JHEP11(2007)076 hep112007076.pdf August 23, 2007 es. November 6, 2007 November 26, 2007 universe cosmol- Received: plest model relies on two Accepted: for a wide range of initial a small positive mass term a wide range of potentials Published: rotic theory is successful in andard big bang cosmology. ility along the entropy field ction in field space. All other o exponential potentials, one ponential potential for one of ropose a simplification of the nvariant spectrum of density the initial conditions. In this rent observations. ropriate values well before the he spectral index and possible he spectral index which is easily tly widens the class of ekpyrotic [email protected] , http://jhep.sissa.it/archive/papers/jhep112007076 /j Published by Institute of Physics Publishing for SISSA Cosmology of Theories beyond the SM, Spacetime Singulariti New Ekpyrotic Cosmology is an alternative scenario of early [email protected] [email protected] Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5,E-mail: Canada Department of Physics, ThePhiladelphia, University PA of 19104-6395, Pennsylvania, U.S.A. E-mail: SISSA 2007 c Evgeny I. Buchbinder and Justin Khoury Burt A. Ovrut Abstract: ° ogy in which thescalar universe fields, existed before whose thefluctuations. entropy big perturbation bang. The leads The ekpyroticdirection to sim solution which, a a has scale-i priori, apaper, appears tachyonic we show to instab that require thesefor fine-tuning can the of be achieved tachyonic naturally fieldconditions, by and adding the coupling tachyonic field toonset gets light of stabilized fermions. the with ekpyrotic the Then, phase.solving app the Furthermore, we flatness, show horizonMotivated that and by ekpy homogeneity the problems analysis ofmodel of in st the terms tachyonic of instability, newfor we field each p variables. scalar Instead field, ofthe it fields requiring suffices and tw to a tachyonic consider massterms a term in along single the the nearly orthogonal potential ex dire potentials are essentially and arbitrary. allows This substantialnon-Gaussianity. grea freedom We in present a determining generalized t consistent expression for with t the observednon-Gaussianity red can tilt. be substantial, We within also the reach argue of that cur Keywords: for On the initial conditions in new ekpyrotic cosmology JHEP11(2007)076 9 2 8 11 40 40 40 18 22 24 31 36 39 12 13 15 39 41 27 38 12 gy 20 – 1 – starts to roll reaches minimum χ V φ oscillations χ about minimum χ 7.3 Mass of 7.5 Energy density in 7.4 Exit happens before 4.1 Contrast with inflation 4.2 Flatness and4.3 homogeneity problems in Flatness inflation and4.4 homogeneity problems in Observational ekpyrotic constraints cosmolo on model parameters5.1 Constraints on5.2 initial conditions Stabilization mechanism 6.1 Converting entropy6.2 into curvature perturbations A bouncing6.3 scenario 18 Summary 20 7.1 Global minimum for 27 7.2 Exit from ekpyrosis: 2.1 Tachyonic instability 2.2 Instability and scale-invariant entropy perturbation 3.1 Generalized potentials 3.2 Spectral index3.3 for general potentials Non-gaussianity in general potentials 8. Conclusion 6. Graceful exit and bouncing 7. A specific exit model 2. Two-field ekpyrosis Contents 1. Introduction 3. Generalized ekpyrotic potentials 4. Flatness and homogeneity in ekpyrotic theory 17 5. Initial conditions and taming the instability 21 JHEP11(2007)076 with ζ plete scenario which densation [20], which ithin a consistent effec- erefore the prediction of [3, 14] — the curvature of near-future microwave standard inflationary big t drawbacks. solved by stringy physics. ζ -mixing and impart ansion shortly after the big lic models [9, 10] to propose f on, many have argued that rly vacuous initial conditions tructure formation: a nearly t the scale-invariant growing ogies [23]. In [18] we showed eziano [7]. nce in dynamics, remarkably ients of the old ekpyrotic sce- continuously, resulting in an usly leads to a scale-invariant ut in a cold state followed by ates a non-singular bounce, all rotic scenario does not [1, 6]. ivable that stringy or higher- ity. In the context of colliding ally true for any non-singular al dimensions remaining finite. tly with the preceding ekpyrotic ty perturbations. One testable for violations of the null-energy ζ ic instabilities [8]. This lead the gularity corresponds to the fifth atter is a useful variable to track m for a scalar field, akin to k- finitive example of a cosmological le-invariant primordial gravitational – 2 – Second, the fate of perturbations through the bounce, and th Until recently, ekpyrotic theory sufferedFirst, from it two was importan unknown how to describe a non-singular bounce w In a recent paper [18], we proposed a fully consistent and com The physics of the bounce exploits the mechanism of ghost con tive theory without introducing ghostsadvocates or of other the catastroph Big Bang/Bigthat Crunch ekpyrotic the [2, universe 3] and undergoesThis cyc seems a plausible big in crunchbranes light of in singularity the heterotic to M-theory mildness be [11, ofdimension 12], the shrinking re for singular to instance, zero theDespite size, sin intense with activity the in recent large yearsbounce three [13], in spati however, a string de theory is still missing. bang paradigm. Instead ofbang, invoking ekpyrotic a theory flash proposes ofa that exponential long exp the period universe started ofthe o slow two contraction. models make Despitescale-invariant, identical this adiabatic predictions stark and for differe Gaussian thedistinctive spectrum origin prediction is of of that s inflation densi alsowaves generates sca whose amplitude inbackground the polarization simplest experiments models [5],The is idea whereas within of the a reach ekpy pre-bigoriginated bang in contracting the phase arising pre-big from bang nea scenario of Gasperini and Ven scale invariance, is ambiguous.mode This of stems the from the scalar fact field tha fluctuations precisely projects out o 1. Introduction The ekpyrotic scenario [1 – 4] is an alternative theory to the perturbation on uniform-density hypersurfaces [15].since The it l isthe conserved outcome on of super-horizonunacceptably the scales. blue bounce spectral For would tiltbounce this amount [16]. within reas to 4d matching While Einsteindimensional this effects gravity, relevant is it near univers is the nevertheless bounce conce may lead to mode a scale-invariant piece [17]. addresses both issues. Whilenario sharing [1, many 3, important 19], this ingred modeldescribable of within New Ekpyrotic a Cosmology 4d gener effectivecurvature field perturbation. theory, and unambiguo relies on higher-derivativeinflation corrections [21] to or the k-essence [22].condition kinetic (NEC) Ghost ter condensation without allows introducingexplicitly ghosts how the or ghost other condensate can pathol be merged consisten JHEP11(2007)076 ang and ility onds of the φ , each 2 , as well φ rom pure φ e through and 1 φ n and generalization ds, meters: the usual fast- pyrotic potential — see ew field variables string theory, remains an to the adiabatic mode by pansion. Whether ghost f the bounce is all within as fundamental fields, the describing the difference in , the model has a tachyonic he field trajectory. This is pply. For our purposes the δ ctral tilt for the most general χ w how such initial conditions if a smooth bounce is realized ugh a rotation in field space, ds, New Ekpyrotic Cosmology bes rolling down an ekpyrotic nt [18, 26 – 28]. See also [29 – essary for ending the ekpyrotic . By having two scalar fields, rbation — corresponding to the ng a scale-invariant spectrum is ation of such a bounce. on phase. and ility is at the origin of the growth rder for the field trajectory to start directions must be nearly the same: φ φ = 0) were found to give a blue spectrum and η χ steep and quasi-exponential potentials becomes – 3 – allows for even greater freedom in the spectral δ two steep and nearly exponential potential along directions. The class of potentials studied in [18] corresp single φ remains fixed at a tachyonic point, thereby making the instab , leaving tremendous freedom in specifying the global shape characterizing respectively the steepness and deviation f and χ χ χ η goes through the bounce unscathed and emerges in the hot big b acquires a scale-invariant spectrum well-before the bounc ζ ζ and , while ǫ φ ) perspective leads us to propose a significant simplificatio . φ, χ ,χχ V = 0. Here we see that turning on ≈ In our model As pointed out in [18, 26] and studied extensively in [32, 33] Our original scenario [18] was cast in terms of two scalar fiel The ( This novel framework greatly expands the class of allowed ek δ , respectively the field directions along and orthogonal to t ,φφ condensation can be realizedopen in issue a [24], UV although see completeby [25] theory, very for such a different as recent physics, attempt.ghost many condensate Even of provides an our explicit key and results tractable will realiz stillthe a conversion of entropyeach with (or their isocurvature) own ekpyrotic perturbations potentials,difference the in entropy pertu the scalar31]. field In fluctuations [18], — weusing is showed features how scale-invaria in the the entropy potentialphase mode which and gets are bridging converted independently in nec into4d the effective bounce. theory, Then, since the physics o phase to generate a smooth transition from contraction to ex χ phase with a nearly scale-invariant spectrum. instability along the entropy direction.of entropy Since modes, this the instab tachyon iswould ostensibly unavoidable require [34]. fine-tuned As initial it conditionsout in stan o rolling along thiscan tachyonic be ridge. achieved In naturally this with a paper, pre-ekpyrotic we stabilizati sho reviewed in section 2.potential along In this language, the solution descri with their own steep,following negative [32] exponential we potential. can instead Thro study the dynamics in terms of n manifest. unnecessary. All we need is a as a tachyonic mass for of the scenario, described in section 3. By treating potential. The only constraintthat which the is curvature crucial of in the generati potential along the origin of the potential in terms of the exponential form of the potential, as well as a new parameter curvature between the figure 1 for a genericNew example. In Ekpyrotic section potential 3.2 we androll derive find parameters the spe that it depends on three para V to tilt. In particular, pure exponential potentials ( JHEP11(2007)076 . These self- χ he arrow indicates the ]. This led [18, 26] to erms in scenario along these lines. . erated during the ekpyrotic atness problems of standard χ ls the tilt can be red by having rotic scenario [1] it was initially ar the assumed degree of initial el of non-Gaussianity of the per- nt and near-future experiments. 6= 0), which does allow for a small gative and quasi-exponential potential η – 4 – direction is part of a more general discussion of χ . δ General shape of the potential during the ekpyrotic phase. T , while remaining perched on top of a tachyonic ridge along φ In section 3.3, we discuss the generic form of higher-order t The tachyonic instability in the initial conditions in New Ekpyrotic Cosmology, in particul flatness and homogeneity. In the contextbelieved of that the the original model ekpy didbig not address bang the cosmology homogeneity and [36]. fl See [37] for a recent critique of the desired solution, corresponding toalong rolling down a steep, ne red tilt. Here, however, wenon-vanishing see that even for pure exponentia in [18, 26, 27],consider deviations in from disagreement the with pure the exponential form recent ( WMAP data [35 interaction terms play a crucialturbation role spectrum. in determining We the find lev phase that the is non-Gaussian generically signal substantial, gen within the reach of curre Figure 1: JHEP11(2007)076 ng cou- s the flatness, homo- uning of initial condi- ergy domination in the e one assumes that over erms the assumed initial ic Cosmology resolves the sponds to drastically dif- words, this term serves to ial is sketched in figure 2. enough e-foldings of scale- , spanning many orders of 0], where homogeneity and ot big bang phase with the d the ekpyrotic phase lasts ver that exponentially fine- ic phase in order for the roll logy. oth. Therefore, making the nd, from this point of view, ion, but by no means is this at early, pre-ekpyrotic times, d homogeneizes the universe ekpyrotic model does in fact ous and flat to allow inflation ncrete and rigorous theory of y without invoking any phase inflation assumes a hot initial nd anisotropic stress. In other lar field blueshifts much faster ed level of flatness and homo- phase makes the universe more e the subsequent generation of roposes a cold beginning, with ning by introducing a positive ility of the two-field model. In ly homogeneous, isotropic, and χ s quickly converted into thermal tic attractor mechanism — during nable. With respect to flatness and – 5 – is exponentially close to the top of the tachyonic χ , which is relevant and stabilizes starts oscillating around its stable minimum. By introduci χ energy density, χ χ In section 4, we argue that New Ekpyrotic Cosmology addresse This is precisely analogous to inflationary cosmology. Ther In this paper, we address in a simple and natural way the fine-t plings to light fermions,radiation. the energy We in find thesemagnitude that oscillations in i for initial a wide range of initial conditions Such criticism motivated the advent of the cyclic model [9, 1 flatness in a givenprevious cycle cycle. is It achievedsolve was these through recently problems a realized, [38], phase in however, a of that way dark the akin en to inflationary cosmo geneity and isotropy problems ofof standard cosmic big acceleration. bang cosmolog This reliesthe crucially ekpyrotic on the phase, ekpyro the energythan density any in other the component, ekpyrotic inwords, sca particular contrary spatial to the curvature naive a expectationinhomogeneous, that here a contracting the universenatural becomes assumptions that increasingly the smo has proto-ekpyrotic curvature patch comparable is to fair for the sufficient Hubble number of scale,high e-folds, then, level the of provide universe symmetry emerges westandard observe in problems today. the of As the h such, big New bang Ekpyrot model. some macroscopic patch the universeto is sufficiently take homogene over.to Subsequently, the cosmic high acceleration levelradius flattens we of an curvature observe is ita much today. larger drawback in of Of ekpyrosis one course,ferent than model assumptions in in about versus absolute inflat the the t state, initial other. with state correspondingly of Instead, high thecorrespondingly this Hubble universe: small corre scale; expansion ekpyrosis rate. p initial In conditions, the either absence starting of pointhomogeneity, a is the co equally proper reaso yardstickgeneity is, compared therefore, to the theboth assum respective models initial are Hubble equally successful. radius. A tions associated with thesection aforementioned 5 tachyonic we instab quantifytuned the conditions must nature be of satisfiedalong the at the the instability tachyonic onset and ridge ofinvariant disco the to perturbations. ekpyrot last We sufficientlymass then long squared solve to term this produce for initial fine-tu ridge by the onset of the ekpyrotic phase. The modified potent but becomes negligible duringset up the the ekpyrotic desired phase.perturbations. initial conditions Thus, In but other does not jeopardiz JHEP11(2007)076 χ direction. χ ss around the original model [18]. s shown in figure 3. This heless the energy density otic phase, the subsequent gle-field ekpyrotic motion tions for the ekpyrotic phase. ical bounce. As described ore the energy density in re not damped by the light s terms which eventually drive . In this part of the story as well, by imprinting the scale-invariant s initially stable in the me positive and flat. Thus, shortly s down to the minimum for a wide range rolls away from the tachyonic ridge, these χ – 6 – keeps on rolling, due to the ekpyrotic attractor entering a ghost condensate phase which leads to φ . φ ζ reaching a minimum of the potential followed by a steep φ rolls to the minimum and starts oscillating around it. Its ma χ A pre-ekpyrotic mechanism sets up the desired initial condi ) picture leads to tremendous simplications compared to our rolls off, we envision . φ χ quickly becomes subdominant as φ, χ The rest of the story consists of In section 6, we turn to a discussion of the exit from the ekpyr Thus χ away from the tachyonic ridge and towards a stable minimum, a a violation of the NEC and results in a non-singular cosmolog in [18], this requires theafter post-ekpyrotic potential to beco along mechanism. The dynamics therefore effectively reduce to sin minimum is generically largeblueshifts compared as to a Hubble, dust andfermions theref component. of the Note pre-ekpyroticfermions that phase become since, the as oscillations heavy and a in can be henceforth ignored. Nevert is necessary to generate a turnentropy in perturbation the spectrum field onto trajectory, there Figure 2: By introducing couplings to lightof fermions, initial the conditions field by settle the onset of the ekpyrotic phase. NEC-violating phase and the resulting non-singularthe bounce ( To bring the ekpyrotic phase toχ an end, we include correction With the addition of a positive mass-squared term, the field i JHEP11(2007)076 φ, χ , which allows eld keeps rolling φ ential which is small rent status of New tory parallels that of our eads to a number of con- e. None of these conditions here. From the onset of the tic energy in on brought about by the te that although the potential eed one ghost condensate field, an be satisfied for a wide range anges only by a factor of order chyonic ridge. The field rolls down .2 and summarize in section 6.3. oscillations must be small enough to allow the – 7 – χ direction and starts oscillating around it. Meanwhile the fi χ direction. The end of the ekpyrotic phase is triggered by a term in the pot φ The successful merger of ekpyrosis and ghost condensation l In the Conclusion in section 8, we step back and assess the cur to enter a ghost condensate phase. While most of this bounce s φ rise to positive values whereis the flat potential and becomes flat. positive,ghost there (No condensate is phase no untilunity.) reheating, sustained the inflationary Climbing scale up phase factor to ch this plateau greatly reduces the kine along the at early times butto eventually a pushes minimum the in field the away from the ta Figure 3: sistency conditions which we deriveFor explicitly example, in the section energy 6 density in the original scenario [18], a keyas difference opposed here is to that two. weframework. only This n is another considerable simplificati ghost condensate to dominate the dynamicsare and very lead constraining, to however, a and bounc weof illustrate parameters how by they studying c a specific potential in section 7. JHEP11(2007)076 , ζ 1: ropy (2.1) GeV. ≪ e that 2 reaches 18 p . During φ 10 × beg ), c pyrotic phase − 4 t . 1 and ek . t = ≪ = 2 ¶ t 1 Pl 2 Pl p φ M M d in figure 4. Starting 2 2 after ( p radiation blueshifts more ngular bounce. Not long eau, and enters the ghost oth at the theoretical and r e-invariant spectrum for denoted by ation. discussed in [18, 26]. In this is converted into matter and − , each rolling down a steep, ng of the hot big bang phase begins oscillating around the ic terms; all higher-derivative 2 µ ss, let us assume for now that φ φ χ ent powers ) is nearly constant, whereas the t ( exp a 2 and V ) is standard: the universe successively 1 0 − φ t ¶ , the field = i Pl 1 t t φ M 1 – 8 – 2 p r is pushed away from the tachyonic ridge and starts − , eventually the latter takes over and dominates the µ χ φ exp 1 V Sequence of events in New Ekpyrotic Cosmology. , when − direction becomes tachyonic, allowing for the growth of ent end χ − )= 2 ek t shrinks by an exponential amount. It is also during this phas , φ is chosen to be the “reduced” Planck mass where 1 1 Figure 4: φ − are amplified and acquire a scale-invariant spectrum. The ek ( Pl H χ V M ). The rest of the story until today ( reheat t The sequence of events in New Ekpyrotic Cosmology is outline = t radiation, which reheats the( universe and marks the beginni oscillating around a new minimum — see figure 3. Shortly there a minimum in thecondensate potential, phase. climbs up This onafter violates the the the flat universe NEC positive starts and plat expanding, leads the to energy a density non-si in negative and nearly exponentialthe potential. potentials are For pure concretene exponentials, in general with differ 2. Two-field ekpyrosis The ekpyrotic mechanism for generating a pre-big bang, scal undergoes epochs of radiation, matter and dark energy domin comes to an end at as described in [18], relies on two scalar fields, Hubble radius fluctuations in this phase, the universe contracts very slowly, so that perturbations. This marks the onset of the ekpyrotic phase, energy. Meanwhile, the stable minimum and decays intoslowly radiation than — the see figure energy 2. density Since in Ekpyrotic Cosmology compared withphenomenological the levels. inflationary model, b from a cold initial state at some initial time In this paper, Potentials that deviate from thesection, the pure two exponential scalars form can were be taken to have canonical kinet JHEP11(2007)076 and and (2.5) (2.3) (2.2) (2.4) 1 φ φ ; , ! ! t t )) )) 2 2 p p 1, (2.3) describes a + + 1 1 , 1 for the background, ≪ p p 2 2 1 2 an attractor because of 3( 3( φ ≫ V V 1 new field variables, ,p tudied extensively in [32, , in which case the scaling − − 2 p 1 w stant potential energy of a er, as we show later in this p cs and is the origin of the p ich are negligible during the not √ ectory exponentially close to (1 (1 ow for a scaling solution: shifts slower than any of the 1 2 + der these assumptions, ce eld trajectory — the so-called − p p ed expansion is an attractor in th equation of state 1 d anisotropic stress. This is the 1 le-invariant spectrum of entropy p 2 2 Pl Pl . te and produce a smooth bounce, φ ; √ . 2 1 0. For M M p 1 2 ,φ ,φ s s ≫ √ → V V t − − 1 = − − Ã Ã − χ = = ) 2 log log 1 2 ˙ ˙ p φ φ ; Pl Pl much faster than all other relevant components – 9 – + 2 ; H H 2 M M p 1 1 2 2 p p p t φ + 3 + 3 + 2 2 2 3( 1 2 1 2 p ¨ ¨ p φ φ p p p = √ blueshifts + = w + 1 ) = ) = 1 p t t H φ ( ( √ 1 2 1 p φ φ √ ; 2 = p φ + 1 p ) t − ( ∼ ) t ( is chosen to be negative and increasing as a t Ekpyrotic dynamics was first studied for a single scalar field The tachyonic instability is most easily seen in terms of two satisfy the standard scalar equations of motion , given by 2 slowly-contracting universe driven by a highly stiff fluid wi are assumed to be negligible during this ekpyrotic phase. Un self-interactions, necessary to generate a ghost condensa where These equations, combined with the Friedmann equation, all φ the corresponding energy density precise analogue in aan contracting universe expanding of universe. why accelerat slowly-rolling In scalar the field latter comesabove context, to components. the dominate nearly because con it red 2.1 Tachyonic instability In the two-field case, however, the scaling solution (2.3) is — curvature, matter, radiation, coherent kinetic energy an χ solution has the property of being an attractor. Indeed, sin This cosmology isgeneration the of essential a scale-invariant feature spectrum of in this ekpyrotic context. dynami a tachyonic instability inentropy the direction. direction orthogonal This to33]. was the This first fi instability pointed isperturbations out essential to and, in the hence, [18, generation cannot 26]paper, of be and a it removed sca is s [18, easy 34].ekpyrotic to phase, Howev include but further which terms atthe in early desired the times one. potential bring wh the field traj JHEP11(2007)076 (2.6) (2.9) (2.7) (2.8) (2.13) (2.11) (2.12) (2.10) , ¶ , as we review Pl ; δχ /M ) t , ! χ t − ¶ ) χ p ( 2 . . . 3 p . Taylor-expanding (2.7) q trum for + − ´ 1 2 0 2 p p Pl . V ) (1 t p q . χ ¶ e 2 /M Pl 1 2 2 t ¶ p − p V V M 2 Pl 2 1 2 χ t φ direction. This tachyonic ridge is pχ ( + p p s ; , 1 M 2 χ Pl µ Pl − 2 p ,χ ,φ /p 1 ≈− 2 V Ã V /M r p ) pM log − − t − ,φφ ) ,φφ χ p log 2 = V = µ − eff p − direction on the top of the tachyonic ridge in 1+ Pl χ ˙ ˙ 2 χ ( V φ ³ = p µ 2 + p φ 1 exp H H remains constant on the background trajectory. = given by 1 – 10 – pM p Pl q 0 . p ,χχ 2 t exp ; 2 1 χ direction. To see this, note that the potential (2.1) φ V t χ p p V 1 + 3 + 3 t 2( p χ p = − φ/M χ V as [32, 33] q ¨ ¨ φ χ χ ) 2 p | r − 1 = = χ e Pl q )= ) = 1 ,φφ /p χ t p − H φ p 2 V ( M e ( p and 0 φ µ eff ≡ V , we find that φ Pl V t t − (1+ ; χ χ , we find to quadratic order that t p φ/M ) ≡ χ , the scalar equations of motion (2.2) take the form t )= 2 p 0 χ − − V ( q χ − φ, χ ( ∼ e and 0 and ) V V t φ 2 ( − p a + 1 )= p , an essential feature in generating a scale-invariant spec t ≡ φ, χ χ p ( the potential is tachyonic in the = t V In terms of χ χ with the effective scalar potential for Denoting this constant by as a power series in The scaling solution above implies that At at sketched in figure 1. Note that (2.8) satisfies This describes the system rolling along the where Note, using (2.8), (2.11) and (2.12), that χ thereby revealing a tachyonic mass squared in the below. while the background solution (2.3) becomes can be rewritten in terms of JHEP11(2007)076 is a χ (2.14) (2.19) (2.15) (2.16) (2.18) (2.17) tachyonic mass er and other background ssion for the spectral tilt, . l. For the moment, we note . rbations, as we now review. oiling the scale invariance of s that, as stated above, the 2 above. sis relies on the existence of a , = 0 /t χ bations. Moreover the tachyonic 2 k 2 , 2 p 1, the rate of instability is always . H − δχ 2 , ¶ = 0 ≪ ¶ ) by ≈ i k t p kt 2 t ,χχ p − χ δχ 1 V √ ,χχ − = 2( V ¶ χ 1 + | ≈ is governed by [18] √ 2 2 µ t , the fluctuations in the latter by definition 2 2 | a ,χχ k χ = k δχ − ikt V 2 ,χχ – 11 – µ k 2 − V H √ | e k + δχ ≡− µ 2 k -mode starts out in the usual Bunch-Davies vacuum: p = / ˙ k 3 + k δχ k k 1, up to an irrelevant phase factor this reduces to H δχ ¨ δχ 2 tachyon ≪ + 3 ) m = 1 without loss of generality. In this approximation, t k 1, we can neglect the Hubble damping term in (2.15) and treat a − ¨ ( δχ k ≪ p rolls down this steep, negative exponential potential, the . The solution to (2.17) with these initial conditions is k φ 2 √ / . Thus, since ikt t 1. As − χ e gets larger in magnitude. A useful relation between the latt ≪ = The evolution of the Fourier modes of Well within the horizon, a given = p χ χ k free scalar with time-dependent mass: where we have used (2.9), (2.11) and (2.13). Since for corresponding to ascale Harrison-Zeldovich invariance spectrum. of the entropytachyonic It spectrum instability follow along in one two-field field ekpyro direction, denoted by In section 3.2 weincluding Hubble will damping solve and (2.15)from corrections (2.14) and from that derive the a potentia general expre at On super-Hubble scales, space as flat — we can set for quantities is where we have used (2.9) and (2.13) to replace much greater than the Hubble parameter. 2.2 Instability and scale-invariant entropy perturbation Since the field trajectory is orthogonal to coincide during the ekpyrotic phase with the entropy pertur mass uncovered above isThis essential implies in that the amplifyingthe instability these entropy cannot perturbation pertu be spectrum. cured without sp δχ JHEP11(2007)076 2 h. ar /t 2 (3.1) (3.2) (3.6) (3.4) ≈− ,φφ rly continues V , ..., + ¶ rotic potentials. ) 4 t w Ekpyrotic Cos- ) t χ χ − , − hey are hardly the only ¶ χ otential in the ekpyrotic ) ( is most transparent when φ, χ ( 2 4 Pl F φ ich we discuss below. 3 2 ; second, that a nearly scale- φ, χ the ekpyrotic phase: first, the M ( p y (2.8), consider any potential he form (3.3). Since the gen- + χ 3 t exponentials discussed above, orth set the form F p 2 + 2 ) ,χχ and t 3 1 p + V 1 and 1 p χ 2 φ p , they play an important role in the satisfies ≈ and the quadratic term in the Taylor χ φ 6 − χ ) φ F χ φ + ,φφ ( ( as a result of the relations 3 f 2 = 0 (3.3) Pl ) χ t t 1 χ ,φφ χ V pM = 0 (3.5) V = − , V t χ 2 1 | – 12 – 2 χ χ and the quadratic term in the Taylor expansion ( 1+ /t ,χχ φ 3 2 µ Pl 1+ F ) 2 µ Pl M p ) 2 ≈− p φ − 1 ( φ/M p 1 V 2 p ,φφ p √ V q 2 2( / − )= 3 e √ in potential (2.8). From the first fact we conclude that it is f p 0 as the fundamental fields in the theory, which we do hencefort t 3 V χ φ, χ χ − ( = V )= encode the self-interactions for t are identical to those in (2.8), this modified potential clea χ )= χ and t F χ φ at − φ, χ − ( ) perspective suggests an even broader class of allowed ekpy χ ,χχ V φ, χ V ( continue to hold. This new perspective greatly simplifies Ne φ, χ imply that the potential (3.2) satisfies conditions (3.1), t is any constant. Meanwhile, the function F t = t t χ χ χ ,φφ = − V The ( χ χ mology and broadens the class of allowed3.1 potentials. Generalized potentials The simplest such modification isof the the following. form Motivated b expansion in to satisfy conditions (3.1). Notekeeping that higher in order the terms case of in two the exac expansion (2.8) gives at It follows from thephase second as fact long that as the we relations are free to modify the p invariant spectrum arises in the fluctuations of 3. Generalized ekpyrotic potentials From the above analysis we learn two important lessons about physics associated with two exponential potentials for Note that although the exponential in where but is otherwise arbitrary. Since the exponential in analysis of non-Gaussianity in New Ekpyrotic Cosmology, wh in and simpler to treat which manifestly satisfies (3.3).eralized functions Hence, (2.8) is indeed of t functions to do so. For simplicity of notation, let us hencef the theory is rewritten in terms of the rotated fields without loss of generality. Then, consider any potential of JHEP11(2007)076 r = 0. (3.7) (3.9) ginal (3.10) t ) = 1. χ φ ( can be a ) in (3.6) = f = 0, given framework δ χ χ 2 φ, χ φ ( 1. the special case V convenient set of ≡ near and ) ) be approximately y from 1 φ φ ,φφ ( φ theory. This will now ( V f variables to f d ekpyrosis ≈ t, generically, e original picture in terms . and tential (3.6). A necessary φ, χ ion in the Taylor expansion ponding to tuning infinitely he potential in (3.2) — with ,χχ Pl V nction .1) for a nearly scale-invariant = 0 , k , 1 1 is given in (2.15) by δχ /p φ/M 2 ) perspective requires an unnatural ¶ 1 |≪ δχ √ |≪ δ − | η ,χχ | ≪ e φ, χ V 0 2 1 (3.8) V + ; ¶ ≈ − ; 2 2 ,φ ) V a k δ ,φφ V 2 ,φ φ ( – 13 – µ )= V µ f is constrained by condition (3.3). It is straightfor- φ VV 2 + ( − Pl k F V − ˙ , which translates in the M 1 2 δχ )=1+3 φ ≡ ≡ H φ ) and the modified quadratic term in the Taylor expansion ( as ) to be steep and nearly exponential, respectively. Mean- ǫ φ η f ( φ + 3 f and ( V k V 1 ¨ φ is not only more general but, as we will see, allows for greate δχ , can be quite arbitrary. In contrast, in the δ F ) has the property 1 — while this is seemingly automatically satisfied in the ori φ ( ≈ f ) φ ( f requires much more fine-tuning. The generalized potential . For comparison, the analysis of section 4 in [18] focused on 2 φ φ ) satisfies the standard fast-roll conditions of single-fiel φ ( language. This would seem to suggest that the latter is a more ) — to be nearly unity. However, the shape of the potential awa V and φ have a profound impact on the form of the spectral index of the 2 ( One might object that, at first sight, the ( The generalized function The perturbation equation for the entropy field 1 f set to zero — is of this form, with φ χ , φ t 1 while, the function be discussed in detail. 3.2 Spectral index for generalIn potentials this sectioncondition we for derive scale the invariance, spectral as we tilt will for see, the is general that the po fu unity over the relevant range of modes, which ensures that where ward to show that potentialspectrum (3.6) and satisfies is the the most conditionsχ general (3 potential to do so. Clearly t by the general function in which require the potential Therefore, let us parametrize over the relevant range of modes, and where the factor of 3 is introduced for convenience. Note tha of identical potentials for fine-tuning — requires one function to— be nearly exponential and one funct a second function mustmany be coefficients of in nearly a Taylor exponential expansion. form, corres function of fundamental fields to study.of On the contrary, we argue that th φ Turning on a non-zero freedom in the spectral tilt. JHEP11(2007)076 ǫ oll ,φφ = 0 V trum χ (3.11) (3.12) (3.17) (3.13) (3.16) (3.15) (3.14) fast-roll n of state = 0 case. η ) at δ δ , ) recast entirely and , are found to 2 ǫ 1. Moreover, for . . . (1 + 3 ≫ + ,φφ ǫ η ¯ V ǫ/dN , 2 . is up to this point an d = δ = 0 rge: ¯ + 10 roximation, as in slow-roll = 0 k ǫ ,χχ k v and V , rameters. cast in the form his limit, the )) 2 δχ ) δ we note, for example, that H corrections, which are negligible. ¶ . 2 + 12 e sake of brievity we only outline ) − ln dN δ − ¯ ǫ d η ( = 0, in which case the calculation is , ǫη d and so on. Let us emphasize that ¯ ǫ/dN 2 dN − 2 − d ǫ ¶ χ , as shown in [18] for the 2 ǫ ǫ 1 , η = . = ¯ N 4¯ 2 (1 + 3 k , ǫη at 3( k ǫ aH cannot vary rapidly while the observable 2 2 ˙ ¶ 6= 0, however, = H ,φφ − µ δ H , η ,φφ δ V 2 η 1 . . . a δχ V ǫ − + 1 2 (1 − + = – 14 – = 2 2 ¯ ǫ , such as ( k )= − a k ¯ ǫ ǫ ; v d k w ,χχ = µ dN ǫ v V 1 2¯ 2 x 1 η + ¯ ǫ 8 2 k 5 2 = (1 + ˙ x − ǫ 3 2 δχ + 1 ¯ ≡ ǫ H 6 + and, therefore, can be consistently neglected in the fast-r ¯ ǫ k 2 + 3 η v 2+ k 2 are approximated as constant whereas dx − ¨ d , and thus an arbitrary function of time. In order for the spec δχ 2 µ η and φ x 2 ǫ ǫ , we follow the approach developed in [18]. Define the equatio s as = ¯ n and k is a dimensionless time variable. Equation (3.11) can now be ǫ 2 and its derivatives with respect to a δχ ,φφ H V ǫ = ln = 0, we find from (3.6) that δ N can therefore be treated as essentially constant in this app η To calculate Thus far, The next step consists of introducing a new time variable In the ekpyrotic phase, the equation of state parameter is la identical to that presented in [18, 26]. When where When where we have dropped higher-order terms in the fast-roll pa in (3.11) can be rewritten as be higher-order in Terms involving higher-derivatives on ¯ After some algebra, the perturbation equation (3.11) can be parameter and rescaling approximation. To illustrate how this substitution works, to be nearly scale-invariant, however, clearly in terms of ¯ arbitrary function of and thus where the ellipses stand for terms of order Since the calculation closelythe parallels key that steps of and [18], refer for the th reader to [18] for details. inflation. Their time-dependence generate order parameters can be written as simplicity, we assume that it is slowly-varying in time. In t and JHEP11(2007)076 χ . (3.20) (3.18) (3.19) (3.21) (3.22) , which F on = 0, it is easy δ 2, gives a red tilt 6= 0 allows greater δ = 0 in our language acts effectively as a ase δ n > δ = er-interaction terms in is positive and dominant. η 2-point function, and work δ the pure exponential form, of course, which would lead , corresponding to a spectral or simplicity, we shall ignore n ), with , it follows from (3.7) that δχ tial (3.6), we now turn to the e spectral index is determined n − ilt if φ k bility aside and henceforth focus spectral tilt. ǫ, η ). Note that in the limit of flat , − δ ) ∼ . y is completely arbitrary, nevertheless ) k ( − δ v η exp( F (1) n . , − . η − 2), this reduces to (2.18). ∼ ǫ . 2 2 η 1 / yH ǫ ) χ Λ − √ 3 − ǫ φ √ 2( )) Pl ( and ǫ t 2 π → ≈ is positive, for instance, which by itself would ( V ǫ − M φ n r – 15 – η 2 ( ≡ / ,φφ k V V − 3 2 V 1 1 = 4( ) and the form of the quadratic term in the Taylor Λ 1 2 ǫ . Here we provide an estimate of the non-Gaussianity φ √ ≈ − ( χ s V = 3 n / k ) v δ , the solution with Bunch-Davies vacuum is given as usual η is approximately constant over the relevant spectral range − 8 , the amplitude tends to . For example, η δ η − − aH 1 ǫ √ ≪ 8( x/ k − = 0 this reduces to the result of [18]. Thus, ≡ 1 δ y p , the level of non-Gaussianity is associated with the functi χ 2) / 1) and exact scale-invariance ( (3 . What we have found here is that the more generic case with dependent over-all factor → φ ≡ φ a n While it is clear from the above discussion that On large scales, In terms of As discussed in [18, 26, 27], pure exponential potentials — to leading order in the fast-roll parameters encodes all self-interaction terms for level based on thethe generic effects size of gravity, of as we these did interaction in terms. section 2.2 F to estimate the yield a blue spectral index, it is still possible to get a red t expansion in 3.3 Non-gaussianity in general potentials Having derived the spectralquestion index of for non-Gaussianity the of generalized theby poten fluctuations. the Whereas th Thus the quadratic term in (3.6) can be written as based on the form of the mass term. To leading order in we can give some naturalness arguments for the cubic and high space ( to get a redcorresponding tilt to non-zero by considering potentials that deviate from with modes exit the horizon. A small time-dependence is allowed, to a running spectral index.on We the leave this simplest interesting case possi where As a check, for where Λ is an effective cut-off scale: by a Hankel function: index: at large freedom in getting a red tilt. Even if correction to the fast-roll parameters contributing to the — yield a blue spectrum. However, as shown in [18, 26] for the c JHEP11(2007)076 , 2 k ζ 3 k is the (3.28) (3.27) (3.24) (3.23) (3.25) (3.26) (3.29) in our θ = . Since ǫ , 2 2 ζ at the end ) /t 2 ζ ≈ end − p 2 hat , / ek ≈ − | 3 ζ t , where ∆ parameter [41, 42], θ Pl − ∆ of the exit from the end α ,φφ − Λ( NL ly their results to our ǫM V 2 f ek 2 = ∆ itude of density pertur- / 3 H √ ential: 2 β ith the details of how the | s to be suppressed by the ount that 3 i · 3 i f the . However, to simplify the k . k tion that the ekpyrotic phase ǫ α φ i i . 3 ure perturbation 2 s expected to be of order unity. / ¢ form [40]. Q P δχ 1 , 4 2 ) − ζ χ ! t ¡ i x α ∆ , is given by ( k ~ , 2 3Λ local g O 5 ζ 18 δχ i is determined by the interaction Hamil- Pl . + X 5 NL 3 3 ≈− Ã f δχ − ǫM reheat ≈∓ χ Λ 3 5 δ 2 3 3 10 3 ) H ) √ )) Λ δχ t π ≈ β ( )+ end 3 3 α 2 – 16 – φ x (2 − / ( = ( [27], an issue we ignore for the present discussion. )) 1 ζ g α ek α t 2 ζ t ζ , and all modes are assumed to be super-Hubble: ( / ∆ i i≈ 1 ζ φ − k ) ~ ( ∆ )= is the amplitude of the power spectrum, ∆ an arbitrary function of )= of the form V Λ( ), inherited from end x 2 ζ ( − / = F end 1 ζ φ, χ ek ( − − ζ t int ( F ek ∆ 3 t H k ~ 2 ζ a priori ) = √ 18 5 t is end ∓ by − α ζ = ek t ( 1. Moreover, ∆ 2 NL k ~ f ζ ) ≪ ) end denotes the magnitude of − has a Gaussian spectrum. From (3.25) we immediately read off t is a model-dependent prefactor having to do with the details end i ek g − t k ζ β ( ek 1 is exactly of the form studied in [27], we can immediately app t k The level of non-Gaussianity is usually expressed in terms o The three-point function for the fluctuations From this point of view, we expect higher-order interaction The last factor in (3.28) can be expressed in terms of the ampl ~ ζ − h int ( i calculation. We find that the 3-point function for the curvat of the ekpyrotic phase ( defined in terms of tonian [39], which we can read off from the cubic part of the pot H k which is fixed by WMAP: where we have substituted (3.22) and (2.11), taking into acc where Note that in the last step we have substituted (3.20) and used Here the coefficient where analysis we shall take it to be a constant which, naturally, i same scale, leading to a generic Note that the shape of the 3-point function is of the approximation. Note that theentropy above perturbation sign is ambiguity converted has onto to do w ekpyrotic phase. As we willends see through in a section 6, sharp in turn the in approxima the field trajectory, then we have bations as [18] where JHEP11(2007)076 . α c- NL (1). and f GeV, ǫ (3.31) (3.30) 15 ∼ O is at most β in inflation, β = 10 . Using this fact n New Ekpyrotic reheat T reheat lieved that the model for H 2 − ameter at the end of the en smaller values of n the observational bound. | ∼ ) of [18]. Thus ture of New Ekpyrosis com- Cosmology. andard big bang cosmology. will show that both of these 10 se issues later in this paper. ive unobservably small s in scenarios where density discussed elsewhere. i-field models [44], in models he curvaton [48] and modulon ree to choose the parameter end vich spectrum in the curvature , as well as the mechanism for ∼ rotic Cosmology is also in sharp elated to the required degree of , however, necessitates a careful − to be within the range [35, 43]: bations are even more Gaussian p ek H NL | . f 1 . − β ǫ α 3 β α − 5 18 10 · – 17 – 2 ≈∓ 40, which is below, but not far from, the present > ∼ ǫ NL ≈ f NL f is required in order to generate a nearly scale-invariant flu . Furthermore, as discussed in section 6, typically χ 3 − slow-roll parameter. Now, as indicated in figure 4 and as we 10 · inf (1) yields ǫ 5 coefficient, this expression is identical to its counterpart ≈ ǫ β ∼ O α 100. Using the liberal end of this bound, we get < 1, the level of non-Gaussianity tends naturally to be large i level, the recent WMAP data constrains σ NL ≪ replacing the ǫ ǫ particle physics terms. ed by an epoch of dark energy as the energy density in other ess and homogeneity problems , much progress has been made al Hubble parameter is no less patial curvature, but rather on n then blows up this tiny region . In particular, from the onset of s should, therefore, not be based pyrotic theory is equally successful k . 2 H 1 2 a ∼ – 18 – k Ω . It follows that over the same period the Hubble 2 /a 1 ∼ /t 1 ∼ evolve in time, evidently so does Ω H H and a In the absence of a concrete theory of initial conditions, ho Ekpyrosis, on the other hand, proposes that the universe beg In inflation, one envisions the universe as starting at the bi the hot big bang phase until the present time the scale factor expanding and chaotic state.assumption is that While there most existsallow of inflation a to the Hubble occur. patch universe A somewhereinto short is whi a phase wi of large, cosmic homogeneous, acceleratio isotropic and flat universe. vacuous and cold state.point was motivated In by nearly theall supersymmetric context physical or of scales BPS at initial theuniverse, the original radiation onset temperature ekpyro of — ekpyrosis are — very initial small Hubbl in absolut whether the universephysics. started out Thus, with in ournatural a current than understanding, hot a a or GUT-scale tinyfor cold Hubble. initi assessing state From the a belong degreeon cosmology of the perspecti genericity absolute of scale of initialwhether any condition it observable, such is as fine-tuned thecomponents. relative initial We to s illustrate other this observables, pointof such by standard discussing big the bang flatn cosmology. 4.2 Flatness and homogeneity problems inViewed inflation as an additionalthe energy universe component contributes a in fractional the amount Friedmann eq This was viewed as a drawback for ekpyrotic theory as compare This concern motivated thehomogeneity at cyclic the onset extension of ofdomination each at contracting the the phase conclusion scenario is of achiev thesince previous then cycle. and However the timecontext. seems We right will to argue that, revisit contrary theseas to inflation issues initial in in beliefs, addressing ek the standardto problems invoke of any big phase bang c of cosmic acceleration. 4.1 Contrast with inflation Inflationary and ekpyrotic cosmology areinitial based state on of starkly the di universe. Since both Now, assuming for simplicityreheating, that we our have universe has been radi JHEP11(2007)076 ) to 0 (4.5) (4.3) (4.2) (4.4) large as the /a g units 1 rad − 0 ponents at N H e = 0 λ as . For instance, since patch at the onset of . A useful expression /T ve considerations, pro- rad , 1 . N cosmic acceleration. Dur- reheat − ¶ ∼ t t, whereas the Hubble pa- ¶ m. nentially suppressed by the e . a . ving units ( total number of e-foldings of large initial radius of curva- 0 i e proto-inflationary patch. If inf essence of how inflation solves eV, then an initial temperature an initial curvature component a T rad reheat N reheat 3 ion grows by an exponential factor: N 2 T a − 2 − k e µ e µ . In other words, starting from the 10 ) i = ln = k ( ∼ changes by Ω = ln ≈ 60, whereas a lower reheat temperature K < ∼ ¶ until reheating 7 ¶ 2 ≈ . 2 reheat 0 2 2 aH reheat i i t H (0) k H H H 2 0 rad = 2 2 i a 0 reheat i reheat a 0 N is at most a few percent today, we are forced to ). – 19 – H H T H H 2 reheat k i 0 2 reheat a a a a reheat = reheat = reheat a /a a 46. Incidentally, we immediately recognize µ µ ) ≈ i 1 (0) k ( k ln ln − reheat Ω (reheat) k Ω , we find that Ω . In other words, (reheat) k rad ≡ ≡ H Ω Ω N rad rad = N inf rad N 2 (1), a long enough inflationary phase results in a sufficiently N N − > ∼ e reheat measures the size of the proto-inflationary patch in comovin ∼ O inf λ must have been exponentially small relative to all other com ) N i inf GeV yields the standard ( k k N 15 e ) to that at reheating. i , then our entire observable universe lies well-within this /a = 10 1 rad is therefore − i N GeV corresponds to H rad 8 > ∼ This immediately implies that any initial curvature is expo Inflation addresses this fine-tuning through a short phase of Therefore, during the radiation-dominated epoch Ω The homogeneity problem follows automatically from the abo reheat = N T i inf λ of parameter shrinks by Then, as long as the current temperature of the universe is Since we know from observations that Ω where the last step follows from the standard redshift relat By definition time of reheating: ( conclude that Ω of 10 for the onset of the hot big bang phase.ing This is this the phase flatness therameter proble scale remains factor essentially constant. grows by Thusinflationary we an expansion can exponential from define some amoun the initial time that at reheating ( ratio between the size of our observable Hubble patch in como natural choice Ω inflation and is therefore highly homogeneous. radius of curvature for thethe observable flatness universe. problem. This isture, Instead as the of in standard requiring big ancomparable bang to exponentially cosmology, all inflation other tolerates forms of energy density. vided that space-time isN essentially homogeneous within th JHEP11(2007)076 ll (4.7) (4.8) (4.6) ness prob- e. In the New gy ude at the beginning . (1) at the onset of the . ¶ ek ful at addressing the well- universe. As indicated in r, for the flatness problem, pyrotic phase the curvature s by an overall exponential N beg end se, so that 2 ∼ O y for our observable patch. ms in an analogous way, yet lly the same, as shown with urvature by the present time − − es dramatically in the process − owly contracting universe with e ek ek n phase that violates the NEC, in inflation. As stressed earlier, ially constant during this phase, on that the initial Hubble radius his intervening phase, therefore, ally rapid expansion with nearly H H = beg) − µ 2 . ¶ (ek k ln ≈ end beg − − (reheat) k ¶ ek ek Ω the scale factor changes by at most a factor H H end beg ∼ − − (1) initially. Of course, in absolute terms the beg end – 20 – ek ek − − reheat H H end) t ek ek − ∼ O a a beg end k k (ek µ both match continuously from the exit of the ekpyrotic − − and | Ω is just the ratio of the comoving Hubble radius at the ek ek = a a H | implies that our entire observable universe is initially we ek end µ N − end) beg) e ln ek rad − − and t N ≡ a (ek k (ek k Ω Ω ek > ∼ defined by N ek . In this precise sense the ekpyrotic scenario solves the flat ek N rad N N > ∼ ek , with N ek N e vanishes momentarily at the bounce by definition — its magnit This is followed by a bounce and an expanding hot big bang phas Therefore, just as in inflation, starting with Ω The condition H Ekpyrotic scenario, this is achievedgenerates by a a ghost non-singular condensatio figure bounce 4, and however, between eventually reheats the onset and exit of the ekpyroticcomponent phase. is Thus suppressed by by the end of the ek 4.3 Flatness and homogeneity problems in ekpyrotic cosmolo Ekpyrotic theory solves theusing flatness drastically and different homogeneity dynamics. proble constant Instead Hubble of radius, exponenti the ekpyroticrapidly phase shrinking consists Hubble of radius. a Theas sl scale indicated factor in is figure essent factor 4. Meanwhile, the Hubble radius shrink ekpyrotic phase, one obtainsprovided an acceptably large radius of c of order unity. Meanwhile, although— the Hubble radius chang initial radius of curvature is much larger in ekpyrosis than within the proto-ekpyrotic patch. Therefore,is the reasonably assumpti smooth guarantees a high degree of4.4 homogeneit Observational constraints on modelWe parameters have seen thatknown inflation problems and of ekpyrosis standardboth are big bang allow equally cosmology. for success the In particula natural choice Ω This definition implies that phase to the onset of the radiation-dominated expanding pha lem. and end of theexplicit NEC-violating bouncing phase solutions is [18].we nevertheless can Ignoring safely essentia assume the that details of t JHEP11(2007)076 , 1 ∼ rad eV, and = 0 ≪ (4.9) must N 13 t (4.10) (4.11) p β χ − χ , where reheat > ∼ . More- 10 H ≈ ek depends on N < ∼ χ . Restricting reheat ǫ p | H ential case, we beg = 2 heat temperature − | ∼ p ek , end H | | − e amplitude of density ek emain near . end d before and as shown in ponding to a microscopic H − | 4 tudy the associated initial e parameter ate the initial Hubble radius universe, we have -dependent prefactor ek ¶ res. First, using e constant e onset of the ekpyrotic phase. H ptions about initial conditions | invariant perturbations, Pl s a cold and vacuous initial state, reheat M . 0 : t. T p T km. reheat eV, corresponding to an initial Hub- µ GeV, we find T 4 6 8 Pl 10 reheat − = T M 10 | . Notice that the fast-roll condition 0 10 2 T = 10 ∼ − end < ∼ 2 − < ∼ | 10 ¶ ek | – 21 – ∼ beg H reheat | ) factor in (3.6) is characterized by the size of the − beg p Pl T φ − ek reheat ( rad M ek H V N H | H − | µ e from the end of the ekpyrotic phase until reheating in the 10 < ∼ | | 10 H GeV yields | ∼ beg 15 GeV. This is no different than in inflation since, as mentioned − p ek 16 defined in (3.7). In the more specific potential given in (3.2) H = 10 10 | ǫ GeV, this gives , we can solve (3.26) and (3.29) for − ǫ 15 15 reheat = 2 T 10 p = 10 . Substituting this into (4.9), we obtain < ∼ Pl ) factor and the quadratic term are the same as in the two expon φ /M ( reheat To make this point quantitative, in this subsection we estim Note that the steepness of the T reheat V T 2 reheat we find from (4.6) that in the two models:proto-inflationary patch, whereas inflation ekpyrotic proposes theorycorresponding prefer a to a hot macroscopically beginning, large Hubble corres radius. however, this is a consequence of strikingly different assum in the ekpyrotic scenario for a range of reheating temperatu For where in the lastfigure step 4, we the net have change substituted in (4.2). As mentione find that the steepness of the exponential is controlled by th ble radius on the order of 1 m. For the sufficiently long to generate the appropriate range of scale- our discussion to (3.2), we note for future reference that th For instance, T over, from the Friedmann equation in a radiation-dominated fast-roll parameter corresponding to a Hubble radius of the order of 10 earlier, (3.29) is identical to its inflationary counterpar 5. Initial conditions and taming theIn instability this section,conditions. we return We to show that the in tachyonic order instability for and the s field trajectory to r expanding phase amounts to a factor of order unity, and thus combined with the WMAPof amplitude puts an upper bound on the re the reheating temperatureperturbations— through see the (3.26) WMAP and constraint (3.29). on th Neglecting the model again using be exponentially close to the top of the potential ridge at th JHEP11(2007)076 , , φ χ pose (5.1) (5.2) (5.3) (5.4) lasts long be subdom- φ is determined χ in section 2.2, it φ , ) r a pre-ekpyrotic 3 into the fluctuation equation of motion. χ ( 2 ons, thereby bringing χ naturally be satisfied O /t 2 + ositive mass term for y density in not with the most general ly phase, the system will 2 ant compared to that in χ g expression (2.16) and the ≈− odes. e 3.6). Recall that an essential following: the initial value of l potential in rotic theory requires incredible gy density during this phase is tial (3.2). We do this only for ithout spoiling scale invariance. ial conditions associated with the 2 tachyon , 2 tachyon ting a Harrison-Zeldovich spectrum m 2 Pl m 2 1 , − M χ 2 − ρ | = 2 φ H ˙ + χ ρ | φ 1 2 ,χχ ρ = 3 V < ∼ = | = – 22 – χ χ ρ total | ρ total ρ ≈ φ , ρ = 0 so that the ekpyrotic phase driven by ρ = 0. For a wide range of initial conditions, the energy in ) t φ χ χ ( eff V = 0 by the onset of the tachyonic phase. Meanwhile, the energy given in (2.12) and (2.14), respectively. Hence, one must im t + χ 2 ˙ 1. φ 1 2 ≪ tachyon = p m φ ρ during the ekpyrotic phase. From the perturbation analysis oscillations is efficiently transfered to the massless fermi 1, it follows that one can ignore the Hubble term in the χ χ ≪ ) and couple to light fermionic degrees of freedom and allowing fo φ p ( χ eff . This follows from the substitution of V For ease of presentation, we will carry through our analysis A necessary condition for this to be the case is that the energ Happily, as we will show below, this condition can easily and δχ must be sufficiently close to exponentially close to χ density in the radiation produced quickly becomes subdomin where we have defined is clear that the tachyonic instabilityfor is crucial in genera inant throughout the entire ekpyroticgiven phase. by The total ener 5.1 Constraints on initial conditions To begin, let us carefully derive the constraints on the init instability in by the parameter letting phase where this positivepredominately oscillate mass around dominates. Hence, in this ear equation (2.17). Hence, we cannotThe remove this main instability constraint w associatedχ with this instability is the these initial thanks to the ekpyrotic attractor mechanism. potential (3.6) but, rather,convenience. with Our the results more easily restrictedfeature extend of poten to (3.2) the is general that case the ( steepness of the pure exponentia enough to generate perturbations over the desired range of m with without fine-tuning. This is accomplished by introducing a p fine-tuning to achieve a realistic cosmology. Were this to be an initial condition, it would imply that ekpy the constraint for all times during the ekpyrotic phase. In this case, we hav where the last equalityfact is that just the Friedmann equation. Usin JHEP11(2007)076 , | at H χ (5.9) (5.8) (5.6) (5.5) (5.7) (5.10) (5.11) ∼ | marking χ ect to that end − ek t . ek has to be exponen- N 2 nergy density in − must be exponentially , beg e , − χ Pl = Pl ek ow this can be achieved. t 2 strongest at ds, M 5.3), (5.4) and (5.7) that rate the necessary e-foldings p M ¶ | at ek , we obtain χ n to this section, the same result N end beg end − − − − e , this introduces a fine-tuning into ek ek ek . . tachyon = 2 H H H | beg χ Pl m µ , from the top of the ridge at the end of − Pl . < ∼ 2 ek 1 ∼ χ p M − χ = 0 ) ) p M end 2 t Pl 2 tachyon < ∼ χ − ¶ )∆ − beg 2 m M ( 2 ek t − 1 4 – 23 – end beg beg end χ displacement at the onset of the ekpyrotic phase, ek ∼ beg − − − − − t end − ( χ is χ )∆ ek ¨ ek ek ek χ − 2 ≈− t ek χ ( χ H H 2 ek χ end H ∆ ρ µ − H 2 tachyon ek t < ∼ ( m 2 tachyon 2 p beg m − in order for the ekpyrotic phase to last sufficiently long. < ∼ ∼ ek tachyon χ m beg ∆ . Noting from the scaling solution (2.11) and (5.6) that ∆ beg) beg) − − − ek energy density has to be exponentially suppressed with resp beg t − (ek φ (ek χ is the displacement of the field χ ρ ρ ek χ end − = 0 at ek t χ χ at the onset of the ekpyrotic phase. Let us see what this entails for the We conclude from (5.10) and (5.11) that the value of Returning to condition (5.2), we see that this constraint is We can also express this bound as a constraint on the initial e φ we find denoted by ∆ where ∆ It follows that the energy density in the ekpyrotic phase. Using (2.14) to substitute for the end of the ekpyrotic phase. At this time, it follows from ( tially close to theof top scale-invariant of perturbations. the As tachyonicekpyrotic an ridge theory. initial in However, condition as order mentioned in tocan the gene be introductio obtained naturally, without fine-tuning. We now show h whose growing-mode solution is of where in the last step we have substituted (4.6). In other wor close to From (2.11) and (2.14), this equation becomes the beginning of the ekpyrotic phase: Thus the initial JHEP11(2007)076 is , a i has eV, t 4 GeV. beg χ (5.15) (5.13) (5.14) (5.12) − 8 . This − φ 10 ek 10 ) to be a t φ < ∼ . ( eating tem- < ∼ , where χ ) Pl must become ) = 0. Since the , . m . beg χ M . The potential beg χ 2 − χ φ beg Pl − evolves down the 1 m has positive mass . χ ek − ) ek 0 t M φ t χ ek φ ( t ( 3 < ∼ ≤ χ ( al at 2 χ ¶ ) χ ground temperature is . t m m tential (3.2). A typical | ations, m 1 2 und on the allowed value beg ≤ Pl ) are naturally satisfied if − beg i tly small values of reheat + t M − is phase yonic phase will be triggered p ek depends on , xits the horizon at ort a range of scale-invariant T t o damp out these oscillations. n be achieved, for example, by ) ek ¶ becomes tachyonic: ( χ µ φ χ H ( χ | m . . . 40 2 χ m tated by choosing GeV gives − + . Given some initial displacement ∼ m 2 χ grows in time as 2 ) 15 10 χ m when )+ ≈ beg 2 Pl φ − ≈ ( 1 = 10 Pl beg ek tachyon − pM t M 2 eff ( ek m t 3 through (4.10) and (4.11) respectively, we find m 2 tachyon 1+ reheat ¶ – 24 – p T µ m tachyon Pl − Pl , since is tachyonic, is preceeded by a phase in which and m reheat M χ T χ φ/M < ∼ )= m beg µ 2 p ) φ ( − ) is chosen so that effective mass term for q Pl 0 ek beg 2 eff to (3.2) where, in general, φ − T ( − e M H 2 m χ 0 ek GeV the bound greatly weakens to χ 10 V t ) m 8 ( − . For example, − GeV, we get a trivial bound of . The end of the pre-ekpyrotic phase and the beginning of the φ χ ( φ 5 10 Pl 2 χ m 10 )= M m < ∼ = 10 ) 30 < ∼ − φ, χ . Expressing the right-hand side of (5.14) in terms of the reh beg ( 10 − V reheat beg ek reheat T ≈ t − T ( ek χ K t m will undergo oscillations around the minimum of the potenti 7 . at 2 χ χ The pre-ekpyrotic phase occurs during the time interval In order to generate the desired number of e-folds of perturb , i ≈ m χ 0 ekpyrotic phase occur by definition at time decreasing function of exponential potential. However, this transition is facili positive mass squared and couplesadding to a light fermions. mass term This ca can be the case even for constant In the lastT step, we have used the fact that the microwave back whereas for some as yet unspecifiedsquared initial time. given, for By definition, most during of th the epoch, by 5.2 Stabilization mechanism In this subsection, we argue that the bounds (5.10) and (5.11 the ekpyrotic phase, during which is positive at early times but becomes tachyonic for sufficien then is given by After this timepotential the of dynamics the is form (5.12) dominated is by shown the in figure ekpyrotic 2. po ∆ The new mass parameter Finally, for universe is contracting, we cannot rely on Hubble friction t subdominant at a sufficiently earlytoo time. late Otherwise, in the theperturbations. tach evolution of Assuming that the the universe, largest resulting observable in mode too e sh perature by substituting for conservative assumption, we canof use (5.14) to put an upper bo that JHEP11(2007)076 (5.17) (5.18) (5.16) (5.22) (5.21) (5.20) itions. herent hat s that from . Moreover, φ is initially orders of gives i χ t , ρ | i lueshift effect. However, . Let Ψ be a typical light of the pre-ekpyrotic phase H | χ . 2 nt (5.10) and ratio of densi- t must satisfy > ∼ tly effective to allow for larger ct can amplify the oscillations. χ | uality at 2 χ . evolution is overdamped. That is, , beg m ilizing to be independent of ) (5.19) to saturate the bound (5.14) since, χ Pl − 4 1 χ Pl ek beg m M is either constant or increases at earlier pM beg − H |≈ | , − i ek χ χ ¶ | t . ek ρ ( φ H | m ≫ | t | χ ρ beg | | i ¯ m ΨΨ. Then the rate of decay is given as usual − at < ∼ is subdominant throughout the pre-ekpyrotic at the beginning of the pre-ekpyrotic phase. H < ∼ χ beg ek i ∼ χ | , – 25 – − tachyon χ p χ χ H χ as Γ m ek during the entire pre-ekpyrotic epoch. It will soon ρ 2 m Pl | µ )∆ χ H i φ | M ∼ t < ∼ ( 2 i χ χ ∼ χ H m ) m 3 ∆ ≈ at any time during the pre-ekpyrotic phase. beg | − φ ρ ek H t ( , but these are clearly fine-tuned and unnatural initial cond χ φ ≫ | ρ m is coupled to light fermionic degrees of freedom, then its co . As in the ekpyrotic phase, we will assume for concreteness t saturates (5.17), χ χ beg . In this regime, we can ignore any blueshift effect. It follow m − | ek H t is the displacement of field , the decay is rapid on a Hubble time and the i ≫ | | is the initial Hubble parameter. Since component. Of course the mechanism eventually fails if χ i H χ H Let us now calculate the initial conditions at the beginning To simplify the calculation, we henceforth take ≫ | Using (5.17) and (5.19), we find that the initial displacemen times, it follows that Γ that will produceties the (5.11) exponentially at suppressed displaceme where so that the evolution is dominated by the field settles at itsThis minimum is well before easily the satisfied blueshift here effe if we choose fermion and assume that it couples to oscillations will decay into these particles, thereby stab if we assume that On the contrary, they will instead get amplified by the usual b become obvious that our stabilizationinitial mechanism is sufficien If Γ in this case, we have This consistency condition ensures that where ∆ Putting these densities into (5.18) and evaluating the ineq phase. by [51] magnitude larger than the earlier discussion that we assume that so that Γ JHEP11(2007)076 . ) ek beg N beg − − 5.11), (5.29) (5.27) (5.25) (5.23) (5.26) (5.24) (5.28) ek ek . Since t ment at . Thus, i and t /H χ H i p m and H − i e t , throughout the ∼ given in (2.11): /p φ ) . i t i t i Γ to the exponentially − GeV, we found that − ( i p/t | /H e t 15 e parameter ction 4 that = beg beg ∝ at − i − t the pre-ekpyrotic phase ek ek χ χ H = 10 ρ H y densities. From the above H | 2 2 √ cement at the beginning of the √ , tion − − i e reheat . Moreover we have substituted . T . /H < e ≈ . beg ek , Pl ) i beg i − ek ek decays as t N − Pl N ek 2 /H − N ek e t pM ( χ (due to blueshift between 2 M χ √ H beg 4 ) 2 √ m ek i − φ ( − , one must evaluate the ratio √ − i ∼ ek N 1 > ∼ − e ≫ − , by the onset of the ekpyrotic phase this has χ 10 2 H e i 2 t > ρ i – 26 – ≈ √ < ∼ from (2.14). t √ ≈ beg beg ) i . Using (5.10) and (5.27), we find that i i i − − at − − t < ∼ χ e χ H /p beg) i − ) | ek ek beg i ∆ ∆ − ) χ ( beg) χ χ i − H ρ ( χ beg − ) beg) beg ∆ i φ (ek ek ∆ ρ − ( χ − − t ρ (ek χ ek ρ t ek ρ ( (ek φ χ ρ H | ; that is, m given by 2 − ≈ e √ beg − beg) ≈ ≈ ek − 60. It follows that for this reheat temperature the displace beg) beg) ) t required at − − (by assumption). Comparing this against the bound given in ( (ek χ > ∼ ) ) χ beg beg) ρ (ek φ (ek χ i i ( χ ( χ − − ρ ρ ek ρ ρ ek N (ek χ t > ∼ ( ρ ) i ( φ and by assumption, the cosmological evolution is dominated by ρ 2 ) i tachyon − φ ( ρ m 10 To get a feel for what this expression entails, recall from se To determine the allowed values of ∆ It is also useful to consider the ratio of the displacement of It is convenient to rewrite this in terms of the initial Hubbl < ∼ ∼ ∼ ) χ i χ ( m The last inequality follows from pre-ekpyrotic phase. Therefore, we can substitute the rela where in the secondis step long we compared have to assumed, for simplicity, tha depend on the reheat temperature. For example, for Putting this into (5.22), wepre-ekpyrotic get phase an given expression by for the displa and thus This is most easilydiscussion done by it considering follows the that ratio the of energy the energ density in starting with some energy density as well as decayed to a value representing a wide range of natural initial conditions. p the beginning of the pre-ekpyrotic phase is given by ρ we see that it is satisfied if fine-tuned value of JHEP11(2007)076 (6.1) (5.30) (5.31) , is proportional is given by σ χ gets stabilized near on acquires a nearly , defined with respect s χ , is proportional to the s and ˙ lement of New Ekpyrotic ch ˙ φ on of choosing the Hubble big bang expanding phase. and r of e-foldings, it suffices to ondensate. We begin with -invariant entropy spectrum d velocity, ˙ σ al two phases required for a namely the conversion of the of Even before the bounce, the e, for the reheat temperature e-tuning problem, as claimed. ocity, ˙ 1 and 6.2 respectively, before phase and 4) the bounce from e onset of ekpyrosis, the precise ) must eventually come to an end ˙ χ , , θ ˙ χ . ek θ 2 N given by (5.27). This will be the case as 2 10 χ + cos √ > ∼ ˙ φ + sin > ∼ ˙ θ φ , it follows from (5.26) that – 27 – beg θ i , the curvature perturbation on uniform-density t − sin beg ζ i p/t t ek − − t to be close enough to the top of the tachyonic ridge ek = t = cos = χ ˙ H ˙ s σ to satisfy the ratio (5.26). Using the fact that during the i t and beg − ek t with the large, natural value of GeV this ratio becomes i t 15 = 10 The remaining issue is to understand the physical implicati We conclude that in order for In general, it is convenient to consider new field variables, = 0 and 2) the ekpyrotic phase in which the entropy perturbati t -dominated pre-ekpyrotic phase reheat a natural and physically reasonableT assumption. For exampl φ Clearly our mechanism naturally removes the exponential fin parameters at by the onset of thestart ekpyrotic phase out to at ensure a sufficient numbe normal vector. In other words, their decomposition in terms to the vector tangent to the curve, while the entropy field vel χ to the field trajectory. As shown in figure 5, the adiabatic fiel scale-invariant spectrum. Inrealistic this cosmology, section, that we is, turncontraction 3) to to the the exit expansion fin from enableda the by general ekpyrotic a discussion NEC-violating ofturning ghost these to c last a specific two example phases, in in section sections6.1 7. 6. Converting entropy into curvature perturbations The ekpyrotic phase described by theif scaling solution the (2.11 universe is toBefore undergo a turning smooth to bounce the andCosmology enter bounce, which the also however, hot occurs we duringscale-invariant first the entropy discuss exit spectrum a from into ekpyrosis, keyexit the e from adiabatic the ekpyrotic mode. phaseonto can the be adiabatic used mode, to imprint described the by scale long as the pre-ekpyrotic phaserelationship begins being sufficiently before specified th in (5.30). 6. Graceful exit and bouncing Thus far, we have discussed 1) the pre-ekpyrotic phase in whi hypersurfaces. JHEP11(2007)076 ) + = 0. (6.4) (6.2) φ, χ (6.5) (6.3) ,χχ θ follows θ V 2 δχ , respectively. χ and = cos , and thus , φ δφ ,ss and ectory. k V φ Ψ to 2 2 ong ile a k ropy perturbation satis- ˙ ˙ θ σ rue. In general, the ( 2 Pl M ution equation for the adiabatic = 4 coincide with k s θδχ. , δs ,σ θδχ ¶ V and . 2 ˙ ˙ ˙ − θ σ φ χ + cos = + sin = + 3 ˙ σ θ θ δφ – 28 – ,ss H θ δφ V sin tan + + 3 ) directions at a given point along the field trajectory in − s 2 2 ¨ σ a k = cos = µ δs δσ + . This result will come in handy later on in the discussion. k ,χ ˙ δs θV H ) and entropy ( + 3 σ + sin k ¨ ,φ δs θV cos Adiabatic ( is the curvature of the potential orthogonal to the field traj ≡ ,σ ,φφ V During the ekpyrotic phase, the background field motion is al The relation between the adiabatic and entropy fluctuations ) space. θ V 2 φ, χ Evidently, it follows from (6.1) that in this phase Using the standardfies energy [52] and momentum constraints, the ent immediately from (6.1): Once we exit the ekpyrotic phase, this is of course no longer t where Ψ is the curvature perturbation in Newtonian gauge, wh where Figure 5: ( equations of motion given by (2.10)field imply the following evol where sin JHEP11(2007)076 ic 0 is (6.7) (6.6) ≈ t , so that χ , which is framework δχ ζ so that it is = istency check, φ φ, χ ntified with the δs , the field trajectory tion to the curvature ion of . δs e is sharp and therefore e simple potential (5.12) ¶ d a term to the potential 2 = 0 is only an approximate e potential that depend on direction. We will shortly ure 3. ) ing a minimum followed by , the exponential prefactor the exit from the ekpyrotic 0 t in the field trajectory at the p perturbation as follows [52] for simplicity. φ χ χ = 0 at the end of the ekpyrotic es important after a sufficient riant spectrum on away from the tachyonic ridge t ic phase, and therefore − ≪ χ . χ ′ χ k ( p . In the simplified 2 2 Pl φ ˙ θδs 1 = 0 tachyonic point. This example will M ′ ˙ H t σ p 2 χ and away from 1 + 1+ φ k χ . Subsequently the classical field evolution is = 0. For µ Ψ δσ – 29 – = 0 and therefore prevents the latter from being χ 2 2 Pl a k χ . A much simpler option suggests itself, however, ˙ χ H H φ/M ′ 2 p = near becomes sufficiently small, however, the correction (6.7) is seemingly negligible at long wavelengths. As pointed q and away from the k ˙ χ − 2 ζ φ e χ φ k 0 U with the entropy perturbation only holds during the ekpyrot , we find that (6.5) indeed reduces to (3.10). Note, however, = ,χχ with a scale-invariant contribution from δχ V V ζ ∆ = ,ss V indeed correspond to entropy perturbations. As a quick cons , and, evidently, fluctuations in the latter are no longer ide χ χ = 0 during the ekpyrotic phase, as mentioned earlier, we have θ = 0 and ˙ θ To be precise, we want to modify potential (3.2) so that Clearly, to endow To illustrate this, let us describe the ekpyrotic phase by th Since Let us therefore turn to the conversion of the entropy fluctua must undergo some turnrapid at on the a end Hubble of time.a ekpyrosis, sharp which In we [18], rise assum this in was the achieved respective by introduc potentials for since studied here, this wouldspecific correspond to linear adding combinations new of terms in th fluctuations in that the identification of solution to the equationswhich of is approximately motion. linear in One can, for example, ad Clearly, this has non-vanishing slope at phase. namely to introduce some term which pushes out in [33], however,phase. this For our is purposes not we necessarily will neglect the this case term during altogether makes this correction vanishingly small during the ekpyrot an extremum ofnegligible the during potential. the ekpyrotic Thisnumber phase of linear but e-foldings eventually term becom is must achieved. be a At this funct point it tips marking the end of the ekpyrotic phase. Thisand is add sketched in a fig correction of the form The first term proportional to introduce a feature inend the potential of that the causes ekpyrotic a phase, sharp thereby turn imprinting a scale-inva entropy perturbation. perturbation spectrum. The latter is sourced by the entropy partially along phase; that is, as long as the field trajectory is along the still a valid solution then. Once proportional to the adiabatic fluctuation becomes important and pushes be studied in detail in section 7. JHEP11(2007)076 : e c 2 ǫ Pl H , a re 4 M (6.8) (6.9) 4 = 2 is not (6.10) (6.12) (6.11) er the p hase for σ /H evidently 2 ,χ ˙ V σ H ≡ n using χ her exit dynam- ǫ , pecific correction given . We henceforth denote on. The 3 iving force is sufficiently jectory as instantaneous given in (6.3). If ˙ φ to match continuously as Pl , but not a delta-function uire that the driving term σ ,σ , δs ǫM V k reheat or 2 , n makes it possible to proceed δs H oduced thereafter are no longer √ -roll parameter lues of Pl ) θ . . | M ∆ end 2 | − with a delta function and treating ǫ end ≈ H 2 , we can integrate (6.9) and obtain ˙ end ek − θ / | ǫ t Pl 3 − δs ek Pl 2 √ ek − M end H | t − H ǫM ( | > ∼ 2 and ek 2 / its value at the exit from the scaling solution. | 3 θ δ σ = H √ k | term will generate a delta-function contribution. δs – 30 – ∆ θ Pl = = 0) σ ∆ end k end χ − ǫM ( ≈ − δs ˙ σ ek — see (2.10) — is comparable to the Hubble damping ,χ 2 √ k ek t / 2 V ζ 3 away from the tachyonic point, corresponding to a sharp χ | H 2 k / 2 = χ 3 k ˙ during this phase, we read off from (2.11) σ ≈ is constant during the transition by assumption. Similarly ˙ k = φ ˙ is constant during the instantaneous exit from the ekpyroti . ζ when (6.8) is fulfilled. This notation is consistent with figu H 2 ǫ = / σ 1 ζ φ is continuous, and we can substitute its value at the end of th σ ∆ = 2 σ away from the tachyonic ridge marks the end of the ekpyrotic p p χ changes by at most a factor of unity during the exit, and we ref δs up to the exit, this can be read off from (2.11) and (2.19), agai the value of δχ is the total change in angle during the exit. Of course, smoot θ = end − δs ek Coming back to general considerations, we assume that the dr Similarly Substituting the above expressions for ˙ In the limit of fast-roll, we can treat the turn in the field tra As argued in [18], ˙ in the equation of motion for φ ,χ where we have used turn in the fieldV trajectory. To make this more precise, we req reader to [18] for a detailed argument. Thus we approximate strong to generate a fast-roll of Equivalently, this condition amounts to requiring the slow where ∆ term: by as constant. Hence to be comparablein or (6.7), greater this than condition unity. is satisfied for For sufficiently example, small for va the s Since since the rolling of all practical purposes. Inscale-invariant. particular, fluctuation modes pr on a Hubble time. This amounts to approximating well, for simplicity, and can substitute for ics could do equallyanalytically. well, but this rapid-exit approximatio phase. This follows by inspection of the equation of motion f sudden change in field direction can at most yield a jump in cannot do the trick, since constant during the transition, then the ¨ But this cannot be compensated by any other term in the equati discontinuity. Thus ˙ ekpyrotic phase. Since ˙ JHEP11(2007)076 , θ ζ at fixed ǫ with a scale- ζ corresponds to a χ the potential which is identified with ∆ as we do here, leads to ude of perturbations is , as studied in [32, 33]. β e are advocating is that onal terms in the poten- χ ting the ekpyrotic phase f a ghost condensate [20]. ion in n acceptable perturbation ussion below (3.29). It is or ave only one field entering bation is converted onto 23], a necessary ingredient lacement is larger, then the 3.26). In the next subsection or independent reasons. For ilities. The analysis closely underscores, to generate the bly narrow spectral range of d range of initial conditions. of the ekpyrotic phase must ntial fine-tuning in the initial ires a nearly flat and positive he dynamics of a non-singular -invariant entropy perturbation tion 7 with the correction (6.7), ation and how it can be matched he reader to our earlier paper for ility in er hand, then the ekpyrotic phase = 0, thereby endowing χ (1). ∼ O as indicated in figure 4 and discussed earlier. θ – 31 – away from χ reheat H 0 is a good approximate solution, but eventually becomes ≈ | ∼ χ = 0 tachyonic point. This term is assumed irrelevant during end (1), for otherwise one needs even smaller values of t − χ ek ∼ O H | θ away from the χ To summarize, the end of the ekpyrotic phase is set by a term in Instead of dialing the end of the ekpyrotic phase with additi On the other hand, adding correction terms to the potential, displacement — see, for example, (5.10). If the initial disp in our sharp turndesirable approximation, to thereby have ∆ confirming the disc ekpyrotic phase will bescale-invariant perturbations. too short, If it resultingwill is in smaller, last an on for unaccepta the too oth long. That is, by the time the entropy pertur reheating temperature to maintain the WMAPwe normalization ( will discuss the conditions for which ∆ the ekpyrotic phase, so that pushes important and triggers the end of ekpyrosis. This sudden mot turn in the field trajectory,spectrum which in onto turn the imprints curvature the perturbation. scale tial, one could alternatively exploit the tachyonic instab invariant spectrum. We willshowing illustrate how this it in leads to detail asmoothly in scale-invariant curvature to sec perturb the NEC-violating ghost condensate phase. 6.2 A bouncing scenario In this subsection, we completebounce the along scenario the by discussingwith lines t a of phase [18]. in whichThe the This ghost energy is condensate density is can achievedto dominated by violate produce by that supplemen the a o nullparallels bounce, that energy of without condition [18], introducing [ anddetails. thus ghost-like we A instab will key be brief, difference, referring however, t is that here it suffices to h Comparison with (3.29) shows that the model-dependent fact However, as the discussiondesired of number section of 5.1 e-foldings mostχ in emphatically this case requires an expone where we have used the Hubble parameter willamplitude. be too In large other intied words, magnitude in to the this yield WMAP latter a case constraint to on a the tuning amplit ofrobust initial predictions conditions. for theMoreover, perturbation we know spectrum that forinstance, corrections a to the broa (5.12) NEC-violating are ghostpotential. necessary condensation f phase Thus requ theeventually have negative, a steep minimumthese exponential and corrections rise potential will generically to push positive values. What w JHEP11(2007)076 . 2 χ 0 / χ 2 ˙ φ . For ≈ (6.17) (6.13) (6.16) (6.14) (6.15) p min χ min ≈ χ . φ he production ) ρ quickly becomes min . χ is a standard scalar ρ ,χ , where the subscript χ min χ onsequently, its energy end end : negligible at early times − − ek .12), this parameter enters ek min theless t a large change in angle is φ , the change in angle in the ll practical purposes. φ ound the ekpyrotic potential drive χ ( the field in the pre-ekpyrotic θ rge ratio of densities is allowed tes a minimum at V = tic attractor mechanism, we can . can dominate momentarily as it (1) is achieved provided . st: χ − ; meanwhile ) χ end φ 0) − , ∼ O end , 2 ek − θ t in this post-ekpyrotic phase. Instead end ek H . 2 from (6.13) whereas t χ − = ( / 3 = 0 and 2 ) t ≫ , which keeps rolling down its steep quasi- ek − ˙ φ χ φ φ ) a ( end at > ∼ − ∼ min V Before entering the ghost condensate phase, we ) – 32 – ek χ χ t ( ρ ( end )= 2 φ as it rolls away from the ridge is of course just given − ˙ χ ,χχ ρ rolls towards a stable minimum, denoted by ek χ V end t will dominate again within a few e-folds of contraction. is massive compared to Hubble, ( − χ ≫ ≈ 2 φ ek χ ˙ χ χ t acquires a non-zero expectation value. Thus they become ρ ) ( 2 χ ˙ smaller than some critical value in (6.2), we see that ∆ χ end since 2 1 θ φ − ek t end ( )= − χ ek ρ t end − towards minimum. at ek t χ gets stabilized. As described in the previous subsection, t φ ( ρ χ χ ρ ≫ χ ρ These considerations impact on the allowed range of ∆ From the definition of Subsequently we assume that The kinetic energy acquired by away from the tachyonic point. Such corrections are assumed the ghost condensate phase, namely the ekpyrotic field exponential potential. In other words,tolerate thanks to the ekpyro The discussion of the previous paragraphby shows that the such ekpyrotic a la attractor mechanism. In other words, negligible compared to the energy density in field with canonical kinetic term throughout. (i) Rolling of field trajectory as wein exit the the amplitude ekpyrotic of phase.indeed density As possible. perturbations. shown in Here (6 we argue tha If we assume for simplicity that from (2.11), we need Equivalently, since then the fielddensity, undergoes averaged many over many oscillations oscillations, in blueshifts as a du Hubble time. C assume that Although this grows in time as the universe contracts, never of scale-invariant modes comes toχ a halt once corrections to Note that the fermionicphase degrees now of become freedom heavy which stabilized as indicates that this marks the end of the ekpyrotic phase for a irrelevant and cannot be relied upon to stabilize but become important for rolls down and undergoes undamped harmonic oscillations ar instance, our fiducial correction term given in (6.7) genera by the difference in potential energy between JHEP11(2007)076 , 2, V / 2 ˙ , and φ (6.18) (6.19) φ 2+ − / and rise 2 ˙ = φ V ˙ = . Very soon, H . To see this, θ 2 Pl 2 Pl ingle field M M 2 onential potential. H positive , which takes the form . φ can be achieved with our χ ation, as we will see, requires becomes t. Our next consideration is the ng a significant ∆ al is shown in figure 6. Region a) nding to the ekpyrotic phase. A egion c) corresponds to the ghost at fixed erm for = 0. And since φ ith NEC-violating ghost condensate ˙ , H ) X 2 ( 4 ) P m 4 ∂φ 2 ( gM – 33 – − ≡− √ entering a ghost condensate phase. This is achieved X = φ L Effective potential for are some mass scales to be determined by the fundamental 0. Thus the field must reach a minimum in M = 0. But then the Friedmann equation, 3 V > . 2 ˙ and φ m Figure 6: m ≫ M is subdominant, the dynamics effectively reduce to that of a s will take over as it keeps rolling down its steep negative exp χ φ Once where is dimensionless. Here theory. As successful mergera of large ekpyrosis hierarchy: and ghost condens however, by invoking higher-derivative corrections to the kinetic t rolls off the ridge and starts oscillating, thereby generati to positive values (regionekpyrotic b)). potential (5.12) Such plus a thecondensation minimum correction phase (6.7). and where sharp The the r rise potential is approximately(ii) fla Ghost condensateepoch phase of NEC and violation, triggered NEC by violation. immediately tells us that recall that the onset of NEC-violation occurs when this also corresponds to the story is virtually identical toand single-field non-singular ekpyrosis bounce. w The effectivedescribes single-field a potenti steep negativenecessary exponential condition potential to correspo violate the NEC is that the potential JHEP11(2007)076 φ y and (6.21) (6.20) ) by the . In the 2 X ( m P − = ˙ φ is assumed to have 2) = 0 without loss .) equation is given by / sate phase which for φ V ˙ (1 H the kinetic energy in P V . In order for fluctuations the X + ˙ ) having a minimum at some π , since its contribution to the ≈ right of the minimum, i.e. for , . 2 π ) X ˙ K φ π ( 4 2 m X K P ( m M 4 P 2 M 2, on the other hand, the field evolution ≈− ≈ / 1 ) V V , P < ∼ )+ + X P ρ ≈ − ( – 34 – − 2 1 V Kinetic function for is degenerate with a shift in ? During the ekpyrotic phase, − 0, corresponding to ghost-like fluctuations [18]. The X − P P ) = < , X X P ˙ ( H ,X (2 P 2 P is brought to the vicinity of the minimum of Pl 2 is an exact solution to the equations of motion, even in a 4 4 / Figure 7: M = 0. Then, near the ghost condensate point we can expand X M M t = 1 , leading to the following expressions for the energy densit = = 2 without loss of generality, corresponding to π / ρ X P + t ) is sketched in figure 7. 2 X m ( . See [18] for details. (Notice that we have taken − P = ,XX φ P = , taken to be 1 2 [18]. If the quasi-linear part lies at reaches the plateau. This marks the onset of the ghost conden K / X φ 1 In the process of climbing up the sharp rise in the potential, The ghost condensate relies on the kinetic function What about the global form for The culprit in violating the NEC is the term linear in ˙ > ∼ must go through a region with decreases dramatically, and desired form of to have positive norm, thisX quasi-linear part must lie to the time convenience we will set at the field as approximately standard kinetic term, corresponding to absence of a potential, cosmological background. finite pressure: where of generality since an overall shift in energy density does not have a definite sign. Put another way, JHEP11(2007)076 a χ d φ ≈ ρ 2 / 2 (6.24) (6.23) (6.22) ˙ φ ) changes . For our t . This is ntaneously 4 ( χ . As shown a is stabilized. ρ m , we can take π 2 χ − + e, evidently this t | ≫ 2 ) . And since 4 ondensate/ekpyrosis m . For instance, for a = 10 e factor beg m p m − ek ≈ − roughout the subsequent ≫ φ to an NEC-violating ghost r, a few more e-folds of ) ( φ onal constraint to consider, on the value of the potential V . | beg ic potential. 60 and eld scenario of [18], however, is π − f the ekpyrotic phase makes . > ∼ ek t ( K ek 2 4 p ˙ . φ N ). From the scaling solution (2.11) . M 4 as the universe contracts — see (6.15) 2 throughout the ekpyrotic phase, that 3 m K min since it violates the NEC. Nevertheless, / 4 p ek 1 /a The successful merging of ekpyrosis with |≪ ,χ N , as ghost condensate, while . Therefore we must require M 2 | -oscillations, denoted earlier by > ∼ φ e min min does not blueshift appreciably throughout the χ V min φ X χ , the value of the potential at the minimum: ( – 35 – |≪ V ρ | |≫ V ≪ | ek is exponentially larger than decreases min min 4 min N V V ≡ 2 | V m | M − e ek N min must lie in the vicinity of the ghost condensate point by by the onset of the NEC-violating era. Moreover, from the 2 V e |∼ ) satisfies this property. Clearly this is the case from the en GeV corresponding to φ φ t ) ρ ( GeV. 15 a falls within the range 3 beg | − 10 ek min φ > ∼ V ( | reaches the plateau, since everything happens almost insta m V | φ . Because the kinetic energy grows during the ekpyrotic phas 4 enters the ghost condensate phase until reheating, the scal GeV and m φ 16 non-trivial since the latter blueshifts as 1 ≫ 10 during the ekpyrotic phase, this immediately implies: 2 It remains to show that So far everything we have said is a review of aspects of ghost c To summarize, the ekpyrotic field is successfully morphed in Firstly, by assumption we must have Secondly, by assumption ˙ > ∼ φ V of ekpyrosis until on a Hubble time, by assumption. That this is also satisfied th by at most a factorNEC-violating of epoch. order unity. Thus moment negligibly small compared to rolling down its steep exponential potential after the end o M merger first discussed in [18].that A key here difference we with the shall two-fi use only one field, namely priori While this greatlyhaving simplifies to the do with picture, the energy there density is in the an additi condensate provided Evidently this is satisfied provided reheating temperature of 10 which clearly can take either sign depending on the sign of ˙ (iii) Constraints forghost successful condensation imposes bounce. some constraints on the ekpyrot —, whereas the ghost energy density this is not a big concern for two reasons. As mentioned earlie condition is most stringent at the onset of this phase: is, − purposes we find it convenientat to express the this minimum, as a denoted constraint by and (4.6), we have the time it reaches the plateau, so that we can approximate in [18], this gives an upper bound on JHEP11(2007)076 until (6.25) (6.26) end − ek t , = 0, which occurs | a min en ˙ V | een to change by a factor p he captions for details. 8 yrotic phase through the < ∼ ike instabilities intrinsic to densate phase. In this case r changes by approximately y density remains essentially as it rolls towards its stable ) ig bang phase: for simplicity that the scaling A numerical solution obtained chanism between χ occurs within at most a Hubble during the bounce is min χ = 0 to be the beginning of the ghost ρ through the bounce, as described ,χ t χ . end = 0, which as shown in figure 9 occurs ρ | − ˙ H ek min φ ( V | V p − . ≈ – 36 – 0) min ) during the bounce generated by the ghost condensate. , t V ( ghost a ρ end − ek φ ( V )= end − ek t blueshifts by a factor of 8 during this phase. ( χ χ ρ ρ 4. The point is that in this time interval the scale factor is s 2 in these units. Meanwhile the bounce occurs by definition wh . . 2 1 Evolution of the scale factor ≈ ≈ t t Meanwhile, it is easily seen that the ghost condensate energ , a sufficient condition to ensure the subdominance of c Figure 8: around around The NEC-violating phase begins by definition when of 2, from approximately 0.4 to 0.2. Note that we have set Therefore, ignoring the effect of the ekpyrotic attractor me condensation phase. NEC-violating phase follows from requiring thattime the bounce or so, which,ghost condensation as remain argued under control in duringin the [23], [18] bounce. and ensures plotted that ina the figures factor 8 Jeans-l and of 9 2Consequently, shows that over the the scale facto course of the NEC-violatingconstant during phase. this phase. See Tosolution estimate t (2.11) its value, holds we all assume we the have way to the onset of the ghost con above. In other words, the potential drop experienced by where the factor of 8 comes from the blueshift of minimum must be small compared to non-singular bounce and onto the radiation-dominated hot b 6.3 Summary Let us summarize the sequence of events from the end of the ekp t JHEP11(2007)076 . χ Pl M es to 2 (6.29) (6.27) (6.28) H f the pre-ekpyrotic = 0 to be the beginning y density acquired by . We also impose that t χ d of the ekpyrotic phase 5.12), for instance of the the value of the potential and undergoes oscillations. y perturbations. See figure 3. min min V ), blueshifts effectively as dust . χ Pl , min starts being comparable to . 2 M | ,χ 2 H ,χ = 0, which from the plot is seen to occur end H V end − | ˙ ≫ H φ = 0. For simplicity, we assume that the ek > ∼ ) during the bounce generated by the ghost ) ( t | φ ( χ V min H . < ∼ χ 4. Note that we have set − φ . ( = 0) 2 – 37 – at which min 0) χ ( , ≈ ,χχ φ φ t V ,χ end V | the corresponding field value. Since stabilization of φ hits the sharp rise, for consistency we must have ( φ V min φ ) = the value of is eventually driven away from the tachyonic ridge, thereby 0 occurs rapidly on a Hubble time, which requires a fast-roll end = 0) occurs at χ ≈ end − reaching a minimum in the potential, which subsequently ris − H ek χ rolls to a stable minimum denoted by t φ ek ( φ χ χ ρ acquires a non-zero expectation value, the light fermions o Evolution of the Hubble parameter χ is heavy at its minimum, 2. The bounce ( , given by . 1 The entropy field at the minimum, and by We denote by is assumed to occur before departure from condition to be satisfied bringing to a halt the production ofThis scale-invariant densit is achieved by aform correction (6.7), to which the has ekpyrotic potential non-zero ( slope at Subsequently positive values where it becomes flat. We denote by As phase become heavy and cannot be relied upon to stabilize χ and quickly becomes subdominant to is triggered by The evolution then reduces to single-field ekpyrosis. The en so that its oscillationsχ are rapid on a Hubble time. The energ ≈ • • • t Figure 9: condensate. The epoch of NEC violation starts when at of the ghost condensation phase. JHEP11(2007)076 ek V (7.1) (7.2) (6.31) (6.30) added is. kinetic χ . We focus V order terms in ensate, given by , . Since the latter decreases substan- χ . ¶ ρ | to the constraint: of ekpyrotic and ghost . . . sful exit from ekpyrotic is negligible during the min + V , we illustrate how these | he mass term for ¶ 2 V ) p 0 8 ever after. The first term , . . . , onsisting of the ekpyrotic po- χ section with an explicit choice ) the ekpyrotic potential (3.2) ns to the generation of density ) i < ∼ K + emblance to the ekpyrotic piece. − e correction term ∆ oscillations, 4 p ) 2 χ χ χ ) the correction term (6.7) which φ, χ ( M ( ii min 2 Pl 2 Pl V early on, it becomes subdominant by 1 ; ,χ 1 M φ |≪ χ ′ pM p end min − )+∆ V 1+ ek 1+ µ φ φ, χ ( µ ≪ | away from the tachyonic ridge. Thus the full ( to fall within the allowed window Pl – 38 – Pl V 4 ek decreases, however, eventually this term becomes χ V m − φ/M min φ φ/M 2 p ek V ′ 2 0) p N q , which guarantees that ∆ )= , 2 p − q e e − end 0 φ, χ e − ≪ V ( 0 ek ′ − U V and forces to the vicinity of the ghost condensate point. This marks the p φ ( φ X is large. As V ) = ) = φ away from the tachyonic point, triggering the end of ekpyros parametrize the location of the minimum and curvature of the φ, χ φ, χ χ ( ( ), respectively. M are both positive, and, as earlier, ellipses denote higher- ek V X V ( 0 ∆ and P U m climbs up the sharp rise of the potential, its kinetic energy , must dominate over the energy density in and φ min 0 V function In order to have a bounce, the energy density in the ghost cond pV blueshifts by a factor of 8 or so during the bounce, this leads where beginning of the NEC-violatingcondensate phase. cosmology requires A successful synergy As tially, which brings These five equations are the main constraints to have a succes The exponents satisfy • • which are irrelevant for our purposes. Note that we neglect t phase followed by aconditions can non-singular be bounce. fulfilledtential by (5.12) In studying with a the the specific fiducial next potential correction section term c (6.7). 7. A specific exit model We now illustrate the constraintsof described potential. in the Our previous fiducial potential consists of two terms: which we assume hasgenerates exact a exponential minimum form in in potential is given by with relevant and pushes ekpyrotic phase where here on this simple form for the latter given its manifest res is the most constrained partperturbations, of whereas there the is potential considerable since freedom it in pertai th the onset of the ekpyrotic phase and remains subdominant for where in (5.12). While this term is crucial in stabilizing χ JHEP11(2007)076 ) 0 p 2 min χ and ≈ ≪ ( φ (7.6) (7.8) (7.5) (7.4) (7.3) (7.9) (7.7) ′ O χ p Pl 0. This M , thereby 2 ≈ . (We will Pl 0 H χ M χ 2 sits idle at the < ∼ H | ∼ , we can use the χ term lies at small ,χ χ . > ∼ V end is just | V 0 | . First note that the < ∼ − χ ,χ χ ek V min | ≈ . φ . χ . We find that this condition is not ′ : ions we can derive explicit . p ¶ p = Pl 0 nge 0 . . min χ 0 Pl end 0 if we assume that the q φ and min M φ ¶ χ φ Pl χ ′ χ M 0 0 from the ∆ χ 2 − p 0 V 0 U /M direction while min 1 χ Pl ′ V U . φ p p ′ φ end 2 = /M − 2 pp r H ek Pl end φ µ . µ Pl − 2 p ′ , however, we have /M ek ln p ln M q φ ′ ′ ′ end − min 2 p p p p e φ φ ′ ′ 0 |∼ √ ” √ by substituting the above expression for q moves a small distance in field space as it rolls ≪ 2 p pp − pp − pV − – 39 – e χ q √ 0 Pl √ 0 p p min = 0) − 0 ≈ χ ′ U χ √ 2 M √ p χ χ 2 − Pl reaches ( . As described in section 6.3, this occurs at a value 2 q Pl 2 Pl 0 ,χ “ M φ √ √ starts to roll away from the tachyonic point at χ e M V M so that this is indeed the range of interest.) Since | 0 0 , a good approximate solution for end 0 V U χ ≈ = = starts to roll = 0 for − ′ χ p p =0)= end 2 ek χ V φ ≈ χ min end min − H ( φ − χ > ∼ 1 ,χ ek min . In other words, V φ φ ∂V/∂χ χ = Pl 1 this says that the minimum along M plays the dominant role in determining min that χ = 0 can be made nearly independent of ≪ V p ≪ ′ p ∂V/∂φ at which a posteriori end Next we can solve To summarize, for − is pushed away from its unstable point. Subsequently we have ek towards the minimum at φ The next step is to find when From (7.2) the left-hand side is given by 7.2 Exit from ekpyrosis: Combining the above two expressions allows us to solve for It follows that ∆ 7.1 Global minimum for This potential has a global minimum. Under some mild assumpt analytical expressions for the corresponding field values And since everywhere: top of the tachyonic ridge. Once is the ekpyrotic phase, characterized by rolling in the values compared to Meanwhile, since the ekpyrotic phase is still going on until χ condition scaling solution (2.11) to determine the right-hand side: check necessary but is only imposed to simplify the expression for away from the tachyonic point towards the minimum. We stress satisfying the first condition (6.27) of section 6.3. terms are small compared to unity, at least in the relevant ra it is sufficient to impose JHEP11(2007)076 ond (7.10) (7.11) (7.13) (7.12) contains V reaches the has reached φ φ 1. . This ensures as it rolls to the is satisfied if ≪ . χ end p − end ¶ starts to roll towards ek 2 φ 2 ≪ , before p φ .13). Since ′ χ ζ < ∼ p − < ∼ nsity in is generically much bigger . For simplicity we assume min 0 hat Pl nto is large compared to Hubble, ation after the φ is just given by the difference χ essary — nothing prevents us min M 0 . since φ χ 2 µ 2 p ≈ Pl − /M . min 0 Pl ′ end χ p χ . − M to be a stable point by the time we reach 2 p ek 2 2 p φ = ′ / 2 p 3 ) min < p q χ around − min – 40 – > ∼ 0 Pl e χ would probably require numerical analysis in this χ ,χ M 0 2 0 Pl end Pl ζ χ − 0 amounts to pV M M min 2 acquires a large mass at the minimum and oscillates ek φ > ( H χ ) )= ,χχ to be small in Planck units. reaches minimum min V min 0 φ φ oscillations ( rolls to the minimum, so that all relevant quantities can be χ , it undergoes oscillations about this stable point.The sec ,χ 0 χ χ ,χχ χ min V φ ≈ ( . ,χχ min end V χ about minimum − ek χ φ reaches is nearly still as . The condition χ φ end Using (7.9) we obtain Using (7.8) we obtain − ek 7.5 Energy density in The last condition (6.31) puts an upper bound on the energy de minimum and starts oscillating.in As argued potential in energy section between 6, the this ridge and the minimum — see (6 Let us turn to the third condition, which requires that 7.3 Mass of rapidly. 7.4 Exit happens before Evidently this must be positive if we want Much like (7.3), this forces Note that this condition is consistent with (7.3) and (7.11) Once Barring some fine-tuning betweenthan these unity. two It terms, then this follows ratio that condition (6.28) ensures that the mass of consistency of the sequence of events assumed here, namely t its minimum. However an estimate of φ evaluated at case. In any case, comparison of (7.9) with (7.4) shows that its minimum, thereby imprinting the entropy perturbation o so that the fieldthat undergoes many oscillations in a Hubble time steep rise in the potential.from converting Again the this entropy assumption mode is into the not curvature nec perturb JHEP11(2007)076 (7.14) (7.15) . Thus end − n as ek φ is achieved for , the direction χ → end − min ek and couplings to light se since (7.3) implies φ φ inflation in solving the χ oncrete theory of initial ncretely, for GUT-scale logy. Indeed, the model = er all revolve around the ity along eneity and isotropy. Note ¶ ′ cm big, whereas the initial p p min nning of inflation, ekpyrosis se can be naturally achieved blems — one envisions a uni- ss for φ rum. We have shown that the 27 suring that the NEC is violated mparable to that of the ekpy- r tionary patch, thereby allowing − . . : n the limit tial states. , the universe emerges in the hot, by definition coincide with entropy g a scale-invariant spectrum. We 1+ Pl ek 2 oscillations is parametrically small Pl χ V µ , /M M 2 0 χ ′ 2 Pl χ Pl 2 0 min 0 pp χ φ χ M 2 p pM √ 2 q Pl Pl ≈ − e /M /M | 0 – 41 – V min min V min φ φ V 2 2 p p ∆ | |≈ p q q − − min e e 0 0 V | V ≈ V < V ∆ 2. Thus the relevant ratio for condition (6.31) can be writte / is to a good approximation given by Pl M min ′ V p . It follows that the energy density in √ Pl 2 / M 3 p The two-field ekpyrotic model displays a tachyonic instabil We have shown that ekpyrotic theory is equally successful as Now from (7.4) and (7.9) we see that the limiting case ≪ = 0 0 perturbations, this instabilitycannot is get essential rid in of generatin desired it without initial spoiling conditions the atwith perturbation a the spect pre-ekpyrotic stabilizing onset phase. of By the adding a ekpyrotic small pha ma Hubble radius in ekpyrosisconditions, is however, of both order represent 1 equally plausible m. ini In theflatness absence and of homogeneity a problems c allows of for standard initial big curvaturerotic and bang field. anisotropic cosmo Starting components fromexpanding co these phase generic with initial conditions athat high this degree is very ofverse similar spatial that to flatness, is how essentially homog inflation smoothcosmic addresses and flat acceleration these over to pro the take proto-infla over. orthogonal to the field trajectory. Since fluctuations in compared to the ghost condensate energyfor density, thereby sufficiently en long to cause a non-singular bounce. 8. Conclusion The many aspects ofissue ekpyrotic of cosmology initial discussedproposes conditions. in a the Unlike pap cold thereheating and hot temperature, nearly and the vacuous chaotic proto-inflationary initial begi patch state. is 10 Put more co exponential factors, evidently this difference is maximal i Meanwhile which has toχ be much less than unity. But this is clearly the ca χ we find JHEP11(2007)076 . ]. /ǫ 1 lead ∼ χ NL f and ire a scale- φ hep-th/0103239 -AC02-76-ER-03071 e for a broad range of ch of E.I.B. and J.K. and one coefficient in a From big crunch to big tic scenario on a stronger d variables ario. Instead of starting ul discussions. This work φ gued that self-interactions with the figures. We thank rfere with the generation of ines the tachyonic mass of a ial, we can think of one field kground experiments will be t of Canada through NSERC times — it becomes negligibly (2001) 123522 [ smology. Turok, . There is, therefore, enormous he level of non-Gaussianity. The . Assuming coefficients of order high, corresponding to ,φφ ]. leads to a slightly blue spectrum, in Pl The ekpyrotic universe: colliding V , however, can make the spectral tilt D 64 δ φ that characterizes the deviation from ≈ ǫM δ √ ,χχ V has important consequences for the predic- χ Phys. Rev. – 42 – , — to be such that the curvature of the potential and hep-th/0108187 χ φ ) framework required two steep exponential functions 2 , φ 1 φ ) language allows for far greater freedom in specifying the , a pure exponential for δ φ, χ (2002) 086007 [ D 65 . This occurs in such a way that fluctuations in the latter acqu χ gets stabilized well-before the onset of the ekpyrotic phas . With vanishing χ Phys. Rev. ,φφ , V have a natural cut-off scale of order Λ = = bang branes and the origin of the hot big bang The analysis of the tachyonic instability in terms of new fiel The simplified picture in terms of These achievements, we believe, together put the New Ekpyro χ rolling down an ekpyrotic direction whose evolution determ ,χχ [2] J. Khoury, B.A. Ovrut, N. Seiberg, P.J. Steinhardt and N. [1] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, invariant spectrum. The ( potential. Whereas the earlier ( in the potential, here we only need one steep potential along initial conditions. The mass termsmall is during only the important at ekpyroticdensity early phase perturbations. and therefore does not inte Taylor-expansion — the mass term for fermions, is approximately the same in either direction: us to propose afrom simplified two scalar and fields more eachφ general with their version own of exponential the potent scen second field unity, the resulting level of non-Gaussianity is therefore freedom in specifying the global shape of the potential. disagreement with recent data. Allowing for non-zero V former now depends on an additional parameter tions of the model, in particular for the spectral index and t The consequences for currentdiscussed and elsewhere. near-future microwave bac footing as a genuine alternative theory of early universe co Acknowledgments We are most grateful to AndreaK. Sweet Koyama, at P. PI Steinhardt, forof invaluable A. help B.A.O. 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