4 + 3 2 2 + 2 ¯ ¯ ,W G a 1 G A 2  ) and (0) 2 3 ¯ ˙ t ,W G B S ( 2 − − ¯ B A G + 12 + 3 ,Y = 2 2 ,X 1 2 4 ¯ ¯ ¯ G G X ¯ ,W  G − − 2  vanishes on the background, since ¯ is the comoving volume. There are two dynamical degrees of fr t G ,Y = 2 ∂ + 2 Z 2 V ¯ 4 G ˙ . H In order to obtain an inflationary solution, we specifically l  t H 4 + 2 − 4 n The word “physical” means that the value of ¯ t ¯ G G 14 ∂ 4 6 is in the slow-roll phase, i.e., vev does not survive against the spatial derivative on it. Th One can realize that background equations have ( 3 Inflationary background The class of models given by ( ground, namely where bar denotes background quantities, andtime dot denotes de Sitter expansion, namely, depend on the normalization of the scale factor drives inflation. WeRobertson-Walker assume (FLRW) metric, the background geometry to be the where being consistent with the vector vev configuration. We then h Note that the choice of of motion and of state for the Proca theory minimally coupled together with the constraint equation where we have defined the Hubble parameter JCAP03(2017)058 , ) ) ˙ H B ) is ) and 3.5 3.6 ˙ ) and ) and B 10 3.4 0 )–( 2 ), ( 0 δB B ) can be 3.5 B 3.11 2 + 3.4 in ( 3.4

0 , = 0, where 3.6 H,B, 2 2 0 0 ˙ ( )–( 2 0 (3.10) 0 (3.11) B 0 H B h H with constant 2 B 0 δB = ≈ ≈ 3.8 2  0 4 0 = H 4 0 2 2 W,Y ) are simply alge- B B H ), the background δB ¨ B B  G H H
,Y Y 2 att 0 4 as 3.8 2 ,X + ¯ 3.11 2 C 0 B G 4
G ) simplifies ( satisfy ( H
,Y Y 4 B 0 ¯ + G 2 2 0 0  only. Solving this results B ˙ + 6 4 W,X 3.9 0 G + 3 B B H as in ( 2 2 δ 0 and + 6 re of a constrained system. G B B ) and one of the remaining 0 ) and ( ,Y B ) can be algebraically solved 2 ,XX H H 2 2 ,XX B 4 ˙ H and 2 2 H 0 H 3.6 G 6 0 G H ,XX and δ 0 3.10 + 9  G ,Y Y = 0, which is the consistency of 2 ) and ( H  B 2 + 3 2 8 B 0 2 2 2 0 ), we impose the conditions under ,( G H − ¨ + 9 m provide the de Sitter background ¨ B . By substituting them to ( can be expressed as functions of G B B B 3 3.8 B H ,XX ¨ H 2
0 δ  B 4 ¨
(and  + 3 B 2 0 + 3 4 B 3.11 0 ) can be written as ,XY 2 G 0  ,W 4 2 2 B 0 4 0 B 2 0 ,X 2 ,W 2 and B 2 9 0 )–( δH 4 3.4 G B 2 B ¯  and G H 2 H ¯ 0 G 2 B − G 0 ) ˙ 3.8 ,XY + 9 4 ) such that H B
3 ,XX H 2 + ,Y Y 2 = 0. (This is no surprising, since ( 0  2 4 2
2 G 2 0 + 3.6 B + 12 4 ˙ G 0 ,W . Now G – 7 – 2 B 2 ,Y Y 0 W,W 0 B 2 4 + ,W 2 2 B ¯ 2 B )–( G H G  G + 9 = G δH ¯ are given by ,Y − 2 0 4 G = 0. Therefore,  2 2 + 0 3.4 ,XX ˙ + B . Hence, ˙ ,Y Y + 3 2 H 4 G B 2 att B ˙ 2 1 B  + 3 ,XY and G ,XY as an inflationary solution. Moreover, once C ,X G 2 2 3 ) ,XY 1 4 ,XY ,W 3 W,W 3 2 2 G G 2 B are functions of − G + 9 δB ,W G 4 G ¯ − G G 2 4
+ 2 0 = 0 in the background equations of motion. First, ( 2 0 2 W,W 0 ¯ ¯ − G G B ), one observes that ( + 6 + 3 + 3 B ˙ G B + 2 − , giving the master equation with respect to 2 4 H 0 1 2 0 ,X 2 ,XX ¨ 2 3.5 0 ,X 4 + B G 2 H ,Y ,XX B 4 and δ ,XY 2 2 4 − 0 2 2 H G 2 G ,XX 2  W,W G = 0 to obtain the expression for ¯ G 2 H G 3 ¯ G ,Y  G G ˙ W + 2 2 0 G + 9 ,XX W,W H  and − ¯ 2 G G 4 H  G + 3 4  +( ,Y ˙ G + 2 G B 2 ,XX 3 3 ) is non-singular at G 5 δ 4 ,Y ˙ )= G + 3 ,X 2 + 3 B , − 2 + G W,X 2 1 2 4 3 2 in the original equations ( 0 G att ¯ 6  2 G  ). Therefore, barring fine-tunings, given a constant G ,Y B ¯ δB C   G − 2 = 0, we obtain an algebraic equation involving H W,W 4 δH ,X ,Y ( G H,B, of W 4 2 W,X W,Y ,Y ˙ G ) to G ( + 2 2 2 . H B 3 0 G G G G G h  , where we set 3( G  3.6 H Given a solution to this set of equations, ( = B +6 − +3 + +5 +3 + +3 + = 0, is automatically satisfied by = B (Num ˙ coefficients. This second-order equation takes the form This can be seen by setting and then linearize them in terms of solved algebraically with respect to setting equations only admit a constant form a constrained system and thusand any ( valid solution of ( braic equations of these variables,in and the the solutions considered to model. the which it is an attractor of the system. To this end, we perturb replaced with the use of ( two should automatically satisfy the other.) Imposing ( where H the numerator and denominator of B in favor of in with the third equationBearing being in mind a that redundant one, due to the natu JCAP03(2017)058 , , as ¨ B o are δ
odes att (4.1) (4.2) (4.3) (4.4) 2 0 0 plays (with C (3.12) B B 3 2 ) in the , 0 and att − ] ˙ H a C B ia . δ t 3.11 for the spa- ∝ , ,Y Y 4.5 2 ˙ )–( B )+ δB G δ the approximate U ). The characteristic 3.8 b ∝ ∂ + 3 δB i,j,... ) modes have trans- ( , t + ,Y ij analyses, we therefore ∂ 2 δB b , h G ). The condition for this U = 0 ( ia one metric, and thus the and t respecting the background ii 1. The deviation from the a graceful exit, we require − iab h i . This parameter 2 ǫ 0 for the model regions where δH 0 ). In this section, we analyze ˙ ell and therefore the attractor µν = B led separately to the true scalar B/HB H | ≪ g ons given by ( , , ii )+ W 2 3.12 att G , is shown in section ) in the conventional models, and −h M C 0 and a | ij = 0 lent version that leads to the same attractor and its derivatives, but we can instead solve δ ∂ , +2 µν ˙ H i 0 n terms of H/H 4 g 0 , t = 0, leading to ( ) 1 in ( + E B δB i being the eigenvalues and with the same ) is particularly useful in that, already at B 4+10 = 22. A convenient decomposition ˙ ij i∝ a O ≡ ∂ 2 B 0 λ h ) and “tensor” ( ij δ × = 0 i M 4.1 H g 0 = , | ≪ ( µν + h  i ij i H i
,E 2 ) 0 ∂ att δg h i 2 0 B E j i B B C – 8 – j i B ∂ | + ∂ and ) + 3 ,B ∂ ∂ = 0 with 3 a a ,X i ai ¨ = and = B 4 + δ + are written as linear combinations of att i δ i ,U for different vector fields, while ), and thus deviations from de-Sitter solutions ij i j do not transform as scalar, vector and tensor, respec- G and constant ) W,W C a ˙ 0 U h a i H E B } δQδ i ( µ 3 G i + B i∝ δ ∂ ∂ ) ia ∂ + 2 A )( )[ ,M λ ] a ,W t + t t ( i a 4 2 = { ( ( = 2 + Y 88 ¯ i A a a +3 G −h and G h ij 2 a, b, . . . E ) t M j λ = = a + W,W i j and ( µ + ∂ i ) ∂ ∂ δH ) is [ 2 i G 0 a A 2 } 0 ( i ∂ a ,W = = B 3.1 2 ≡ + 3 i ) under the attractor condition ¯ ,U ij + 2 1 ,X G ) Y t a 2 4 i a i ( µ ij are symmetric. We denote other modes as “scalar.” While the m ∂ ∂ G 3.11 + 3 W,W 4 ,M ij δA Y, δA ψδ a ) and ( 1 G − h a Y )–( ( ∂ ,W { 2.1 2 ) (2 − , t + 3.8 ¯ )= 0. In this limit, we have ( G } and Hence we conclude that inflationary expansion is achieved by a 2 φ, δg ). Some other versions are also possible. ) and their attractor condition att Y a W,Y ij ≡ → t 15 C 1. Whenever we consider the de-Sitter limit in the following G 2.2 = 2 = = 3.12  W ) att of ij a 00 G . 12 ( 0 C | ≪ δg n The procedure described here has another different but equiva δg δA att 15 × Y,δQ,M,U C (Den attractor to stay isotropic, i.e. pure de Sitter is expected to be at slow-roll orders take for perturbations verse/traceless properties, i.e. the linear order, they are decoupled from each other and coup 4 Perturbations We have obtained intheory the ( previous section inflationarythe soluti linear perturbationsit around is this stable. backgroundstarting number and The of search variables concerned of perturbations system is has 3 three vector fields and solutions of ( in order to| realize a sufficiently long period of inflation with cial coordinates. The “vector” ( condition. We have solved the constraint equation in favor of it in such a way that we have two coupled first-order equations i with configuration ( where we reserve the indices the roll of the effective mass square (in the unit of quickly washed away. Since the constant solution absorbed into this guarantees an exponentialbehavior. decay of these variables as w equation of this system is given as and also given in ( { tively, in the standard sense, the decomposition ( JCAP03(2017)058 } `ala (4.6) (4.7) (4.5) Y,φ,B { , 4 vector sess gauge . } ] ) against per- ij , h 3.11 ia t , [ δQ,M,U )–( ] { non-dynamical modes (2) ) can always be done, thout time derivatives, T ij 3.8 S , h 4.2 -transform the perturba- ia ]+ t i [ roceed the analyses of each )–( e current work, we are only om. There is a gauge freedom r-fields’ internal space, which by constraint equations. We Gauge-flation. The quadratic dy the stability conditions in 3 scalar , py by demanding the stability he remaining modes, ,E (2) abilities only hint the ones tabilities and the conditions to T 4.1 i gainst anisotropic deformations ) equence of the background field nding the action up to quadratic S t the decomposition is done with y taking the spatially-flat gauge, olution ( k the form of ghost and/or gradient ,B m limit, as this would immediately lain that the stability of the tensor sor) are decoupled from each other a t, ]+ ons. This amounts to expanding the ( ) reduces to a δ ,U a 4.5 x ,U · a ]. This subsection is devoted to outlining k i ,M e M a [ 143 2 ′ Y / [ modes in the metric. This is therefore an ex- 3 k ). (2) ) V 3 } (2) V π S d – 9 – ij S 3.1 in the vector) give the constraint equations, which (2 h 0. } { ]+ i ]+ Z → ,B k a )= Y { x modes. The action ( t, δQ,M,U ( } [ ′ δ and tensor ij } (2) S , h i S ia t ,E { i = B { (2) S ) and FLRW spacetime ( Y,δQ,M,U,φ,B,ψ,E [ 2.1 = 0. Note that the vector fields under consideration do not pos (2) S are non-dynamical, i.e., they enter the quadratic action wi i S ) denotes each perturbation. The variations with respect to , vector } E } x i in the scalar sector and = and 4 tensor t, } = } ( ,B a (2) a δ E S Y Not all of the modes are physically relevant degrees of freed In performing the computations, it is convenient to Fourier ,U { a = Y,φ,B M φ,B,ψ,E { invariance and no further gauge fixing is applicable. Among t and { up to totalare derivatives, then and left therefore with can the be truly integrated propagating out degrees of freedom, expressed in termsinterested only in of studying the physical linearizedequations degrees theory of of of motion perturbati up freedom. toorder. first In order, In or th either equivalently way, to(their the expa couplings three sectors emerge (scalar/vector/ten each only sector; at the wesector first nonlinear separately level). outline in the the following Wemodes general subsections. stu in procedure Also the we long-wavelength and exp limitin then is our equivalent setup p to and theof demonstrate one the to a tensor ensure the sector background in isotro the this limit 4.1 General procedure Our goal is to investigate the stability of the inflationary s { turbations. Disastrous breakdown ofinstabilities. the solution We focus arises in in particularlead on to their the high momentu breakdown of vacuum states while low-energythe inst procedure of howevade to them. identify those ghost and gradient ins but its usefulness comesconfiguration from ( this linear decoupling, a cons regarding general coordinate transformation,ψ which we fix b where tions as Let us note in passing that the decomposition of the type ( ceptional case to the standardrespect decomposition not theorem in only tha tois the also coordinate the space caseaction but for can also models be to written like the as Chromo-natural vecto a inflation sum and of the three sectors, ( Jeans instability and may thus be under control [ JCAP03(2017)058 . , ) n ∝ † D T (4.9) (4.8) D 4.14 = → ∞ es are (4.13) (4.10) (4.12) (4.11) (4.14) ) from 3 † k a T eld space for tensor, } , ij , in the i , h i 2 D k ia D t δ ∆ { ∝ 2 2 ticularly interested in ¯ Ω 2 Ω ) with respect to ∆ he properties † D † D δ ” takes the form ∆ ∆ 4.10 − and Ω − , D the volume element ( ntegrate out those modes. As ˙ 1 δ D is Hermitian by construction, , ) rm, and therefore, the no-ghost k ˙ , -sector) is written in terms only ∆ of motion in this limit read N X ¯ is an array of dynamical modes, T 0 . or ¯ are square matrices with the size for vector and † D X 0 and therefore negligible. Eq. ( D δ ≈ 2 } k δ † D ˙ a − B D ∆ ,...,λ ∝ → ∞ 2 ,U ) becomes, again after properly adding ∆ D , or − a ¯ k
X ,λ and Ω 2 D ˙ 4.8 D 1 M Xδ H ¯ Ω { ∆ λ ∆ X for † D ˙ + δ R ¯ , X t limit. T + = – 10 – † D no ghost condition k ¯ ˙ D , X∂ ∆ D ˙ δ are derived by varying ( δ = diag( 0 + k T + 2 D for scalar, , D † D 2 are of order t ∝ TR ˙ } ˙ δ 0 † ∆ i h ¯ λ X R ¯ ¯ > 3 T∂ T i ≡ † D λ ka ˙ . It is convenient for our later purpose to choose the rotatio 3 ¯ ∆ and i T h , and ) and to the fact that ¯ 2 δQ,M,U 3 T dtd ¯ { . The non-vanishing off-diagonal components of these matric Ω 2 ka 4.12 Z = 3 ) is satisfied in large † 2 1 = Ω is in general not diagonal. We thus perform a rotation in the fi 2 dtd † ¯ = Ω is diagonal. Then the action ( T 4.13 2 , Z ¯ (2) X 2 1 S TR are real for all − † = ) so to have (we have this freedom for constant rescaling) i and Ω R = λ the number of dynamical degrees. The matrix † (2) 4.9 X . As mentioned at the beginning of this subsection, we are par ¯ S i X − N The equations of motion for ∆ The matrix = † matrix ( and thus with the cases where ( for all where such that are algebraic in thea Fourier result, space, the and action theyof of can dynamical each be modes, decoupled solved i.e. sector to (sometimes i sub Ghosts refer to the modescondition that is have the a negative requirement time kinetic te X the definitions of the matrices. is an operator equation, and we assume that the “eigenvector total derivatives, corresponding to the number of the dynamical modes and with t Thanks to the choice ( dagger denotes Hermitian conjugate, and the indication of a coupled system. Notice that we factor out up to addition of appropriate total derivatives, where limit for all the cases of our current concern, the equations where time derivatives of and can be cast into the form JCAP03(2017)058 , 2 0 2 s c G is a > ctor 16 , 2 s (4.19) (4.20) (4.15) (4.17) (4.16) e c 0 as ≥ ij h 2 ǫ , . , 2 s 27 c ) given by − k t, 2. ( to the metric tensor 4 in the function / γδǫ λ / ) t 2 system × nvenient to work in the  ) requires ,µν ih ) × ˆ k + 18 , and therefore negligible a (  + ǫ ˙ B 3 ˜ 4.14 F + is translated in terms of the left- λ ia , , and gradient instabilities, all γ ) h a 4 Π or and scalar sectors with the × ij or ( is the sound speed and µν ). In the following subsections, e fy both of the conditions of no- =( − tors individually. The conditions F × the decomposition of s ˙ = 0 0ina2 c H R,L h 3 L −  ns to ( = 0 (4.18) δ h X = 4.19 , > imposes . 2 + 4 λ ǫ = ¯ 2, 2 s ) leads to a quadratic equation for Ω , with 2+2 = 4 degrees of freedom. 2 s H + ij 3 system / c )/( c − e − ia Z ) − = 0 t + 2 )= 2 s × × h δ s 4.15 k for β 2 c ih 4.17 δc = t, a k + − ( and + ij 2 s + ¯ ia 2 h , γ X ij h , ) reduces to a cubic equation of i
h ) determines the sound speed of the dynamical 0ina3 0 2 s αc – 11 – 0 2 case, ( c =( , t are of order − > ≥ + 2 4.15 ≥ R × γ 4.15 2 ) 2 s 2 s δ )), and so we restrict ourselves to showing the explicit h a
c c β k 3 2 s − 4 2 2 c t, 2.2 3 a k − ( − for
and 2 λ 2 s ¯ for this equation are both real and positive, we thus require 2 T c h s  2 s c c (see (  ˆ k 3 system, ( , γ 4 , α det  ). For our current study of perturbations, each decoupled se 0 G × 0 λ ij in the deep subhorizon, where Π 4.15 , where we demonstrate some illustrative examples. e and 5 3 system. For a 2 i ′ 2 R,L × , ǫ> G X = dt , β> λ 0 ) limit, and the condition to evade gradient instabilities is ′ 0 t ( in this limit. Solving ( k )= 2 or 3 k k α> k/a × s t, ) and positive squared sound speeds ( , δ> ( c is not a tensor quantity under a spatial rotation, it couples at the linear level. Due to the presence of t 0 ij )), parity violation is involved in this theory. Thus it is co R ia ij h 4.13 t i h 2.4 − γ > h The usual decomposition of tenor perturbations 16 (see ( On the other hand, for a 3 constant vector. Then the existence of non-trivial solutio To ensure that the two roots of and then the reality and positivity of all the roots of mode exp In order for thethe system dynamical to degrees be ofghost completely freedom ( must stable simultaneously against satis ghost handed and right-handed canonical modes by we apply this procedure forfor the positive tensor, vector sound and speedsgeneral scalar functions sec are of extremelyresults lengthy only for in the section vect right- and left-handed basis of the tensor modes, following 4.2 Tensor sector The tensor sector perturbations consist of is either a 2 modes in large for all the solutions of ( where the time derivatives of While compared to JCAP03(2017)058 . ! ility R/L 2 3 (4.29) (4.23) (4.22) (4.21) (4.24) (4.28) (4.25) (4.26) B B 1 . ,X 2 ,W 4 2 B tion, corre- ¯ G ). After this G 1 ,X ˆ k 2 2 4 ( ,W . − ¯ 2 G and mass matrix 00 (4.27) ¯ 2 G R/L ij to be positive, i.e., 4 T . However, they do NG1 > > − Π T ¯ −
X 2 ¯ ,Z C ), yielding 2 2 T ± 2 2
4 2 B B ¯ B G

B B 4.25 , . )= . 2 2 ,X 4 ˆ k ,X s and transverse properties ,X 0 ] B B ( 2 4 4 ¯  G rated, namely L > 2 1 ereafter omit the labels G G 0 ,X ,X ,t 1 4 4 een the right- and left-handed R/L lj onal to i is indication of parity violation, 2 L , ) and ( + 2 G G
,W Π y demanding the positive roots h mponent of + 2 B 2 2 [ 4 k  4 ,
¯ ˆ k G B G + 3 + 2 3 4.23 G 0 (2) 8 L 4 4 ikl ,X S ,W 0 2 4 NG1 1 > − 2 iǫ G G i ¯ C G NG1 G ,W 2 ]+ 1 1 2 2 2 C R B + NG2 NG2 NG2 B G ,W ,W 3 − 1 2 2 ,t ≡ C C NG2 ) and ,X B ,X limit, the mixing matrix 2 4 R ˆ k G G C 4 ,W + 4 ( h 2 2 2 k ,X 0 G ¯ NG2 [ , C G 4 ¯ 2 NG2 – 12 – G C ,X − − 0 ¯ NG1 G have kinetic mixing. Following the procedure de- 2 4 L/R ij − (2) R 1 C 2
> S G ,X ,X 2 ,X 2 4 4 ,W , C 4 − 2 B R/L G G  2 ¯ t G 2 ]= NG1 G ) = Π NG1 ,Y 1 ,X 4 B ˆ k 2 . In large ia C = 4 C ( h ¯ NG1 NG1
k limit of the matrices, ( ,W − G ∗ ,t G T 3 2 and C C ¯ k ij T 4  G ,W h 2 +3 NG2 2 [ R/L ij B 4 limit and do not arise in either the no-ghost or gradient stab = ¯ C +4 1+ 2+ G R/L G (2) k T T h + ≡ = ≡ S R 1 ) = Π ), becomes diagonal. Its explicit form is T T T ˆ k ,W NG1 β β α , we diagonalize the kinetic matrix by the rotation 2 − 4 C ( ¯ denotes tensor modes. Then the kinetic matrix after the rota G h 4.10 4.1 − T 2 2 T R/L ij given the large is independent of in ( NG1 α − 2 s T C , the modes ¯ c T ¯ ,Y 2 T 2 is the polarization tensor satisfying symmetric, traceles

¯ (2) G R/L , 2 2 λ ij S a ) for k 0 ≡ In ≃ ≃ read 4.15 2 T T NG1 ¯ 2 T ¯ Ω X C whenever causing no ambiguity. scribed in section where where Π where subscript together with Π decomposition, we observe that the two handed modes are sepa sponding to The actions of the twoand sectors the do difference not appears exactly only coincide, in which the terms that are proporti Notice that Ω Then the no-ghost conditions are given by requiring every co conditions for tensortensor modes. modes Therefore is this irrelevant difference for betw our current analysis, and we h not contribute in the high and the no-gradient-instabilityof conditions ( are obtained b JCAP03(2017)058 ) ± ± (2) L Y Y can 17 S ) = have ± 4.29 ˆ k (4.30) (4.31) (4.33) (4.32) V i 6= − E )–( ( ing sub- ( ± i (2) R i e S 4.26 have kinetic ,B ). Therefore, a ± .  U 2.1 ] ,U
a − 2 2 B B ,B and rees of freedom, i.e., ,M a − ,X ,X up to addition of total 4 ± Y . As we have used the sor stability and shown 2 4 roperty and i ¯ ,U G ¯ on-dynamical modes, -axis, the components M G B z − , + esults for the most general as, in the Fourier space, − lar polarization states. This ] 4
,M } ± d plug the expressions for me procedures to obtain their ¯ ince i 2 G and − , B Y ,U he right- and left-handed modes, a  [ ) ,B relatively compact in length. We i ± 2 Y ,X k n we study some specific examples ed by satisfying all of ( eds for them are extremely lengthy sectors decouple, i.e., 4 to them lead to constraint equations (2) B − M t, ,U ¯ [ G i
( S ± ′ 3 λ V (2) ,W ,M ± + 2 ]+ i 2 S  4 + Y ¯ do not coincide. This is due to the fact ˆ G k ¯ { G is oriented along the  ,B 4 = 8 perturbations, − and rotate the system by ]= i = λ i (2) ±
+ V breaks parity (the same reason as 1 ± e i 2 × S 2 V ± ,W ,U 4.1 B – 13 – ,B 2 ). Then the 1 = G + X ¯ ± λ ˆ k 2. G ( / ,W 2 ) ± i ,M ,U 2 2 ¯ e mode behaves as a pseudo-vector due to the presence G )= + ± , hinting parity violation, comes with the terms either ), with the background configuration ( . iV 2 ± − Y k a [ ± (2) − ∓ t, − ,M U 4.1 U ,Y 1 S ( 2 (2) ± + V i )= ,Y ¯ Y S ˆ k V G 2 [ (  ¯ =( G and ± k 1 k (2) ± ± e ]= dependence in i j V S ∓ (2) ˆ k + Z a as ,B S 1 . Instead, we only show the no-ghost conditions in the follow 0 2 a ijk i 4 or linear in V iǫ G  ,U a ,Z = 2 ¯ G and ± ,M ) and a 2 R ˆ k Y [ G ( . ∓ i is the circular polarization vector satisfying traceless p 5 e (2) V back in, we obtain the action only in terms of the physical deg S λ i . They enter the quadratic action without time derivatives, e in front of its definition, ( ± ± )= basis in the tensor sector, it is convenient to work the circu To proceed to stability analyses, we first integrate out the n In the above, we have demonstrated the calculation of the ten ˆ k B B ( iab More explicitly, provided that momentum vector ǫ 17 ± ∗ i 4.3 Vector sector The vector sector is initially composed of 2 in the tensor) and also that the derivatives, and thus the variations ofthat action with are respect only algebraic in the Fourier space. Solving them an As mentioned previously, these conditionsand apply therefore to the both stability of in t the tensor sector is guarante of proportional to the difference between simultaneously. e where been gauged away), including the non-dynamical modes be expressed in terms of the explicit expressions fornow all proceed the to conditions, thestability vector as conditions, and they but scalar are the sectors expressionsand and for not follow the the sound illuminating. sa spe functions of Thus we refrain from showing the full r and that the non-vanishing One can observe that the two sub-sectors R/L sections and discuss thein gradient section stability conditions whe and where the equality holds on the constrained hypersurface. S amounts to decomposing the vector modes mixing, we follow the procedure in section JCAP03(2017)058 . , . 5 al U V β part. (4.34) (4.39) (4.38) (4.40) (4.35) (4.37) (4.36) → ∞ } modes. k and , and for ± , V 2 − α ¯ Ω  gauged away. δQ,M ˙ ). B { , i.e. = ! E } B 2 + ), 4.26 ,X 2 ¯ Ω 4 NG5 and ! ¯ 4.9 G C ψ  , denoting ˙ t B and β HB ∂ NG4 0 δQ,M,U , − { 2 C , ¯ ] X B 2 , with and 1+ ne can then observe that 2 2 − NG2 B B

   α H C ,  1 k = ,X 2 he constraint equations, obtained 2 4 0 + or some specific cases in section ¯ and + B ) 1 G ka ˙ 4 the same for the two sub-sectors and ¯ 0 0 >
δQ,M,U B X [ 3 B H − − ′ B , 1
,W 2 − 2 2 ,W 2 NG5 (2) ¯ S B 2 2 T ¯ B B . Notice that the kinetic matrices of both G S C ,XX G B 4 2 ,X = NG1 ,X ,X 4 + ¯ − G . 4 2 ,X C 2 G ]= NG4 2 4 + ¯ ¯ , and so it suffices to rotate the 2 + G G B + ¯ ) for sound speed is identical for the 0 0 T 4 G 4 ,W + 6 B ,Y 2 G M Y,δQ,M,U,φ – 14 – G 2
2 + 2 ¯ + 2 G , C 3 G 4.15 4 4 or NG3 B 2 + ˙ 0 0 2 2 ,W ¯ k 4 H C G G NG6 , we rotate the system by, as in ( 2 2 2 + 2 G C > H ¯ ( δQ 1 U H G
should be strictly positive at all times. As argued at the
2 1    2 ,W are non-dynamical and enter the quadratic action without − . The no-ghost conditions in the vector sector are to ensure − NG4 2 ± vector modes. Then the kinetic matrices after the rotation, B NG3 1 B ¯ C 1 − = ¯ T G B ¯ C T ± 2 ,W ,X S ,W Y,δQ,M,U,φ,B 2 4 2 [ NG3 R = ¯ ¯ ¯ C G G G with respect to them, we can integrate out these non-dynamic ), become 2 2 (2) + S 2 , and + 2 a k ¯ + ) is used to replace 2 T ) with corresponding expressions of S φ in terms only of the dynamical variables, (2) − S 4 , we pay particular attention in the large momentum limit, 4.8 , ,Y − ¯ S 3.5 2 G 1+2 ,Y Y (2) ¯ 4.17 S 4.1 2 G ,Y in ( denote the S 2 2

¯ G − ¯ ¯ 0, already imposed from the tensor no-ghost condition (  − G T ± = limit, one can observe that ), eq. ( > ≡ ≡ ≡ ± ¯ k T 4.35 NG2 NG3 NG4 NG5 C C C C In large Among these modes, any time derivatives (up toby total varying derivatives). the By action solving t Together with a trivial rescaling for modes and express where subscripts where where the equalitydoes holds not on have a the kinetic constrained mixing hypersurface. with O that both of the components of sub-sectors are identical, provided 4.4 Scalar sector The scalar sector contains 6 perturbations, beginning of section Therefore, the conditions to ensurecan gradient stability be are obtained by ( corresponding to this reason, the characteristic equation ( respectively. We discuss these conditions in more details f Therefore, the vector ghost stability conditions are To obtain ( JCAP03(2017)058 g , 2 also ion. B = 0, a (4.41) ), off- ,X (4.44) (4.42) (4.43) U ˙ 2 4 H ¯ G W 2.1 does not ¯ G ) is acting = 3 +4 , spoiling the )] limit on the ˙
i . ), they do not 3 W B 2 5 . The system ) in the long- B NG7 ∂ W 2 s B 2.1 ∂ C c + ,X 4.2 . We show some + 4 → ∞ 2 2 S ¯ G ǫ e Sitter solution of efore the vector sector W W k in ( ∂ ) for ∂ +2 . ). , 4 ij +3 2 and ¯ h G 1 + 3 4.18 B   ), the “vector” mode S W 1 4.26 δ ∂
( W 3 4.1 , W take the 2 tric vector mode ∂ S ¯ NG5 is concerned, ( sotropic configuration of the G ,W B ), for reality and positivity of γ 2  2 ith the configuration ( C calar sector then requires that ntee the stability of the tensor ) + ly with ess and transverse components ¯ G 3 uation ( tion ( ) 0 B Y rs in the context of the inflation 2 W at the linear order. Therefore, in ∂ 4.19 + + ∂ ) by , − − a i [ 2 k Y + ns explicitly in section 0 ( 4 2 ∂ ,W limit, δA B W 4.18 O 2 > − 2 ∂ ¯ k G in + H 0 0 +3 1 NG6 ia +4 + 2 t W 2 NG7 1 ∂ B ( C 2 2 ,W 2 , C 2 H B – 15 – behaves as a pseudo-vector on the vev ( ¯ 0 H G + a 2 ,Y Y Y > U 2 ∂ It is worth emphasizing that our approach is slightly ¯ G NG6 be positive at all times, namely −  + C + 2 S ), is stable against anisotropic deformations. As opposed NG5 2 4 0 0 0 18 , and the operator ¯ T B B C 3 0. ,Y 2
3.1 2 NG2 3 ,W ). Then the kinetic matrix after the rotation, correspondin H ¯ 2 C G → ,W ¯ might be excited dynamically and spoil the isotropic expans 2 G k   ¯ +12 i G 3.11 + 2 a i ≡− + 2 ) and ( B 2 A is a vector while )–( 2 = 2 h ,W i ,W 2.1 2 0, which is ensured by the tensor stability ( H S 2 in the de Sitter solutions we are currently interested in, NG7 B 3.8 ¯ ¯ ¯ G T G C > 0 at the linear order, which we have confirmed explicitly. Ther ,Y Y + 2 denotes the scalar sector, 1 +3 → ¯ NG3 G 1 S in the limit k ,W NG2 C 2 +3 ,W C ¯ } G 2 2 = on its right. Note that this expression already assumes the d ia ¯ B without any spatial derivatives and thus could source the me G 2 ), takes the form, in de Sitter and large 3, and therefore we impose 5 conditions as in ( ,t a i W − W values. We denote the three coefficients in ( ≡ ¯ ij NG7 G × 4.8 ¯ ,Y G 2 s δA 2 h 2 C c W { ¯ − G ≡ ¯ in ( In order to compute the stability against gradient terms, we G ,Y 2 One may speculate that due to the specific form of the decomposi ¯ T ¯ NG6 G 18  C provided that give any new condition.the The remaining two stability components against in ghosts in the s and therefore as far as the stability around such background Notice where concrete examples and illustrate these stability conditio is now 3 where subscript The anisotropic part inin the the homogeneous metric background corresponds and therefore to coincide its exact tracel would not induce any background anisotropy. with all the diagonal components of wavelength limit. This tensor mode couples to couple in the limit to the case of a scalar inflaton, even if the system starts out w different from a conventional analysis of attractor behavio to 4.5 Attractor conditionIn against this anisotropic subsection, expansion background, we given derive by ( the condition under which the i order to have thesector isotropic configuration, it suffices to guara enters in only on the background, see ( de Sitter background and then compute the characteristic eq isotropy. However, since JCAP03(2017)058 , ) n 2, to / R/L λ nary 3) 4.21 h (4.47) (4.46) (4.45) / m the ) in this 1 h, 4.45 C 4 4 all the B , ) reduces to a −
,Y  3 9 2 4 and left-handed , ¯ ,Y G ,W 4.45 B p has only constant 2 } 2

¯ ¯ R 2 G ± 3 G = 0 without time deriva- ), (
3 ,t B R/L 2 ,W + tial equation for 2 R − 2 h 2 ( B h ¯ R/L R/L ,W λHt G , 2 { 2 in the super horizon as in h h ,W 3 t ¯ 2 ,W + G d ∂ − 2 ¯ izon and do not disturb the ing an additional dynamical 2 G , 1 exp( ¯ − G R/L ng from an anisotropic metric, ds at an accelerated rate. The h, al to ,W h ∝ 2 + 3 − C ,X espect to time, we can substitute = 0 m become more and more indis- 2 ¯ ctory at least for small, linearized 4 3 sing that G l, simply to consider the stability 1 λ ¯ B ,X G H n 2 stant tensor modes in super horizon R/L 4 ,W 2 ,X + 3 ¯ h B G + 4 2 3 ¯ 1 G ¯ ,X G 4 3 2 + ,W

2 ¯ R/L 3 3 0 limit. By varying the tensor action ( G ¯ + ,Y h G
2 2 2 ,W t ,W 3 . Therefore, the necessary and sufficient con- 2 → ¯ 2 ∂ ,Y G
2 ¯ 2 2 ¯ ,W 2 3 G G k 4 1 h, ¯ 2 G h, ¯ ,W C – 16 – + G 2 C 4 1 − + 2 2 2 ¯ 4 G , and therefore it would not lead to any appreciable growth. 2 − 2 ¯ ,W + 2 H G and 0 limit, 2 Therefore, in order to have isotropic expansion as an + 4 ,W 2 are degenerate. One can show that the solution to (
2 ¯ 1 ¯ + 2 G Ht/ 3 G ¯ 3 λ → ,W G h, 19
2 − ,W ,W 3 e k C 2 ¯ 2 0. + 2 G R/L ¯ ¯ + 3 G ,W 1 , t G h 2 2 ≥ , we obtain a set of two second-order differential equations i 1 3 t ¯ ,W + 3 + G 1 2 + 2 ,W Ht/ 1 1 2 ¯ h, ), and the expansion stays isotropic throughout the inflatio 3 R/L 1 G + H∂ . t ¯ − ,W C ,W G 1 e 2 2 ,W 2 2.1 3 , 2 ¯ ¯ ) between + 6 ,W G  G ¯ + 9 2 G 2 Ht 3 2 and , 3 ¯ 1 G − B − − 0 3 3 4.46 0. This prevents the background system from running away fro R/L h, NG2 e 3 4, then two roots of h 12 C C / ., R/L − − ≥ ≡ 4 t to have only non-increasing modes, i.e. for the real parts of > h ∂ = = iso = 27 2 1 ) implies that it admits a constant solution for const C and its derivatives to obtain a single fourth-order differen R/L 1 h, h, NG2 ∝ h h, C C C C 4.45 , and their actions are identical in the R/L R/L t } h L The tensor sector consists of two sub-sectors, right-hande ,t When L 19 h or decreasing modes in this limit. By taking the ansatz the standard slow-roll inflation incan general be relativity. regarded Con isotropic as attractor, gauge since modes thetinguishable for background from local perturbed the observers unperturbed bystability inside the one against hor as anisotropic the expansion universe is expan ensured by impo using the relation ( isotropic configuration ( which shows the FLRW attractore.g. solution in Bianchi the type-I system cosmology.variable starti responsible Our for method anisotropyof at is, the the long-wavelength instead tensor background modes. of leve anisotropic This add deformations. should be satisfa provided case is fourth-order polynomial equation, which can be solved as evolution. attractor against anisotropy, we require a single conditio with respect to dition for Let us note in passing that the absence of the term proportion be non-positive, imposes { which reads, in the de-Sitter and each sector. After differentiating themall once the and twice with r where tives in ( JCAP03(2017)058 ) 0 S or he ,ǫ ≈ 2 S 4.51 ˙ (4.48) (4.51) (4.49) (4.50) ected. (4.52) G H ,δ 6= 0, at S 2 ≈ ,γ ˙ ,W B and de Sit- V 2 ¯ G k ,β , , , , , ) is V 0 0 0 0 0 in the (quasi) de α 2.1 ≥ ≥ > ≥ > 2 S S T V tively. While the ex- 6=0 or NG3 ǫ β β 1 C 4 4 NG6 27 me large , ). One immediate obser- ,W ≈ − − 2 −
ticular, first we take a few 2 T ¯ 2 V 2 xt section. For a stable in- G S ies in the high momentum , C ǫ 4.41 B NG7 o stabilize the system around 0 S δ nd the (quasi) de Sitter back- C ,X ffice to impose the following 5 S ), all of the conditions in ( > 2 4 0 (remember to take a quick evidence of ghosts in the γ ), ient instabilities. As the attractor G ion away from ( > 2.2 ) and ( 2 NG5 4.43 + 2 + 18 B ) S 4.37 2 ǫ ), respectively, those for , , C 3 NG2 S , ,W )–( , α , α γ 1 , ǫ 2 0 C 0 4 0 ¯ 0 0 G 4.28 > ≥ − 4.35 > > | ≪ > 3 S + 2 NG1 V T δ iso S – 17 – NG5 att 1 ), ( NG3 4 C C C C 2 ) and ( ,W | 2 − ) without specifying any functional forms of 4.24 ¯ + 2 S G , C δ 4.27 2.2 2 0 S ). As noted below ( 2 NG2 = 2( ), one must have a model with > C 3.11 NG3 = can be expressed as ). NG3 NG2 )–( C C , γ , β , β , δ 0 for the pure de Sitter. Also for the background evolution, t 3.8 0 0 0 0 + NG4 4.47 NG4 → , C are given in ( C ≥ > > > C ) for 0 T NG1 S S 0 is granted as long as the other no-ghost conditions are resp T att V β δ γ > α C α C ) in the theory ( 4.35 3 > − and NG1 3.11 2 S T correspond to the tensor, vector and scalar sectors, respec C γ NG4 α )–( C is defined in ( ) need to be satisfied at all times. 3.8 iso C T,V,S 4.52 Regarding perturbations, we are concerned with instabilit On the other hand, the gradient stabilities demand, in the sa . In this section, we consider a few concrete examples. In par 4 condition to prevent from rolling to anisotropic configurat and take the limit Therefore in orderinequalities to ensure stability against ghosts, it su where the definitions can be found in ( where G 4.6 Stability conditions We here collect all thethe conditions de Sitter we need solutionscondition to against for impose anisotropy, ghosts the in and background order grad dynamics, t we impose pressions for 5 Illustrative examples In the previous section, we have analyzed the stability arou ground ( and ( vation is that since limit on the background ( are to be shownflationary and solution discussed to with be concrete realized examples in in the the class ne of models ( Sitter limit; moreover, the very least, forexisting a models no-ghost of inflationary vector-driven solution. inflation. This is where and hence in the expression ( ter limit, JCAP03(2017)058 ,X by 4 2 (5.3) (5.1) (5.2) G k and A sence of or gradi- , so we study . ξ is the trace of the ], and (iii) fixed 1 ) K ds and their non- , 42 ), and its de-Sitter er-derivative ghost, ≪ , h the longitudinal mode

), read ,X 4.48 2 40 V 3.12 , B 3.11 ≈ 39 2 ,XX )–( H V ) that eliminates a non-linear, ) l is characterized by the choice 2 ξ ) ξX , 3.10 is the lapse and ξ n solely by vector fields without 2.3 given in ( ], the term proportional to ally be combined with the choice ]. Although these original works t. The original model is known to of quadratic action, as is evident from ul example that evades all the sta- + N re we include it for consistency and 42 43 2 with some constant p in ( can be classified into three main cat- att , 2 , C ν M 2(1+3 NG6 ) ,X 40 42 c 4 ξX ( C , = , 4 , where G A +3(1+4 3 4 µ 39 40 2 ) and demonstrate how ghost and/or gradient . We also impose the condition under which V , 2+ b , The third category fixes the value of ( 1 with / H + 2 K/N p ) A 5.2 39 t 5.1 , G ) ξ 2 ), driving inflation; however due to the presence ] ∂ 21 a – 18 – M 2 ) ν ( is integrated out, with our gauge fixing choice. Thus this 2 0 B | ≪ 2 = 0 on the background, the ghost becomes relevant only for A A X M φ = 2 ( µ i att [ H 0 − 4 ∂ V C ) A ]) and focus on the first two models combined. | h 2 4(1+3 ξ ], (ii) non-minimal coupling [ G ⊃ − 6 was taken, but we leave it arbitrary in our analysis. − abc / 46 A ǫ 1 43 R ( − Y 2 ) in section 2 λ − A = ] already, and we here demonstrate that the same conclusion = , and after . Since ,XX ) and 2 ξ 4.52 3(1+6 48 V N , this model does not fall in the domain of the class of models δN X G ], 2 3 , ( λ

/ ≈ 42 ij V 2 47 2 2 , g ≡ − t ]. Here we translate their results to the ones in our framewor B H H 40 ∂ − φ ) and ( 2 ). Therefore, we retract our concern from this model (the pre p 48 0. In order for this to be an attractor, one needs ( ⊃ = Y – ]. The reason is a technical one: this ghost is associated wit M | ]. All of these models have been found unstable, by ghost and/ One note to make is that in [ 2.2 ij ≈ 4.51 3 . The background equations in de Sitter, ( 48 46 = K ξ att 0. 45 ˙ [ H , where 20 2 C | φ µ → t G ≈ ∂ A 2 ˙ µ ) att B and arises from the term χ A C t χ ∂ ( ) is set to be zero by hand, but this is expected to lead to a high µ ∂ The model in question is with a potential term of the vector fiel ∝ 2.3 This higher-derivative ghost mode is not excited at the order In the original works [ = R 21 20 L µ 2 extrinsic curvature A A them together. the term of Lagrange multiplier should occur in the cubic action or higher. of an additional variable, characterized by ( with a constant existing, therefore unstable, cases in section the full analysis in [ as well as in ( egories: (i) potential driven one [ or Ostrogradsky instability, at theshow nonlinear that level, and instabilities he appear nonetheless. instabilities is studied in detail in [ minimal coupling todid gravity, motivated not include by the [ higher-derivative counter ghost, terms we proportional do to sosuffer for from a instabilities consistent [ treatmen of functions holds even with the inclusion of the counter terms. This mode the background is an attractor, i.e. norm of ent instabilities [ instabilities appear. Then we provide a simple yet successf bility conditions ( limit by illustrate the cause ofof instabilities. functions The first two can actu 5.1 Existing (unstable)Existing examples models in the literature in which inflation is drive the non-Abelian-specific structure JCAP03(2017)058 . 0 V s 2 c ies > and B (5.4) (5.8) (5.9) (5.6) (5.7) (5.5) 2 6, we S 2 p ξ / , β and 1 near de 4 M  T s − + 6 4 − c ≪ 2 S B = 2 ) α NG2 ξ ξ B 4 C NG6 − C = and 0 and , while we need to (1 . We postulate the 3 2 3 > NG2 , X NG6 W ξ 1 C ), and thus we do both. S 2 , indicating anisotropic X , ∝ W β S 4 − β c n 0, , C V 4.51 1 unity. The other two can be + 1 tion and identified the causes > 3 r and vector modes, − and ient stability conditions simul-  c the cases where S 2 with α = t instabilities in the scalar sector. l coupling to Ricci scalar is always S = B 2 ) α 2 p xistence by demonstrating a simple 4 , . H , = 0, where , M NG3 2 ,XX 2 4.49 2 S 0 V β B NG4 , G ≫ NG2 NG2 2 ≥ ξB C − 2 ξ , C 2 + C C 1 , 1 2 2 W B 2  S s − B 4 c c 1+ 1+ ) NG2 ξ = S ξ , , + 6 C ; however, the system suffer ghosts in the vector – 19 – 4 α S 1 ξ , 2 − = − W = 1 = 1 iso 1 2 (1 iso c ) , β 2 2 C ξ to be 2 C 4 T s V s 4 s S c c + ) in this model are satisfied by making the following quan- c B )+ G ) 2 p (vanishing on the background) and ξ 2 2 M 4.51 NG6 Z X,Y ), one can see that the conditions for no gradient instabilit and − ( C = 2 F 5.8 (1 G 3 for ghost stability, as mentioned below ( NG2 = ξ )–( NG2 C 2 2 2 G ,W 5.3 6 is necessary. − 2 / , C 2 1 G 1, one vector and one scalar modes are always ghosty, at least B 2 ≥ − with reasonable values of or = 1 H ξ 2 1 2 = ,XX H ,W V 2 NG5 G − ≪ C NG3 2 C = The no-ghost conditions ( By looking at ( B = 2 S NG1 ,XX α C additionally have negative values of following simple forms of and therefore The stability against anisotropic expansion requires ( and their positivity and reality can be ensured by requiring background expansion, ghosts in allTherefore, the this model sectors with and vector gradien potentialunstable and around non-minima inflationary backgrounds. 5.2 A simpleWe successful have example so far consideredof instabilities. known models In of thisexample subsection, vector-driven infla we that provide can a satisfy prooftaneously. of all e First of we attractor, turn no-ghost off and grad and scalar sectors. Note also that in the case around attractor solutions can be achieved in, for example, We then consider a class of models that contain terms linear i tities positive: Also the squared soundobtained speed by of solving one the of equation the ( scalar modes is also preserve Since Sitter. On the other hand, the squared sound speeds for tenso simultaneously. V respectively, can be solved easily, yielding JCAP03(2017)058 . , . 2 0, 0 /B → ≈ ,XY . We (5.11) (5.17) (5.14) (5.15) (5.16) (5.13) (5.10) (5.18) (5.12) ¯ 4 F  c , att NG2 2 NG6 0 C , B C + 6 ,C 4
≥ 3 c 2 , 2 2 ,
2 2 B ,Y ,c 2 and 2 ¯ ) ), i.e. 1 4 F NG1 ,c +12 2 c B , B 1 4
c 3 2 4 C 2 4 c 2
c c c NG3 0 2 , 4.49 2 O B ≫ + 3 C 2 O ,c 2 +8 − c + 2 + 8 1 1 4 , 4 ctor, we take c B , c +   B 2 4 − B | 2 2 c 2 2( ) 4 NG2 B NG3 c B O 2 c 8 = 0 c C 4 −
NG3 C 2 + 8 c 2 + B
8 , 2 3 C c  2 4 ,Y +3 B 2 c + |≫| ¯ 1 4 + NG2 F 2 4 3 2 4 ) now reduces to c ,XY 3 ns, and they are now all written B nd, which is given by the solutions c NG1 c + 6 c NG2 NG2 C B ¯ 2 c 2 | 3 2 ,Y F
C 2 4 1 c C C ¯ 2(  + 2 2 F c B c c 8 + 2 3.12 ) is preserved if ( 128 . To further simplify our analysis, we 2 −
NG1 2 4 1 c . To illustrate the stable region in the 2 2 ,X 2 NG1 NG1 Y C c 8 ¯ + 2 NG1 H B − F 3.1 + 6 B 3 C C ,Y Y 3 C , which can now be treated as indepen- 1 ¯ c F ,Y c + 2 1 2 B = = = ¯ and , = ,XY F . c and 1 ¯ 5 F 2 S H c T
V are eliminated by using the background equa-  + 2 − 3 X 1 2 in terms of ) and ( B 2 2 NG2 and 4 ,c 0 simultaneously. Among the parameters in this 3 NG3 + 2 ,X c – 20 – B +3 1 4 − c ¯ H 2.1 ,XY F c  i ¯ > , δ ,C 4 F 3 ,Y = + 9 c c 3 , β NG6 ,XY
¯ F 3 O ) ¯ 4 1 2 and F ,
 and 2 NG2 , C 1, the expression ( ,C 1 2 NG6 B ) c − ,c + − 1 2 ,XY ¯ 2 2 1 4 c ) F ¯ 2 ,c − ,C 6 F c B 2 , β ,C 1 4 , ,Y B B +3 2 c NG3 . Then all the stability conditions are expressed by the in- ) B c ¯ | ≪ B 4 1
1 F ,Y 2 ) 2 4 c c 0 2 c H ¯ c O NG1 2 c F ,C 2 NG3 att is a function of ,XY 0 and
 + 3 c 2 2 O ,c + NG6 ¯ + 2 + + C c F 1 4 1 − ,C | NG3 3 F C ,C ≈ c ) c 1 + 4 has properties , B +4( c NG2 C c + 3 4 ˙ ,Y ,c ,Y c B 3( ,Y (2 1 ¯ ,XX F NG3 F 2 ¯ ,C O NG2 ¯ c NG3 ¯ F F c F 6 + NG3 NG2 − ≈ C 4 C + = 6 +4( c 2( ,C 3 2 C B C ˙ NG1  c 2 4 H − B B ) 2 c ,Y 4 NG2 2 4 +3 2 ¯ 2 2 ,C c NG1 F 2 NG5 NG2 NG2 NG1 ,Y C c c B 4 2 2 2 ¯ ,Y 2 4 2 1 C C − F c C C C NG1 ¯ c ,c F 32 8 3 2 C ), i.e. 2 + 2 h NG1 = = = c c + 3 1 4 NG1 NG1 C 8 c )). Further, by considering the de-Sitter limit of the attra 2 2 28 ( C C 3 ( 3.11 ¯ 2 are constants and F NG1 NG3 NG6 B 3 2 H i = = = = C C C )–( c 5.11 − = S S T V ǫ = γ α α ≈− 3.8 iso 3 att C c C consider the case where replace them in thein expressions terms of of other stability conditio where For the attractor condition and we require is satisfied. The parameters for the no-ghost conditions are where bar denotes quantities on theto quasi de ( Sitter backgrou model, we can express 6 The isotropic configuration described by ( tions ( dent model parameters (note that with which we can replace dependent parameters finding parameter space, we expand those expressions in the limit JCAP03(2017)058 ) 0 0, > > 4.48 2 (5.19) c 2 V α tor model 0, are then ≈ ≈ V fulfilling ( ˙ β 0. Moreover, in H 4 ), are demanded, ). This is, to our > − 2 S ). For these solutions 2 V δ 4.49 4.52 have presented a sim- 2 S α the stability conditions, γ depend on at least one 3.9 . Notice that the model 2 0, ). Proceeding the analysis 2 ≈ is set of conditions. The gauge symmetry is broken, respect to the vector-field l global O(3) symmetry as paper, we have explored a G /B decoupled sectors, “tensor,” 2 S > ) and ( . Within the class of model instabilities and applied it to or fields are solely responsible ǫ ) and ( belian gauge fields. which inflation is driven only ρ e 4.47 onary backgrounds, spectra of outlined the general procedure 2 T ν ns can be satisfied when ational data. This implies that 27 ) onary solutions driven solely by NG2 t after ( sustain a sufficiently long period α b s it visible that the simultaneous 4.48
( 4.51 tudinal modes of the vector fields ry attractor that is stable against − 0 2 ired background. While some non- ,C F ≈ in most of vector-driven inflationary 3 S 0, immediately follows, unless there ,c , ǫ ) and ( µρ 1 4 ) T S c ≈ a δ β ( NG1 4 S ˙ 3.12 F B O C γ ) − b ν ( + ≫ 2 T A 2 ) + 18 α 2 a B ( µ S B 2 4 ǫ – 21 – 2 c A NG1 3 c S 2 c C γ = 4 4 2 | ≫ − = 4 W ), 3 c S | δ iso 4 C 5.13 and − ρ 2 S ν δ ) ). Quasi de-Sitter solutions for the background, 2 S b ( γ are also fulfilled, as F 2.2 2 s µρ ), these terms are therefore necessary ingredients for a vec c ) b ( 2.4 , provided that 0, and F ) a )–( > ( ν NG3 ) that has an inflationary solution as an isotropic attractor A 2 S ) 2.2 γ a 5.9 ( µ > C ≈ A ) and that satisfies all the stability conditions ( S = δ NG1 4.49 3 Albeit some required hierarchy and tuning of parameters, we In order to perform a general analysis and then to narrow down 1 C − W 2 S with inflationary solutions without ghosts. we have taken theour starting Generalized point, Proca see theory ( with an additiona and thus the isotropic condition is also satisfied. Hence we can immediately see that all the stability conditio is some fine-tuning of the model functions, as discussed righ sought for, and the “slow roll” of the vector vev, ple model ( and we have provided theirof explicit linearized expressions in perturbations, ( “vector” we and have decomposed “scalar,”internal them where space into as the 3 wellto decomposition as obtain to is the the conditions done spatialeach to coordinates. sector with evade separately. We both have The thefulfillment set ghost of of and no-ghost gradient them conditions make requires (not necessarily suffices) to hav and ( to be an isotropic inflationary attractor, two conditions, ( knowledge, the first exampleby of vector fields stable without inflationary relying models on in the terms specific to6 non-A Conclusion We have demonstrated that, in afor new inflation, class of quasi models de-Sitter whereall solutions vect the pathological can instabilities. be Itmodels, an has inflationa violation been of known that, gaugeof invariance is accelerated invoked expansion, in but orderthat it suffer to leads from ghost to and/or propagatingAbelian linear longi gauge instabilities on field the models des perturbations are in known to those provide modelsthe stable gauge are inflati symmetry incompatible should with bea observ broken more in those general models.new class class Once of of the model modelsvector Lagrangians fields. is that allowed/motivated. allows for stable In inflati this and of Lagrangian ( has sufficient numberreality of conditions independent of parameters to achieve th γ the same limit, we have, from ( JCAP03(2017)058 ,X ass 4 G des are ex- edom in the ] and the large , is motivated by 144 models of inflation ξX 2+ / 0 and any non-negative nary models of the early 2 p terms, as the vector mode the one with non-minimal )). Mass eigenstates with e CMB [ ility conditions for the gen- & y provide the origin of the erturbations are dynamical, his may have some impacts M Z parameter window to satisfy ous non-standard interaction nsor-to-scalar ratio can con- case the attractor condition, . The first example resembles onsidered, there is no helicity essful model without instabili- lso helps to make the analysis espect all of the attractor and = ational waves are expected to 4.22 e linear combinations that are att 6= 0 on the background, where X r-order correlation functions are the terms proportional to 4 s. Cross correlations, especially ut successful. Phenomenological erturbations. Two examples are 4 C he second example is truly stable the high momentum limit — and he unique behavior of the scalar, ially leaves distinctive signatures f interest for further studies. The ts such as binary systems excite c hese counter terms, the model has G ,Z nce of 2 ons are quite lengthy in expression + ], one of the distinguishing features G 3 c ) and as they propagate towards gravitational 108 = X , 4 ia ( t G V 107 − and , they are not mass eigenstates but are linear Y – 22 – ij 2 h = W 2 2 c G + 1 is allowed. Compared with the previous model of dark W ], coming from this model would be distinguishable from the 1 att c C 148 )+ – ], may also provide a unique arena to test the models of this cl 145 86 X,Y ( F 4, parity is broken, and the left- and right-handed tensor mo ), those that eliminate the Ostrogradsky ghost degree of fre / = 2 µν 2.2 ) G a ( ˜ F ) 1 for the inflationary attractor, is replaced by a ( µν F − While the primary interest of the present paper is in inflatio We have obtained the attractor, no-ghost and gradient-stab Our aim of this paper is devoted to the demonstration of viable | ≪ in fact behaves as a pseudo vector). = att a C or slightly-negative value of energy based on the generalized Proca field [ U universe, the quasiaccelerated de expansion Sitter of solutions| the that universe we today have as found well. ma In this (notice that parity violation exists even without the prese scale clustering fossils, [ parity-odd correlators [ different masses thenexhibit propagate oscillations differently. with the Thus other the tensor gravit modes ordinary realizations of the scalar, vector and tensor mode of the current modelon is gravitational-wave the physics. existencethe of While ordinary extra astrophysical gravitational tensor objec wave modes. modes T eral case. In orderall to those show conditions, that there weof exists have gradient a chosen stability model simple more that examples.for has illuminating, the a This since general a the case,considered: conditi especially the those first of one, specified vector by and scalar p nonlinear orders. It isno shown parameter that region despite to the have inclusionmodel stable of — inflationary t at solutions. least T againstwe ghost have and explicitly gradient shown instabilitiesstability that in conditions. it admits a viable scenario to r wave detectors. This implies that the detectors will observ Z that are healthy atconsequences the of theoretical such level, pathology-free whichobservational models has are turned bounds therefore o on o strain/falsify the these scalar models. spectraldependence In index in the and the simple the tensor example te sector, we but have c for models with combinations of them (see the off-diagonal component of ( pected to be producedas by opposed different to amounts. scalar-tensorin Also, theories, the the CMB/LSS and vector and theirterms p other fate should observable arise potent spectra. at cubicexpected Moreover, and vari to higher have orders, interesting andvector thus features. highe and In tensor addition, modes due in to this t model, we would expect that th a few existing, unstableties, models, given and by the second is a new, succ in the action ( the combination of thecoupling to vector field gravity, while model the with difference a is potential the and inclusion of JCAP03(2017)058 ] , ]. The Four year 149 , (2002) 103522 , hep-th/0103239 [ D 65 .N. was supported in -Lifshitz gravity , , ]. r International Research (2009) 084008 orted by the NSFC grant d basic results ITP for their invitation as , ty through the CRF Grants Gill, as well as by the World JSPS) Grants-in-Aid for Sci- SPIRE alogous to neutrino oscillation il (NSERC) of Canada and by Phys. Rev. ant No. 2016M590134, MEXT D 79 IN ]. , EXT, Japan. The work of SM (2001) 123522 kson, P. Keegstra et al., ]. s a new theoretical basis for the e ]. avity in the literature [ ] [ of this work and their later collab- The Ekpyrotic universe: Colliding SPIRE ]. D 64 SPIRE SPIRE IN IN IN ]. ] [ Phys. Rev. ] [ ] [ , SPIRE IN New Ekpyrotic cosmology Galilean Genesis: An Alternative to inflation SPIRE ] [ IN Phys. Rev. – 23 – , ] [ arXiv:0904.2190 ]. 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Jac different from those at thehas source. been This considered kind in of the phenomenon an context of bimetric theories of gr quasi de Sitter solutionsgravitational found wave in oscillation. the present paper provide Acknowledgments R.E. and Y.Z. are verywell much as grateful their to both very warm oforation. hospitality Kavli during IPMU The and the work Y initial of stage of R.E. the was supported Government of by the Hongpart Hong Kong by Universi Kong the SAR Natural under Sciencesthe HKUST4/CRF/13. and Lorne R Engineering Trottier Research Chair Counc inPremier Astrophysics International and Research Cosmology Centerwas at supported Initiative Mc by (WPI), Japan M entific Society Research for (KAKENHI) the Promotion No.Center of Initiative 24540256, Science (WPI), and ( MEXT, by Japan.No. World The 11605228, Premie work the of ChinaKAKENHI YZ Postdoctoral Grant was Science Nos. supp Foundation Gr 15H05888 and 15K21733. References JCAP03(2017)058 , , th the , ] , ]. 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