On Perturbations of a Quintom Bounce J JCAP03(2008)013 ] 2 1 6 2 4 ]

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JCAP03(2008)013 hysics $30.00 24 P , 3 , 2 , [email protected] 5 , 1 and stroparticle stroparticle public of China A public of China , Robert Brandenberger 1 [email protected] , public of China [email protected] , and Xinmin Zhang 4 0711.2187 ,TaotaoQiu 1 A quintom universe with an equation of state crossing the cosmological perturbation theory, cosmology of theories beyond the osmology and and osmology stacks.iop.org/JCAP/2008/i=03/a=013 [email protected] C Department of Physics, McGillKavli University, Institute Montreal, for QC, Theoretical H3A Physics, 2T8, Zhong Guan Canada CunCollege East of Street Physical 55, Sciences, Graduate School of ChineseTheoretical Academy of Physics Center for Science Facilities (TPCSF), CAS, Institute of High Energy Physics, Chinese Academy of Sciences, Yun-Song Piao cosmological constant boundaryquintom can bounce provide a andwe bouncing thus investigate solution resolveanalytically the dubbed the and the cosmological big numerically.mode perturbations bang in We of singularity. the ﬁndperturbations post-bounce in the that expanding the In quintom phase pre-bounce the contracting this couple ﬂuctuations phase. bounce to paper, in the both the growing mode dominant of the SM, physics of the early universe Keywords: ArXiv ePrint: doi:10.1088/1475-7516/2008/03/013 Abstract. 2 3 4 5 Yi-Fu Cai 1 Beijing 100080, People’s Re Sciences, Beijing 100049, People’s Re People’s Republic of China E-mail: PO Box 918-4, Beijing 100049, People’s Re [email protected] Received 20 November 2007 Accepted 27 February 2008 Published 17 March 2008 Online at

ournal of ournal An IOP and SISSA journal 2008 IOP Publishing Ltd and SISSA 1475-7516/08/03013+ c

On perturbations of a quintom bounce J JCAP03(2008)013 ] 2 1 6 2 4 ]. 20 22 22 14 − 16 , .In 6 ] 13 4 ) w> ]–[ 1 ]and[ 1to − 12 ]. Also, in loop 8 ]–[ w< 10 ]. In [ 9 On perturbations of a quintom bounce ...... 4 ...... 11 ...... 16 stacks.iop.org/JCAP/2008/i=03/a=013 ...... 5 d within the standard four-dimensional on is crucial for comparing the predictions ional action is modified by adding high order 03 (2008) 013 ( ] or the cyclic/ekpyrotic scenario [ ] that the null energy condition (NEC) is violated for 7 ...... 17 15 ...... 10 ...... 12 in our model is able to transit from ], a subset of the present authors considered a bouncing w 15 ], it is possible that the evolution is singularity-free. 6 ...... 8 ...... 6 3.2.1. Heating phase. 3.2.3. Bouncing phase. 3.2.5. Numerical results. 3.2.2. Slow-climb-contracting (SCC) phase. 3.2.4. Slow-roll-expanding phase (SRE). ]. If inflation arises as a consequence of modifications of the gravitational part of the action, as in 5 In some previous work [ 3.1. Background Acknowledgments References 3.2. Perturbations 2.1. Equations of2.2. motion of Equations the of double-field motion quintom of perturbations An initial singularity occurs in inflationary models if scalar fields are used to generate the accelerated expansion Journal of Cosmology and Astroparticle Physics Starobinsky’s original proposal [ of space [ 4. Discussion and conclusions 6 For years, it has beento suggested the that problem bouncing universe ofafflicts scenarios the both could initial provide standard singularity a big solution in the bang evolution cosmology of and the inflationary cosmos, universe a models problem which [ 1. Introduction 3. A concrete example of a quintom bounce 2. Review of the quintom bounce in a double-field model 1. Introduction Contents the literature there have beeninstance a the lot pre-big of efforts bang towards scenario constructing [ bouncing universes, for quantum cosmology the evolution might be singularity-free [ terms. Since,are in general, very in involved,fluctuations, these it and models is the the knowledge difficultof of equations the to the model for reach evoluti with cosmological observations. definitive perturbations conclusions about the evolution of models were considered in which the gravitat and then connect with normal expanding history. universe model (‘quintom bounce’) obtaine Friedmann–Robertson–Walker (FRW) framework, but making useIn of this quintom model, matter [ ita can short be period shown in around(EoS) the [ bounce of point. the Moreover, matter after content the bounce the equation of state JCAP03(2008)013 ) 3 ], φ μ 29 ]. In φ∂ 26 μ ) ]) forbids φ, ∂ 24 ( L ]–[ = 19 modes enter the k L ], [ 16 1atpresent.Thismodel ]. The quintom model is − 17 On perturbations of a quintom bounce ], models with vector field [ w< ] (also see [ modes feel the resolution of the 28 18 1. It is not easy to construct a k − stacks.iop.org/JCAP/2008/i=03/a=013 ], of ideas from non-local string field = 30 osmological perturbations. In singular in non-singular cosmologies, and features cenario, the EoS crosses the cosmological the EoS is crossed. Interestingly, we find w ] realized a quintom scenario by considering ry different in the region of large comoving oint and have no time to complete a single the decaying mode in the expanding phase), m matter can provide a bouncing cosmology ions is not under control classically. The 27 phase with those in the expanding phase. In n evolve smoothly from the contracting to the ns through the bounce possesses features found ]). Because of the brand new properties, quintom 1 in the past and 03 (2008) 013 ( ical reason is easy to see: small − 34 ]. This model has been studied in detail in [ 25 ]–[ , w> he commonly used variable Φ, the metric perturbation in 32 16 modes undergo many oscillations which are in the bounce k ] was initially proposed to obtain a model of dark energy with an modes evolve as if there still were an abrupt transition between 16 k which satisfies w ], and others (see e.g. [ 31 ], is called the quintom bounce. In this s A quintom model [ In order to be able to compare the predictions of a quintom bounce with observations, 15 is mildly favored by the current observational data fitting [ EoS parameter a traditional scalar field model with Lagrangian of the general form a dynamical scenariocross of over dark the energyconsistent cosmological quintom with model. a constant A salient barrier no-go theorem feature proven that in [ its EoS can smoothly the non-perturbative effects ofsingle a scalar DBI with model. higher Moreover, derivative there terms are in models the action which involve [ a recent years there havemotivated been from a string theory, lot the of authors of theoretical [ studies of quintom models. For example, Journal of Cosmology and Astroparticle Physics from having its EoS crossadd the cosmological extra constant degrees boundary. of Therefore, freedomtheory it if with is we unconventional necessary expect features to to totheory. realize the a conventional viable The single-field quintom model simplestlike in the quintom and framework another model of Einstein’s phantom-like involves gravity [ two scalars with one being quintessence- of the evolution inan singular bouncing evolution similar models. to Forof what small Φ comoving occurs in wavenumbers the in one contracting singular finds phase bouncing couples cosmologies only (the to growing mode making use oftheory an [ extended theory of gravity [ both in some previous analyses of fluctuations whereas the transfer of the perturbations is ve models give many unexpectedpaper predictions. is that Most a important universe for dominated the by subject quinto of the present that our result for the transfer of fluctuatio wavenumbers (the growing modein in the the contracting phase phase of couples expansion). to the The dominant phys mode which allows us toin avoid [ the problem of the initial singularity. This scenario, developed bouncing models thefluctuations evolution diverge at oftheory of the fluctuat linear cosmological bounce perturbations,connect and the point, secondly fluctuations making firstly ina it the impossible invalidating quintom contracting to the bounce reliably result model applicability is the that of cosmological evolutionexpanding the perturbations of phase. ca fluctuations We show is thatlongitudinal t gage, well passes behaved. smoothly bothwhere Our through the the first bounce cosmological main point constant and boundary through the of points sub-Hubble region very close to the bounce p it is crucial to investigate the evolution of c constant boundary twice around the bounce point. oscillation, whereas large phase at sub-Hubble scales. Thus, whereas the large singularity, the small JCAP03(2008)013 4 (7) (3) (5) (4) (6) (2) (1) , . ) ) contains 7 ψ 4 ( W − of this model: ) φ p ( ]) conditions V has a canonical kinetic 36 − 2 φ ˙ is given by ψ ng matter action, we easily 1 2 w On perturbations of a quintom bounce . − of a universe which crosses the ) 2 and pressure ˙ ψ φ w , ( 1 2 ρ ) ψ W ( = , we review the equations of motion 2 stacks.iop.org/JCAP/2008/i=03/a=013 W )+ φ , − ( ). Here the field on–Walker (FRW) universe, the metric is erturbation equations for this model. In ) ill follow the fluctuations both numerically approximations valid during those periods, V − φ 2 ,p oints, and finally numerically. The numerical , W ( ˙ ) ground and the perturbations can be matched ] (initially derived in [ . + ψ t − ψ V + 2 , ( 35 ˙ − ψ ns consistently for all times. Section − − V is the Hubble parameter and the dot denotes the 2 W 03 (2008) 013 ( , ˙ 2) φ ψ + / μ 2 )+ a/a ˙ (1 , ψ , φ ψ∂ ) ( . By varying the correspondi 2 1 2 =˙ − μ , i ˙ ∂ 2 ψ V x ˙ =0 − 1 2 H φ d 2 =0 + − i ˙ ,ψ 2) − φ 2 2 x ,φ / ˙ ˙ 1 2 φ , we investigate the perturbations, first analytically by solving φ W ψ V ( )d μ 1 2 (1 t − ( + 3.2 − 2 ˙ φ∂ πG ˙ 3 ψ πG φ μ a 4 2 1+ 8 ˙ ∂ φ H H − 1 2 − − 1 2 = 2 ]. = t = = 2 +3 +3 = 37 ˙ ¨ ¨ ρ w ψ L φ H H , we determine the complete evolution of the background using numerical =d has a kinetic term with the opposite sign and thus plays a role of a ghost field. 2 3.1 s ψ ) is the scale factor, , we study the behavior of a specific quintom bounce model systematically. t 3 ( a It is well known that this model can realize the EoS This paper is organized as follows. In section For a discussion of the dangers in applying the matching conditions at the transition point between contraction cosmological constant boundary. Namely, the expression for and expansion see [ 7 To start, we take a quintom model consisting of two fields with the Lagrangian 2.1. Equations of motion of the double-field quintom and analytically. In thebounce analytical point, analysis, in regions we where applyconsistently both matching via the conditions back the away Deruelle–Mukhanov from [ the the contracting to the expanding phase. We w section In section 2. Review of the quintom bounce in a double-field model of the double-field quintom and derive the p integration. In section and matching the solutions at theresults transition support p ourdiscussion analytical and calculatio conclusions. the equations in separate periods by means of and the background equations of motion are given by obtain the following expressions for the energy density Journal of Cosmology and Astroparticle Physics derivative with respect to the cosmic time where where the signature of the metric is (+ In the framework of a flat Friedmann–Roberts term, but given by d JCAP03(2008)013 ] 5 (9) (8) 39 and (10) (12) (11) δφ ) + φ ’ perturbed ij represent the → φ δp , the perturbations δψ and and δq for convenience. The fields , ’ equation, and the third is δφ On perturbations of a quintom bounce i t/a δρ , ] i x =d . d re is no anisotropic stress to linear η i egular not phantom matter alone can )Φ x stacks.iop.org/JCAP/2008/i=03/a=013 p δp, 2 + etric fluctuations (gravitational waves) do ρ 2Ψ) d cuss them here. By expanding the Einstein rough the cosmological constant boundary. tric which couple to matter fluctuations (see δρ, πGa − 2 =3( (1 defined by d η πGa δq − 03 (2008) 013 ( 1 and the universe lies in a quintessence-like phase; 2 )Φ = 4 1 and correspondingly the universe is phantom-like. 2 2 − ∇ η − 1 H a + Φ) = 4 − ) H H πGaδq, δp + 4 + − +(2 (Ψ )[(1 + 2Φ) d is less than δρ η is larger than ( is the comoving Hubble parameter and the prime denotes the H Φ ( w 2 3 Φ= w H H a η − H d = +3 +3 Ψ + 2 2 ’ equation. For scalar field matter the s a/a δρ d Φ Ψ ∇ ii . To linear order in the scalar field fluctuations ]. If we work in the context of a spatially flat four-dimensional background 1, but only around the bounce. Neither r =d is negative, is positive, 15 δψ − 2 2 ˙ ˙ + ] for a comprehensive survey of the theory of cosmological perturbations and [ H ψ ψ ψ 38 − − From the conservation of the stress tensor, we also obtain the analog of the continuity In the double-field quintom model, the scalar fields are perturbed: 2 2 ˙ ˙ → φ φ equation for the matter fluctuations: Einstein equations that Φ = Ψ. Therefore, a quintom model is the only possible solution for resolving this difficulty. achieve a transition in the EoS parameter th In recent work,scenario it [ was foundFRW that metric, the there quintom mustif model be we can a are period providephase. to a when obtain In bouncing the a this null universe less smooth case, energy than we transition condition need from (NEC) a a is kind contracting violated of universe matter into which an admits expanding an EoS parameter which is if Journal of Cosmology and Astroparticle Physics 2.2. Equations of motion of perturbations Now let usNewtonian) consider gage metric linear perturbations perturbations are given of by the metric. The longitudinal (conformal- If order in the matter fluctuations, and hence it follows from the off-diagonal ‘ where we introduced the comoving time not couple to matterequations and hence to we first will order, not we dis obtain the following perturbation equations: Φ and Ψ dependfluctuations, on the space degrees of and freedome.g. time in [ and the contain me thefor information a about recent overview). theby scalar We scalar will metric field not matter. discuss To vector linear fluctuations order, since tensor m they are not induced ψ perturbations of density,the momentum perturbed and ‘00’ pressure,the equation, diagonal respectively. ‘ the The second first the equation perturbed is ‘0 where derivative with respect to the comoving time. Here JCAP03(2008)013 6 will =0 (19) (14) (13) (15) (16) (17) (18) ψ w ) . δψ, δψ, ,ψ ,ψ Φ , W W ψ Φ − φ Φ) + Φ) On perturbations of a quintom bounce Φ+4 n the initial phase of evolution ψ ψ ,ψ Φ+4 − − W ,φ 2 V a 2 δψ δψ a stacks.iop.org/JCAP/2008/i=03/a=013 ( ( 2 +2 ψ ψ − 2 2 , d yields a normal (i.e. non-phantom-like) . Thus, the kinetic energy density of the 1 1 δψ ) a a but also realize a scenario where the EoS 3 φ δφ − ( ,ψψ − − a ,φφ V W V δφ δφ 2 2 − a ,φ ,φ 03 (2008) 013 ( a ψ V V + μ − , − ] δψ ψ∂ . Therefore, the contribution of the ghost field δφ 2 μ 6 δψ 2 Φ) + Φ) k ∂ − k φ ψ φ 1 2 a − − . + − − − 2 φ φ δψ δφ 2 δφ μ δφ δφ ( ( H H φ m 2 2 φ∂ 1 2 and hence also the EoS of the universe will oscillate about dominates only close to the bounce point. It must be able to decay φ φ dominates by a large factor. Thus, i − μ [ 2 − 2 − 1 φ 1 ∂ ψ a a 1 φ a 1 2 = = )= evolves proportionally to = = = φ = ˙ ( ψ δρ δp δq δφ δψ L V In this model, the evolution of the universe is symmetric about the bounce point. In We thus take the Lagrangian of the quintom model to be (period 1) theand field the average state is similar to that in a matter dominated period. Since the universe Journal of Cosmology and Astroparticle Physics 3.1. Background We will assume vanishingthus potential consists for of the kinetic ghostit energy field, exclusively. follows whose From that background the energy equation density of motion of the ghost field of energy, momentum and pressure can be expressed by the region long beforedensity the in bounce the point, we field choose initial conditions for which the energy ghost field is proportional to respectively. The linear termswith in respect the to equations the arefield obtained two perturbations, by matter varying fields; the we matter obtain action the equations of motion for these scalar dominate at the bouncewhat point, we need but in will order decay to quickly both before achieve and a afterward. bounce This and connect is to just the observed universe. evolution of thethat universe the at ghost very field early and very late times. Consequently, we demand after the bounce. In order toa make specific the example analysisprovide of more a a quantitative, bouncing quintom in model. solution this of section Moreover, the we our universe, would quintom like model to should take not only 3. A concrete example of a quintom bounce crosses the cosmological constant boundary an and, to be specific, we choose a simple form of the potential JCAP03(2008)013 7 ). pl w 18 M 6 − .The =0. climbs 1 ) 1. This − w is rolling φ 10 φ − × w 414 . in the model ( H is still subdominant. =1 ψ ,m 73 − On perturbations of a quintom bounce which we took to be 10 10 × M 62 . stacks.iop.org/JCAP/2008/i=03/a=013 500. Note that in this and the following sion parameter increases sufficiently, the =4 − ˙ d the EoS returns to a non-phantom-like ψ , = 2 t 10 inflation model), it will start to oscillate about × 2 03 (2008) 013 ( φ 2 56 . m =2 ˙ φ , 3 1. This triggers the onset of period 3, the bouncing phase. 10 by dividing by a mass scale − decreases below a critical value (the same critical value which . Plot of the evolution of the Hubble parameter × we plot the evolution of the Hubble parameter and of the EoS, φ ψ pl = 6 . 2 M 5 6 w − approaches negative infinity at which time the bounce occurs. After − and to the energy density is increasing. We take initial conditions such that and = w φ ψ 1 , Figure 1. initial time was chosen to be In the numericalφ calculation we choose the initial values of parameters as: figures with results fromof numerical simulations, 10 all masses are expressed in units 1. Once H − , m w> In figures rises exponentially and soon starts to dominate. At this point, the EoS parameter slowly but thevalue ghost field decayssignifies the rapidly, end an of inflation in an transition is the timeDuring reversal of this the second end ofslowly period, the up the slow-rolling its period period potential.contribution in of of scalar During field quasi-de both inflation. Sitterat periods the contraction, 1 transition the point and between field Due 2, periods to but 1 the and in 2 exponentialψ particular the decrease contribution in of of period the 2, scale the factor during period 2, the contribution of Journal of Cosmology and Astroparticle Physics the bounce, theagain time enters reverse a of quasi-denearly exponentially, Sitter the i.e., phase evolution a (period period before of 4), the inflation but bounce takes this place. occurs. time During the this The period, universe universe is expanding universe will enter a nearly de Sitter contracting phase (period 2) with crosses the barrier The parameter is contracting, it is heatinglarge up. and the Once absolute the value amplitude of of the the Hubble field expan oscillation gets sufficiently the minimum of its potential, and the EoS starts to once again oscillate around calculated numerically. Insuch this as calculation, we normalized the dimensional parameters JCAP03(2008)013 ) 8 11 (20) with ). The φ ) 18 )and( 10 ), ( 9 is oscillating around δψ. in the model ( ψ w w . 1 φ φ On perturbations of a quintom bounce + and its initial conditions only H m 2 which is similar to matter with its 3 − stacks.iop.org/JCAP/2008/i=03/a=013 πG a The transitions between the phases are ∝ ned above is rather generic and does not ude that a bouncing solution is obtained if ˙ al conditions. The reason is the following. he potential of the quintom field to vanish, ψ Φ=8 2 SCC) phase, a bounce, a slow-roll-expanding −∇ 03 (2008) 013 ( Φ 1. 2, which leads to an EoS parameter smaller than 1. If / 2 φ φ − φ 2 w , we can see that in this model the universe undergoes a m 2 −H H )= φ Plot of the evolution of the EoS parameter ( V +2 Φ initial values of parameters are the same as in figure Figure 2. φ . From figure φ G √ / ) = 0, the phantom field evolves as H− ψ ( =1 Note that the background evolution obtai W +2 pl Journal of Cosmology and Astroparticle Physics To study the evolution of the perturbation, we combine the equations ( 3.2. Perturbations Φ and obtain the equation of motion of the gravitational potential where the PlanckM mass is determined in terms of Newton’s gravitational constant by the initial conditions arethe imposed parameters during the of period the when canonical the field phantom such part is as negligible, the mass affect how long the0 universe and is in staying the in SCC the phase heating when phase when a quadratic potential EoS parameter equal tothe 1. absolute value Thus, of we thethan can phantom that concl energy of the density normal inthan matter. the 1. contracting This In universe will our grows occur concrete faster model if we the have EoS chosen of normal the matter normal to matter be is a smaller canonical field heating phase, a(SRE) slow-climb-contracting phase, ( andsmooth. finally a phase of cooling. depend sensitively on the details of the initi i.e. In our model wea have bouncing introduced a solution. phantom Since field which we makes have it chosen rather t trivial to obtain JCAP03(2008)013 , 9 k (21) ) . Perturbations 3 , the curvature fluctuation ζ On perturbations of a quintom bounce stacks.iop.org/JCAP/2008/i=03/a=013 diabatic fluctuations only in this paper. ast. Since in the initial heating phase the he scale factor shrinks nearly exponentially ariables can be determined from Φ, knowing s since it is conserved on super-Hubble scales is ill-defined near the bounce point, as already . . ζ Φ) 03 (2008) 013 ( Φ) H H + 2 + H approaches 0 and also when the universe is crossing the (Φ 9 (Φ H in the quintom bounce. 2 k k 1. Therefore, 2 −H H 2 − A sketch of the evolution of perturbations with different comoving = H = ζ ) w ]. Thus, we will focus on the evolution of Φ which is well defined w Φ+ 41 , ≡ ζ (1 + Figure 3. wavenumbers 40 Since there is only a single physical scalar field degree of freedom for adiabatic scalar Since in the linear theory of cosmological perturbations all Fourier modes evolve in comoving coordinates, which is given by metric fluctuations, all other perturbation v the evolution of the background. A frequently used variable is In the case of anwell expanding universe the with adiabatic regular fluctuations matter, on this variable large is scale known to describe because it obeys the equation Journal of Cosmology and Astroparticle Physics However, this equation becomeswhen singular the both Hubble when parameter phantom the boundary universe isremarked at in the bounce [ point independently, we will follow each mode, labeled by its comoving wavenumber throughout the bounce.Since there We are are tworeturn focusing matter to fields, on this it question a is in possible future to work. have entropy fluctuations. We hopeindependently. to Interestingly, in our modelcrosses there the are Hubble four radius. times when The a evolution perturbation of mode scales is sketched in figure start out inside the Hubble radius in the far p Hubble radius isOnce shrinking period faster 2, the than ‘slow-climb the phase’ starts, scale t factor, modes exit the Hubble radius. JCAP03(2008)013 , φ 10 (22) (23) (26) (25) (24) ). .The 20 3 ) does not exist ]. Therefore, in k 18 Φ 2 , k 41 , such that they enter the can be neglected. In this 40 k ,wehaveanaveragevalue ψ B t On perturbations of a quintom bounce − large enough for the perturbations t . k stacks.iop.org/JCAP/2008/i=03/a=013 B . ollowing equation for the gravitational t 3 4 η ydrodynamical matter is non-relativistic, in outside the Hubble radius throughout ) because this variable is singular both at = uring the rest of this phase, and throughout ζ ill denote the time corresponding to the end verse as being dominated by a single field avior of perturbations from equation ( H , 03 (2008) 013 ( , ) V , 2 2 2 V η 2 a = 0, and the contribution of a consists of modes with small − =0 , + w ) k + 2 = 2 Φ fall into two broad categories, as sketched in figure φ 2 φ 1 2 H k k −H −H − ( + 2 ] (related closely to k H Long before the bounce, when 43 3 0 during the contracting phase. Thus, we obtain the expressions 3 πG HH Φ πG , , 2( 8 2 8 6 η 2 η 42 [ η< = + = , the end of the bounce phase by 2 v k c t φ φ H H H Φ ,with 2 η ∝ a To summarize, scales Making use of these relations, we obtain the f In a bouncing universe, it is unsuitable to use the equation of motion for Mukhanov– In the following we will develop analytical approximations to study the evolution of Journal of Cosmology and Astroparticle Physics 3.2.1. Heating phase. whereas the Hubble radiusenter is the constant. Hubble Thus, radius wavelengths which during are this not phase. too D large will re- the bounce period, thepost-bounce scale remains inflationary sub-Hubble, phase, and exits onlywith the to Hubble an re-enter radius extremely again the in long Hubble the radius wavelength at will late rema times. Modes period 2. Sinceenter the the Hubble Hubble radius radius before the diverges bounce. at the bounce point, even these scales will using conformal time and Hubble parameter. Further we have in momentumequation space. in a Note matter that dominant period this except equation that is the very last term similar to the perturbation of the EoS parameterregion, which we is can treat the evolution of the uni with the Friedmann equations being for hydrodynamical perturbations (ordinary h potential: Sasaki variable first consists of modesto with return comoving inside wavenumber the the bouncing Hubble phase; radius theHubble before radius second the during the bouncingthe bouncing phase post-bounce phase. spectra and We of will escape fluctuations show outside later between after that these there two is ranges a of difference scales. in and the bounce point and at the cosmological constant boundary [ of inflation by the fluctuations in the various phases. We w the following we will directly study the beh JCAP03(2008)013 φ 11 (28) (31) (29) (30) (33) (32) (27) ,then ) |H k | , On perturbations of a quintom bounce )] kη ( 2 , / . 5 2 . Consequently, the canonical scalar / − , φ =0 7 3 stacks.iop.org/JCAP/2008/i=03/a=013 0 J − k k η . Therefore the initial condition for the : 0 3 8 Φ B 1 we can take the asymptotical form of the ρ rt to climb up its potential. We denote 2 2 Ak − Davies vacuum, determine the coefficients. / ial conditions are determined by canonical k η √ 5 π 2 k + ssel functions we obtain the following form of √ | ∼ For the universe in the contracting phase, we πG k − . 4 kη )+ 5 Φ | /a 2 − = 03 (2008) 013 ( ≡ on, the universe is in a period of SCC phase with φ kη η H k ( 2 c ) / 2

7 / 5 − can be determined from the initial condition. − J k k k σ 2 is anti-frictional, and instead of damping the motion of η>η A ,A A B ˙ ,B . 2 3 √ 3 φ 2 / k kη / 5 kη i i H 3 +2( 2 − − − − . k and k e 2 =0wehave .For e √ [ 2 / c Ak 3 Φ 3 2 k k 3 η / w √ k − π 5 A Φ H 2 1 2 − a πGφ √ Ak σ 15 i Aη η 4 √ =i = = +2 = ≡ ]. We will take the Bunch–Davis vacuum as the initial conditions (when k k k k k k k Φ Φ u u Φ Φ A 38 0. In this case, 3 is a normalization factor. When will eventually stop oscillating and sta 1. We assume that at the beginning of this period the ghost field can still be A H< φ To set the initial conditions, we first define the variable − : is sufficiently far away from the bounce point) k We will discuss other possible initial conditions in a follow up paper. neglected. Therefore, the equation of motion for the gravitational potential becomes where the coefficients 8 3.2.2. Slow-climb-contracting (SCC) phase. our scalar fieldmotion matter can is be relativistic). written as The follows: analytical solution of the above equation of Journal of Cosmology and Astroparticle Physics This is the variablequantization in [ terms ofη which the init w When the wavelength of perturbation is larger than Hubble radius with gravitational potential becomes During period 1 when where have field this transition time as From this result, wegrowing can during see the that contracting one heating of phase. the two modes of Φ is constant and the other is by using another asymptotic form of the Be Φ Bessel function and, matching with the Bunch– The result is in the expanding phase it accelerates the motion of JCAP03(2008)013 k 12 C (38) (39) (34) (35) (37) (36) ]) in 35 .Since , ) B when the η 36 − B η , ))] c ˜ ) for the gravitational η 33 − , η ( k )) ( c On perturbations of a quintom bounce ν ˜ η and a mode with coefficient − , J − k k , we can determine the coefficients η 5 D D c ( − c ν η k η k stacks.iop.org/JCAP/2008/i=03/a=013 k B . cos( ) this difference leads to a difference in the ˙

k )) + H c . We can obtain the asymptotic form of sponding quantity is evaluated at the time ¨ D ˜ η H c H 2 H − . / . η 0 ] )) + ( approaches inﬁnity, all wavelengths will be inside k c +3(1+ 03 (2008) 013 ( ≡− ˜ k η +1 D ( k ν ) σ c − 5 A /H J , η ) − k η c )+

( H η c 2 3 C k k (these are the matching conditions derived in [ ˜ / η ≡ ν becomes more and more important, since its kinetic energy 1 B − ζ − c − 3 ψ k η − [ sin( and η (1 α c ( − k ) . 1 k k 15 c k k H C ˜ η C / and ˜ [ A Once the universe enters the quasi-de Sitter contracting phase, the c 1 2 2 2 − π π 2 45 1 2 ˙ ( η H . Therefore, the SCC phase will cease at some moment H

−H 6 c begins to dominate, and then the EoS parameter of the universe will η − = =( = a ν k ψ − k k k k ≡− , Φ Φ

C D η Φ 1 2 1 and fall to negative infinity rapidly. Correspondingly, the energy density ) in the sub-Hubble region as follows: − 36 k = D α w Making use of the above relations, we can solve the equation ( Through matching Φ and As is well known, we have and . k c perturbation: Journal of Cosmology and Astroparticle Physics 3.2.3. Bouncing phase. which is growing. the heating phase and in the SCC phase at the time where we take the slow-climb (slow-roll) parameters as equation ( η where a subscript ‘c’ indicates that the corre C In the latter form the two modes are a constant mode contribution of the ghost field density scales as Therefore, in the SCCinitial phase growing mode the of main Φ contribution to the perturbation comes from the at this moment the Hubble radius 1 way the initial spectrum of fluctuations transfers through the bounce. the Hubble radius.bounce point. Thus, Short the wavelengthare modes perturbations will sub-Hubble will have near time belonger to the once perform intrinsic bounce many period again oscillations point, and oscillatingperform which also whereas a close spend complete long to oscillation. less wavelength the time As modes being we (which will sub-Hubble) have see, will a not have time to contribution of and the Hubble parameter reach zero when the bounce takes place at the time cross and in the super-Hubble region where JCAP03(2008)013 φ ). 13 )to 20 .In (40) (42) (41) 4 φ ) H 2 we present a − , 4 2 ) φ B ) η B η − in the bouncing phase. − η ˙ ( φ η 2 B ( ¨ φ/ 2 B αa + ) αa On perturbations of a quintom bounce ) H /π )) we can solve equation ( /π (2 40 (4 − 1 stacks.iop.org/JCAP/2008/i=03/a=013 = (see ( H, e parameter near the bounce point (we 3 H ation equation exactly from equation ( αt = − , is initially climbing up to the hill. However, as changes its sign which then leads to the field H 2 ) ¨ φ φ 03 (2008) 013 ( B α πG order to simplify the equation. η 4 − − η 2 ( ˙ ψ 2 B B is positive. a αa in the bouncing phase. The values shown in the figure support ) H ˙ 2 φ ˙ is decreasing, /π ψ A plot of the evolution of the factor 3 ¨ φ/ (2 αt H αt, 3 + − − 1 H )= t = ( = ˙ ¨ φ φ H a Figure 4. , we will show that this is a very successful simulation by comparing with the denotes the scale factor at the bounce point. is a positive constant. = 0 to correspond to the bounce point) is B 3.2.5 t α a In this case, we obtain the analytic forms of the scale factor and the comoving Hubble A convenient parametrization of the Hubbl In the bouncing process, the field It is rather complicated to solve the perturb Making use of the parameterization where parameter: choose the absolute value of In order to gainwe some resort analytical to some insight approximations into in the evolution of fluctuations in this region, Journal of Cosmology and Astroparticle Physics around the bounce point. Consequently, we can derive another formula reversing its direction anduniverse beginning has to bounced slowly and roll down the potential.obtain At this stage, the section where numerical calculation. In order to show this point more explicitly, in figure plot of the factor 3 JCAP03(2008)013 2 ). 14 . 3 /k (47) (43) (44) (46) (45) 1 y should 3.2.5 ) | , B η − .Thesetwo 2 ) k η | B , F η ,

k − )] (also see ﬁgure E B is small enough and η η ( 1 k 3.2 y − 3 2 η ; ( 2 1 k ; ), we obtain the following On perturbations of a quintom bounce 2 ¯ sin[ m is large enough that the mode 20 k − k 1. ˜ E 1 F , | 1 0 )] + stacks.iop.org/JCAP/2008/i=03/a=013 ) ) since from the ﬁgure, it can be seen the numerical results in section B k B in the SCC phase and in the bouncing which can enlarge the amplitude of the η η . The solution to this equation can be F 42 ζ 2 B k 2 − ) t after the bounce. We sketch this solution − B η )Φ αa η η ( 1 ) )) + ( − y k B k η | ( η /π 1 + y . 03 (2008) 013 ( (8 − cos[ k 2 k 4) uare term and higher order terms of / k k F η E ˜ (3 ≡ F 2( + − , the spectrum has been oscillating already at late times 2)

+( 1 and two undetermined coeﬃcients / k − 2 1 y th Hermite polynomial and a conﬂuent hypergeometric k ) 1 π B y E ¯ B m/ y m 3 η )Φ √ 3 2) − − B /

η η +1 1 ( ( π Γ( H 1 ¯ ¯ y m m/ ¯ y m 2 − √ 2 m k 4) is big enough, We can make use of the following approximate: ¯ H − 2¯ m η / k − ( 2 (3 √ 1 2 m E − y k Γ(1 e 2 ) = ≡− +3 ≡ B η k k k k − ˜ m ˜ η F Φ Φ E ( 1 y and will find that it is consistent with 2) / 5 (3 − We suppose that the approximations above should be valid only in the neighborhood Substituting these approximations into equation ( In the first case with If the parameter ¯ Note that there are two possible ways for the modes in the SCC phase to evolve =e k Journal of Cosmology and Astroparticle Physics is sufficiently small.blows This up is at to the bemode bounce expected functions point in should and our oscillate, all scales bounce even for become phase sub-Hubble since and the hence Hubble the radius associated the approximation we have made in equation ( of the bounce point, and hence the sq that this factor vanishes around the bounce point. Φ written as Interestingly, we obtain a factor e functions are linearly independent, but both of them show oscillating behavior when in the SCC phase. Through matching Φ and where we define the parameter be neglected. equation: the mode re-enters the Hubble radius only during the bouncing phase. perturbation before the bounce and suppress i in figure which is constructedfunction from an with ¯ ¯ and where The first possibilityre-enters is the that Hubble the radius comoving in wavenumber the SCC phase. The second is that into the bouncing phase, as described at the beginning of section JCAP03(2008)013 )] )] of 15 B B η k (48) (49) η (50) C − − ) − − B B η η ( ( k k sin[ cos[ as follows: , . )] )] c c (1) ˜ ˜ η η k )] ˜ )] , the coefficients of the F B )] )] B − − − c c η η B ˜ ˜ − η − η , B B − − and η − η − )] ( − ( On perturbations of a quintom bounce − B − − B k k B (1) H k η B B η )] η ˜ ( E η η ( k B − ( ( k k η cos[ cos[ k k − k k − B ). D D η . Through matching the spectrum in stacks.iop.org/JCAP/2008/i=03/a=013 − B ( − B η )cos[ )cos[ k )]] cos[ )]] sin[ B η k k c c )] ( ˜ ˜ )] + )] + − η η B D D c c k cos[ ˜ ˜ η − η η − − − − B ∼H k B B − η − − − − sin[ ( D k H H B B − − − k ) η η B B B ( ( + + B − 03 (2008) 013 ( η η η k k k η k k B ( ( ( C k k k C C /π H − sin[ sin[ [( 2 [( )and − c − − − sin[ sin[ cos[ B η

B B k k k η k k 1 k 1 ( k k

− H H D C C k D

kD kD − 2 ) ) ) ) B B B B B − − η η η η η ( A sketch of the evolution of the gravitational potential Φ in the k k 2 2 2 − π π π k − − C C − − we can determine the coefficients 1 B 1 1 − − η − − − 2 2 2 ( − B B ) ) ) B 1 k k B B B B B B y η η η η η η H H 4 ( η ( ( / − − − k 3 k k − − − e B B B of the constant mode in the SCC phase may play an important role when − η η η Figure 5. bouncing phase. + − + + ( ( ( 1 1 1 k y y y − D 4) 4) 4) / / / (3 (3 (3 e e e Secondly, we consider the case when (2) (1) (1) k k k ˜ ˜ ˜ Journal of Cosmology and Astroparticle Physics phase at the time we fine-tune the phases E the bouncing phasefollows: with that in the SCC phase, the coefficients can be determined as E perturbations in bouncing phase depend in an UNsuppressed way on the coefficient F From this result we can read off that, in the case when the growing mode incoefficient the SCC phase. However, this is not an absolute relation since the JCAP03(2008)013 . , 16 . B+ } B+ } (54) (56) (55) (51) (53) (52) )] a )] B B η η ) H − − k , B+ B+ η η ( ))] ( k k B+ ˜ η cos[ of the subdominant ) in the sub-Hubble cos[ − k k k ˜ F 52 η ˜ F ( D k )) ( ν )] + )] + ). B+ − B . ˜ B B η J η On perturbations of a quintom bounce )] η η k B − − H η − − η )] ν ( − B k − B+ k B+ B η η − η η ( ( ( are the same as those which appeared B − k stacks.iop.org/JCAP/2008/i=03/a=013 k k η )) + cos( ν ( − k B k sin[ B+ )] η sin[ H ˜ η k ( B k and ˜ k η sin[ E ˜ − E to the usual dominant mode on super-Hubble { α k . { − η ] k

)) + ( k cos[ D )] − )] k ) Due to the symmetry of the quintom model, the B+ H ( B B ˜ 03 (2008) 013 ( B+ η ν η B+ − ˜ η ( η J k ˜ η B k − k )+ C /π − − 2 − 2 H η G 2 ( ν − B+ . We denote the scale factor at this moment as sin[ π/ k ˜ B η − π/ B+ −

B+ k η η k B+ k η

[ (

( −

( η D α k )). Thus, the solution takes the form k sin( D k ) η ) 2 ( ) k B 33 and the values of k B k B+ η G η ˜ η G − [ 2cos[ 2sin[ − 2 B+ π − 2 2 2 1 − π π 2 ) B ) − H η B η B ( B / η η 2 1 η ) y − − ( B =( 4) η +1 k B+ / B+ − η k k k η ( (3 . We can obtain the asymptotical form of equation ( ( − B+ 1 + e Φ Φ Φ 1 begins to slow-roll down to the bottom of its potential and the ghost field B 1 to above, which provides a link with the normal expanding history of our y η y η ( 4) − 4) φ 1 / / ≡ y 3.2.2 4 (3 (3 / − − 3 B+ e e e η When determining the coefficients of the two perturbation modes, we again need to The equation of motion for the gravitational potential in this phase is the same as in decays soon after the bounce. Therefore, a phase of slow-roll expansion (like slow-roll (1) k (1) k (2) k ˜ F Journal of Cosmology and Astroparticle Physics G and assuming that all the phases are non-trivial, we obtain 3.2.4. Slow-roll-expanding phase (SRE). scalar field ψ inflation) starts at the time match with the perturbation in the bouncing phase in two cases. In the case The second mode is constantscales (it in corresponds an expanding universe) and the first mode is decreasing. mode in the SCCthe phase bouncing which phase, mainly unless we dominates fine-tune the the coefficients phase of the perturbations in in section H From these two equations we find that it is usually the coefficient the SCC phase (equation ( universe. During this period,occurs the in Hubble the parameterfrom usual is below inflationary nearly scenario. constant, Interestingly, which the is EoS similar of to the what universe evolves region and in the super-Hubble region where ˜ JCAP03(2008)013 . 9 17 for –

7 (57) (58) | )] ]and ]that Φ B 7 | η ) 44 − ,

B+ 0. Therefore )] η ) in these four ( B η k 20 ψ − sin[ k B+ ˜ F η 0 holds in the bouncing ( − k )] B cos[ η On perturbations of a quintom bounce /φ k ˜ . From this figure we can find F − φ 6 ) negligible. To make sure that + B+ 20 )] + η H ( B k η stacks.iop.org/JCAP/2008/i=03/a=013 . 4 − exact and precise, we also solved the cos[ k B+ ontracting phase matches (up to correction he dominant mode in the expanding phase. ˜ lly small enough in the heating phase, the η E rovide the numerical results in figures sing mode becomes negligible. The amplitude ( ) only to the decaying mode in the expanding

k k k 2 π 03 (2008) 013 ( sin[ and also assume that all the phases are non-trivial. k cooling phase and hence we have ˜

E B+ ) ] shows that there is a non-trivial mixing between the

B η 2 π 50 ∼H , ] in the case of four-dimensional toy model descriptions of − k 49 , only the decreasing mode at late time. This is similar to the 48 B+ k B+ η ]–[ ( In the above sections we have presented the analytic calculation H 2 2 ) ) 45 B B η η . We can see from this figure that in the period long before the − ] scenarios. In the pre-big bang scenario, it has been shown [ − 8 k B+ B+ . η η ( ( 0 1 1 ) is negligible during the whole evolution. This is a key to solving the y y 4) 4) 20 / / (3 (3 , we plot the absolute value of the Newtonian gravitational potential − − 7 e e Φ appearing in the equation as shown in figure 2 k − − In the other case we take In order to make the previous analysis (2) k (2) k growing mode in the contracting phase and t perturbation equation in ourenergy model density explicitly. of AsSCC is the pointed phase, phantom out the field in SRE previous is phase sections, and usua the the the ekpyrotic scenario, althoughis in defined the in casehigher higher of dimensional the dimensions, context ekpyrotic and [ scenario an the analysis physical model of the evolution of fluctuations in the Journal of Cosmology and Astroparticle Physics 3.2.5. Numerical results. of the dominant mode inin the the expanding contracting phase phase. ison set We non-singular by will the bouncing compare amplitude cosmologies of our in the result growing section with mode those obtained in other works Therefore, in this case themodes two are coefficients both are important of inevolves the the outside sub-Hubble same the Hubble region. order radius, which However, the means as decrea soon that as the the two perturbation the assumption isterm valid during the whole evolution, we plot the ratio of the rhs to the phases. Moreover, we have argued that the relation 2 of our model; duringof the equation process ( we have made the assumption that the right-hand side G We then obtain the growing mode of the fluctuations in the c factors which are suppressed by powers of phase. The amplitudeamplitude of of the the dominant constantsame mode (and result in thus also the subdominant) holds expanding mode [ phase in is the thus contracting set phase. by The the perturbation equations numerically. We p that the value ofassumption the ratio very well. in our model is always much less than 1 which agrees with our phase which makes the right-hand side of equation ( it is a good approximation to neglect the right-hand side of equation ( H Thus, we obtain thecontain, result to that leading order for in behavior these of very fluctuations long in wavelength modelsekpyrotic modes, with universe a the [ singular perturbations bounce, such as the pre-big bang [ In figure different values of JCAP03(2008)013 18 ) On perturbations of a quintom bounce stacks.iop.org/JCAP/2008/i=03/a=013 ne may notice that the initial oscillatory others which only depend on the background 03 (2008) 013 ( ). The initial values of the background parameters are 20 . 1 . 1 Plot of the evolution of the absolute value of the ratio of the rhs to the Plot of the evolution of the absolute value of the gravitational potential Φinequation( . We give the explanations as follows. In our model, the canonical 2 7 k in our model. The initial values of the background parameters are the same | Φ Figure 7. | as in figure term Figure 6. the same as in figure evolution in figure Journal of Cosmology and Astroparticle Physics bounce point, thosewavelength curves is are sub-Hubble,value similar. the oscillates perturbation with At is increasing the near amplitude. beginning its of O Bunch–Davies the vacuum evolution, and when its the behaviors of different modes are the same as JCAP03(2008)013 19 =0 . 1 w ) 1inevery − = w 1and1asshownin − =0and ˙ φ On perturbations of a quintom bounce e contracting period which results in stacks.iop.org/JCAP/2008/i=03/a=013 03 (2008) 013 ( se, the EoS oscillates between . 1 Plot of the evolution of the gravitational potential Φ in our model. Plot of the evolution of the absolute value of the curvature perturbation in our model. The initial values of the background parameters are the same | ζ Figure 9. | as in figure Figure 8. The initial values of the background parameters are the same as in figure . In our analytical calculation we simply take the average value of the EoS initially oscillates around its vacuum in th 2 φ a heating phase. During this pha figure Journal of Cosmology and Astroparticle Physics field and show that there areand two the modes for other the beingwe gravitational potential a encounter with the growing one case mode. being that constant However, there when has we to take be the a numerical point calculation, satisfying JCAP03(2008)013 . 8 20 there is ) .From 9 k in figure ζ -modes becomes much k evolves almost as a constant. On perturbations of a quintom bounce k -modes behave very differently. k dominated by the constant mode; -modes. In the expanding phase, for stacks.iop.org/JCAP/2008/i=03/a=013 k r and lasts longer because the wavelength e point after the wavelength re-enters the tial contracting phase which lasts until to here are oscillations in the evolution of the s into the SRE phase and the wavelengths ngularity problem of standard big bang y oscillations of the perturbations near the again. Now, there are also two very different l relativity. The violations of the null energy quintom bounce. In this model, space–time Since these resonances are determined by the ve the same frequency and features regardless s before the bounce and decreases after the oint, whereas for smaller values of h amplified amplitude) which is caused by the ndeed be a large growth of fluctuations due to d friction in the expanding phase, as shown in , the oscillations begin much earlier with their k 03 (2008) 013 ( , it is dominated by the decreasing mode. k . This issue will be discussed in future work. After the 9 ). When it enters into the SCC phase, it grows monotonically 32 , the gravitational potential is k there is a decrease in the gravitational potential. All these features . For the same reason we can understand the periodic amplifications k k ). On the other hand, the oscillating period for small , the oscillatory behavior starts earlie 45 k as they appear in figure | We also plot the values of Φ of different modes around the bounce point in figure Finally, after the bounce the universe enter During the bouncing phase, the curves of different Finally, we also provide a plot of the curvature perturbation After the bounce, the gravitational potential for large In this paper, we have considered the evolution of cosmological fluctuations in a ζ | Journal of Cosmology and Astroparticle Physics parametric resonance. Therefore it seems that t oscillation. At this moment, there can i gravitational potential, but this isbackground evolution, not those true. ‘oscillations’ ha of the wavenumber of while in the case of small values of For large shorter since these modesphase do has not started. enter in the sub-Hubble region until after the bouncing of such modesescapes enters from into the theoscillations Hubble near sub-Hubble radius the region bounce much much later point earlier increase after before the the bounce. bounceequation and ( The amplitude of the mode exits the Hubbleas radius, is shown the in perturbationat equation will first ( be and dominated becomesHubble by oscillatory radius. its near growing the mode, bounc bounce, which followsanti-friction from in the the Hubble contracting term phase in an the equation of motion which acts as amplitude being enlarged at the bounce p We can see that, for larger values of large enough values of only a very short period of oscillation (wit are consistent with the analytical results discussed above. of the fluctuations become super-Hubble once bounce. evolution scenarios for fluctuations with different a non-vanishing minimal radiusphase, is reached provide and then a smoothly possible transits into solution an expanding to the si continues to be described by Einstein’s genera Bouncing non-singular cosmologies, with an ini 4. Discussion and conclusions particular bouncing cosmology, namely the However, for small this figure we canbounce see point. that there are inevitabl cosmology, a problem which is not cured by scalar-field-driven inflationary models. JCAP03(2008)013 ] ] 21 64 41 , 40 ) ] scenarios, ], that making 8 55 ]–[ 53 -essence. Earlier .Inananalytical K 9 ] for fluctuations since, ]. The first of these is a 63 only sources the decaying 56 , , k 62 On perturbations of a quintom bounce 41 , ] and the ekpyrotic [ 40 7 stacks.iop.org/JCAP/2008/i=03/a=013 ] observed that the way the post-bounce term with opposite sign generating a non- singular space-like surface, which leads to r to what is obtained in singular bouncing bservable perturbations will be larger than m depends sensitively on the details of the e to the dominant mode in the post-bounce he dominant fluctuation mode is determined length modes). Which category of modes 57 ial Φ is completely non-singular. Whereas in 03 (2008) 013 ( through the cosmological constant boundary. We have ], in our case no such problems arise w 37 ]. The authors of [ 57 ]. For long wavelength modes, however, we find that the dominant 61 ]–[ ]. 58 ],andithasbeenshowninthiscontext,usingthetechniquesdevelopedin[ 48 ], [ 52 , ]–[ 12 51 , 44 11 Our second main result is that for short wavelength modes, modes which re-enter the Thus, the fluctuations in the quintom bounce model evolve in a way which combines Let us close with some more general comments related to the quintom bounce. Recently, a ghost condensation mechanism for smoothing out the singularity in the ekpyrotic scenario has been 9 the perturbations mustserious be conceptual matched problems across [ a condition required to getallows a a bounce transition are obtained of by the making EoS use of quintom matter, which studied the evolution ofa the particular gravitational model potentialone both of of analytically a and them quintomthat numerically a the bounce in evolution quintessence-like in of the field, which gravitational potent matter the consists other of a two phantom-like scalar field. fields, We have found models with a singular bounce such as the pre-big bang [ introduced [ Journal of Cosmology and Astroparticle Physics use of iso-curvature modes,spectrum the in dominant the spectrum expanding phase. in the contracting phase can be passed on to the dominant bounce. In lightmode of in this, the it pre-bouncephase is phase [ not does surprising not that coupl other bounces show that the dominant spectrum depends on the pre-bounce spectru fluctuation mode in the contracting phase at leading order in bounce in thehigher context derivative of gravity mirage model, cosmology, and the the second, third, one one in in the the context context of of a specific singular bounce is in [ work on fluctuations in models with a kinetic by the growing modebeen in the found contracting in phase. other non-singular This bouncing result cosmologies is [ similar to what has recently Hubble radius in the slow contraction phase, t approximation scheme, we canare apply matching then conditions well atapproximations. non-singular justified, surfaces and which we can also study the evolution numerically without any mode in themodels expanding [ phase, a result simila aspects found in some other recently studied non-singular bouncing cosmologies [ Although this theory is stillcontext in of the a stage bouncing of development, scenario there related have to been predictions works for done observations. in the For example, in [ the Planck length atignorance. the bounce point and thus never enter the zone of trans-Planckian (the behavior ofcosmologies (the short behavior wavelength of modes) the with long aspects wave found in singular bouncing unless the periodsexpansion of times, slow-climb the contraction wavelengths and of slow-roll all the expansion o last many Hubble determines scales accessible toparameters current of cosmological the observations depends model.bounce on and Our the non-singular model detailed quintom bounce may bounce help more model us clearly. there to is As understand no a trans-Planckian the side problem physics remark, [ of let singular us add that in the JCAP03(2008)013 ] 22 ] 69 ] ] ] , ] 68 ) ] SPIRES [ SPIRES SPIRES [ [ SPIRES 8 [ ] ] ] SPIRES 648 ] [ ] ] hep-th/0207228 ] B 086007 ][ SPIRES [ 65 ] ] ] D SPIRES [ SPIRES [ ] 123522 astro-ph/0404224 JHEP10(2007)071 Phys. Lett. astro-ph/0511562 ][ JCAP03(2006)009 64 ][ hep-th/0610274 hep-th/9211021 123515 hep-th/0111098 ] On perturbations of a quintom bounce astro-ph/0501104 ] ] ][ D Phys. Rev. ][ JHEP01(2000)035 72 ][ astro-ph/0703202 gr-qc/9312022 083513 ] ][ D hep-th/9708046 hep-th/0207130 and 10775180 and by the Chinese ] 66 SPIRES [ ][ ][ SPIRES [ Preprint SPIRES D [ gr-qc/0202077 SPIRES SPIRES [ 35 [ SPIRES ][ [ stacks.iop.org/JCAP/2008/i=03/a=013 Preprint Phys. Rev. SPIRES [ 607 317 SPIRES 1220 SPIRES gr-qc/0102069 [ [ 063522 Phys. Rev. 1 B ][ ] astro-ph/0407107 48 J. High Energy Phys. he National Natural Science Foundation of 1 or the suppression of the low multi-poles of 126003 73 ng Xia and Yi Wang for helpful discussions. SPIRES erivative terms was explored. See also [ 043527 ] 3305 ][ [ Phys. Rev. 712 hep-th/0409055 D ] 65 astro-ph/0410680 373 72 72 ][ ] 57 J. High Energy Phys. ][ D D SPIRES SPIRES [ 2717 467 D J. Cosmol. Astropart. Phys. 03 (2008) 013 ( [ SPIRES ] the possibility of obtaining a bounce in a quintom SPIRES [ [ Phys. Lett. 19 SPIRES 195 99 [ 67 SPIRES SPIRES [ 5227 [ SPIRES [ Phys. Rev. 91 287 Phys. Rep. Astropart. Phys. 1 86 Phys. Rev. Lett. 389 B Phys. Rev. Phys. Rev. 023515 347 265 Phys. Rev. 602 ]and[ 108 Phys. Rev. Lett. B 71 23 B 047301 15 B D D 71 D Phys. Lett. ] ] Phys. Lett. Phys. Rev. Lett. Class. Quantum Grav. ] ] ] ] Phys. Lett. Phys. Lett. Phys. Rev. Phys. Rev. ]) it was pointed out that with a bounce followed by slow-roll inflation it Mon. Not. R. Astron. Soc. Phys. Rev. ][gr-qc] 66 , 65 astro-ph/0612728 0704.1090 hep-th/0508194 hep-th/9910169 hep-th/0108187 astro-ph/0507482 hep-th/0103239 [ [ [ [ [ [ [ Biswas T, Brandenberger R, Mazumdar A and Siegel W, 2006 Steinhardt P J and Turok N, 2002 Khoury J, Ovrut B A, Seiberg N, Steinhardt P J and Turok N, 2002 Gasperini M and Veneziano G, 2003 Bojowald M, 2002 Gasperini M and Veneziano G, 1993 [4] Linde A D, 1982 [6] Starobinsky A A, 1980 [1] Guth A H, 1981 [5] Borde A and Vilenkin A, 1994 [9] Bojowald M, 2001 [3] Albrecht A and Steinhardt P, 1982 [2] Sato K, 1981 [7] Veneziano G, 1991 [8] Khoury J, Ovrut B A, Steinhardt P J and Turok N, 2001 [15] Cai Y F, Qiu T, Piao Y S, Li[17] M and Zhao Zhang G X, B, 2007 Xia J Q, Li H, Tao C, Virey J M, Zhu Z H and Zhang X, 2007 [18] Xia J Q, Cai Y F, Qiu T T, Zhao G B and Zhang X, 2007 [14] Setare M R, 2004 [16] Feng B, Wang X L and Zhang X M, 2005 [13] Biswas T, Mazumdar A and Siegel W, 2006 [11] Cartier C, Copeland E J and Madden R, 2000 [12] Tsujikawa S, Brandenberger R and Finelli F, 2002 (see also [ [10] Brustein R and Madden R, 1998 [23] Abramo L R and Pinto-Neto N, 2006 [22] Zhao G B, Xia J Q, Li M, Feng B and Zhang X, 2005 References is possible to give a reasonable explanationmodel f with a single matter field with higher d Academy of Science underpart Grant by No. an KJCX3-SYW-N2. NSERCand Discovery The by Grant, work an by of FQRNT funds RB Teamwhen for Grant. is a the supported RB large Canada thanks in part Research the of Chairs KITPC the Program, for work hospitality on during this the project period was carried out. the CMB anisotropies. In [ We also would like towork. thank the This anonymous referee work forChina was his/her under valuable supported Grants suggestions Nos in on 90303004, our part 10533010, 10675136 by t It is a pleasure to thank Mingzhe Li, Jun-Qi Acknowledgments for other works. 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