A Nonsingular Cosmology with a Scale-Invariant Spectrum of Cosmological Perturbations from Lee-Wick Theory
aa r X i v : . [ h e p - t h ] O c t A Nonsingular Cosmology with a Scale-Invariant Spectrum of CosmologicalPerturbations from Lee-Wick Theory
Yi-Fu Cai , Taotao Qiu , Robert Brandenberger , , and Xinmin Zhang ,
1) Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, P.R. China2) Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada and3) Theoretical Physics Center for Science Facilities (TPCSF), Chinese Academy of Science, P.R. China (Dated: October 29, 2018)We study the cosmology of a Lee-Wick type scalar field theory. First, we consider homogeneousand isotropic background solutions and find that they are nonsingular, leading to cosmologicalbounces. Next, we analyze the spectrum of cosmological perturbations which result from thismodel. Unless either the potential of the Lee-Wick theory or the initial conditions are finely tuned,it is impossible to obtain background solutions which have a sufficiently long period of inflationafter the bounce. More interestingly, however, we find that in the generic non-inflationary bouncingcosmology, perturbations created from quantum vacuum fluctuations in the contracting phase havethe correct form to lead to a scale-invariant spectrum of metric inhomogeneities in the expandingphase. Since the background is non-singular, the evolution of the fluctuations is defined unam-biguously through the bounce. We also analyze the evolution of fluctuations which emerge fromthermal initial conditions in the contracting phase. The spectrum of gravitational waves stemmingfor quantum vacuum fluctuations in the contracting phase is also scale-invariant, and the tensor toscalar ratio is not suppressed.
PACS numbers: 98.80.Cq
I. INTRODUCTION
Recently, ideas originally due to Lee and Wick [1] wereused to propose [2] a “Lee-Wick Standard Model”, amodification of the Standard Model of particle physicsin which the Higgs mass is stabilized against quadrat-ically divergent radiative corrections and which in thissense is an alternative to supersymmetry for solving thehierarchy problem. The Lagrangian includes new higherderivative operators for each field. These operators canbe eliminated by introducing a set of auxiliary fields, onefor each field of the original model. The higher derivativeterms have opposite sign for both the kinetic and massterms, which indicates how the quadratic divergences inthe Higgs mass can be cancelled.Fields with opposite sign of the kinetic term in theaction have recently been invoked in cosmology to pro-vide models for dark energy. Fields with negative kineticenergy but positive potential energy are called “phan-tom field” [3] and were introduced to provide a possiblemechanism for obtaining an equation of state of dark en-ergy with an equation of state parameter w < −
1, where w = p/ρ , p and ρ being pressure and energy density,respectively. In addition to the conceptual problems ofhaving phantom fields (see e.g [4]), phantom dark en-ergy models lead to future singularities. To avoid theseproblems, the “quintom model” [5] was introduced. Thismodel contains two scalar fields, one of them with a reg-ular sign kinetic term, the second with an opposite signkinetic term. This model allows for a crossing of the“phantom divide”, i.e. a transition of the equation ofstate from w < − w > −
1. When applied to earlyuniverse cosmology, quintom models can lead to nonsin-gular cosmological backgrounds which correspond to a bouncing universe [6] [70].The Higgs sector of the Lee-Wick Standard Model hassimilarities with the Lagrangian of a quintom model: theHiggs field has regular sign kinetic term but the auxiliaryfield has a negative sign kinetic term. Thus, it is logicalto expect that the Lee-Wick model might give rise toa cosmological bounce and thus solve the cosmologicalsingularity problem, in addition to solving the hierarchyproblem. In this paper we show that this expectation isindeed realized.Given that the Lee-Wick model leads to a cosmologicalbounce, the cosmology of the very early universe may bevery different from what is obtained by studying the cos-mology of the Standard Model. It is possible to introducea potential for the scalar field in order to obtain a suffi-ciently long period of inflation after the bounce in orderto solve the problems of Standard Big Bang Cosmologyand to obtain a spectrum of nearly scale-invariant cosmo-logical fluctuations. However, this requires fine-tuning ofthe potential. On the other hand, given a bouncing cos-mology it is possible that the cosmological fluctuationsoriginate in the contracting phase, as in the Pre-Big-Bang[9] or Ekpyrotic [10] scenarios. In this paper, we studythe generation and evolution of fluctuations in our Lee-Wick type model. We consider both vacuum and thermalinitial conditions for the fluctuations in the contractingphase and follow the perturbations through the bounce,a process which can be done unambiguously since thebounce is non-singular.We find that initial quantum vacuum fluctuations inthe contracting phase have the right spectrum to de-velop into a scale-invariant spectrum in the expandingphase. What is responsible for this result is the fact thatthere is a coupling of the growing mode in the contract-ing phase to the dominant (constant in time) mode inthe expanding phase, and that this coupling scales withco-moving wave-number as k . The Lee-Wick model thusleads to a concrete realization of the proposal of [11](see also [7, 8, 12] and more recently [13]) to obtain ascale-invariant spectrum of fluctuations from a matter-dominated contracting phase (see also [14] for an analysisof gravitational wave evolution in this background).The outline of this paper is as follows. In the follow-ing section we introduce the Lee-Wick scalar field modelwhich we will study in the rest of the paper. In Section3 we study the background solutions of this model, tak-ing initial conditions in the contracting phase. We showthat, at least at the level of homogeneous and isotropiccosmology, it is easy to obtain a bouncing cosmology. InSection 4 we study how cosmological fluctuations set upin the initial contracting phase pass through the bounce.The evolution of the fluctuations is well behaved, Section5 contains the computation of the spectrum of gravita-tional waves, starting from quantum vacuum fluctuationsin the contracting phase. We end with a discussion of ourresults. II. A LEE-WICK SCALAR FIELD MODEL
We will take our starting Lagrangian for the scalar fieldˆ φ to be L = 12 ∂ µ ˆ φ∂ µ ˆ φ − M ( ∂ ˆ φ ) − m ˆ φ − V ( ˆ φ ) , (1)where m is the mass of the scalar field and V is its in-teraction potential. The second term on the right-handside is the higher derivative term, involving a new massscale M .As discussed in [2], by introducing an auxiliary field ˜ φ and redefining the “normal” scalar field as φ = ˆ φ + ˜ φ , (2)the Lagrangian takes the form L = 12 ∂ µ φ∂ µ φ − ∂ µ ˜ φ∂ µ ˜ φ + 12 M ˜ φ − m ( φ − ˜ φ ) − V ( φ − ˜ φ ) . (3)We thus see that M is the mass of the new scalar degreeof freedom, the “Lee-Wick scalar” which comes from theextra degrees of freedom of the higher derivative theory.Note that both the kinetic term and the mass term ofthe Lee-Wick scalar have the opposite sign compared thesigns for a regular scalar field. One may worry that thetheory is unstable because of the wrong sign of the kineticterm of the Lee-Wick scalar [15, 16, 17]. However, as wasargued in [18], the perturbative expansion can be definedin a consistent way and the theory is unitary. Buildingon these workds, a recent study shows that Lee-Wickelectrodynamics can be defined consistently as a ghost-free, unitary and Lorentz invariant theory [19]. By rotating the field basis, the mass term can be diag-onalized. However, the coupling between the two fieldsin the interaction term remains. To be specific, we con-sider a quartic interaction term. Thus, the Lagrangianwe study is L = 12 ∂ µ φ∂ µ φ − ∂ µ ˜ φ∂ µ ˜ φ + 12 M ˜ φ − m φ − λ φ − ˜ φ ) . (4) III. BACKGROUND COSMOLOGY
In this section we study the background cosmologi-cal equations which follow from coupling the matter La-grangian (4) to Einstein gravity. For a homogeneous,isotropic and spatially flat universe the metric of space-time is ds = dt − a ( t ) d x , (5)where t is physical time, and x denote the co-movingspatial coordinates. The system of equations of motionconsists of the Klein-Gordon equations¨ φ + 3 H ˙ φ + m φ = − λ ( φ − ˜ φ ) ¨˜ φ + 3 H ˙˜ φ + M ˜ φ = − λ ( φ − ˜ φ ) (6)for the two scalar fields and the Einstein expansion equa-tion H = 8 πG (cid:2)
12 ˙ φ −
12 ˙˜ φ + 12 m φ − M ˜ φ + λ φ − ˜ φ ) (cid:3) , (7)where H = ˙ a/a is the Hubble expansion rate and G isNewton’s gravitational constant. An overdot denotes thederivative with respect to t . Combining these equationsleads to the following expression for the change in theHubble expansion rate˙ H = − πG (cid:0) ˙ φ − ˙˜ φ (cid:1) . (8)from which we immediately see that it is possible forthe background cosmology to cross the “phantom divide”˙ H = 0.Let us take a first look to how it is possible to obtaina bouncing cosmology in our model. We assume thatthe universe starts in a contracting phase and that thecontribution of φ in the equations of motion dominatesover that of the Lee-Wick scalar. This will typically bethe case at low energy densities and curvatures. As theuniverse contracts and the energy density increases, therelative importance of ˜ φ compared to φ will grow. From(7) it follows that there will be a time when H = 0 - thisis a necessary condition for the bounce point. From (8)it follows that at the bounce point ˙ H >
0. Hence, weindeed have a transition from a contracting phase to anexpanding phase, i.e. a cosmological bounce.Let us now consider the above argument in a bit moredetail. For the moment we will set the interaction La-grangian to zero, i.e. we will assume λ = 0. We begin theevolution during the contracting phase when the energydensity is sufficiently low so that we expect the contribu-tion of the Lee-Wick scalar to the total energy density tobe small. For these initial conditions, both matter fieldswill be oscillating, and the equation of state will hence bethat of a matter dominated universe. In fact, as followsfrom the Klein-Gordon equations (6) which in this casereduce to ¨ φ + 3 H ˙ φ + m φ = 0¨˜ φ + 3 H ˙˜ φ + M ˜ φ = 0 (9)both scalar fields will be performing oscillations with am-plitudes A ( t ) and ˜ A ( t ) which are blue-shifting (i.e. in-creasing) at the same rate A ( t ) ∼ ˜ A ( t ) ∼ a ( t ) − / . (10)Eventually, the oscillations of the field φ will freeze out.From studies of chaotic inflation [20] it is well know thatthis happens when the amplitude A becomes of the orderof the Planck mass m pl , more specifically when A = (12 π ) − / m pl . (11)After freeze-out, φ will slowly roll up the potential andthe equation of state will shift from w = 0 to w ≃ − w = p/ρ , p and ρ denoting pressure and energydensity, respectively) leading to a deflationary phase dur-ing which the scale factor is decreasing almost exponen-tially. This phase is the time reversal of a period of slow-roll inflation. However, during this period the Lee-Wickfield ˜ φ is still oscillating with rapidly increasing ampli-tude. Hence, its contribution to the energy density willrapidly catch up to that of φ .Let us give a rough estimate of the duration of thedeflationary phase. It will depend crucially on the ini-tial ratio of the energy density of the Lee-Wick scalar ˜ φ to that of the regular scalar φ . Let us denote this ra-tio by F . In the absence of coupling between the twoscalar fields, i.e. for λ = 0, the ratio will be unchangedduring the period of matter domination when both fieldsare oscillating. However, once φ enters the slow-rollingphase, the amplitude ˜ A will increase exponentially ac-cording to (10) while that of φ will remain virtually un-changed. Thus, the condition on the duration ∆ t of thedeflationary phase is | H | ∆ t ≡ N = 13 log ( |F| − ) . (12)Thus, to obtain a deflationary phase with N >
50 (whichin the expanding phase will correspond to a period of in-flation of sufficient length to solve the cosmological prob-lems of the Standard Big Bang Model) required severefine-tuning of the initial conditions. As we will discuss below, this problem may be even worse if coupling be-tween the two fields is allowed.Once the contribution of the Lee-Wick scalar to theenergy density catches up to that of the original scalarfield, the deflationary phase will end and a cosmologicalbounce will occur. Note that once H = 0, the Lee-Wickscalar is still oscillating whereas φ is slowly rolling. Thus,˙ H > H m of | H | before and afterthe bounce is set by H m ∼ m , (13)since it is determined by the potential energy at the fieldvalue where the slow rolling of φ begins. The amplitudeof ˙ H at the bounce, denoted by ˙ H b , can in turn be esti-mated by ˙ H b ∼ πG ˙˜ φ ∼ πGm m pl ∼ m , (14)where in the first step we have used the fact that thekinetic energy of φ is negligible at the bounce, and inthe second step the fact that the bounce is determinedby having the same absolute value of energy densities of φ and ˜ φ , and that the field value of φ at the bounce isabout m pl . The bounce time ∆ t b can now be determinedvia ˙ H b ∆ t b = 2 H m . (15)This gives ∆ t b ∼ m − . (16)Note from the above that the value of H m is set by themass of φ , not the mass of ˜ φ . Similarly, the bounce timeis determined by the mass of the original scalar field andnot of its Lee-Wick partner.After the bounce, the amplitude of the oscillations ofthe Lee-Wick scalar exponentially decreases while φ isnow slowly rolling down the potential. This is a phase ofinflation which is time-symmetric to the phase of defla-tion before the bounce. As we have seen, without fine-tuning of the initial contribution of the Lee-Wick scalarto the energy density, the period of inflation will be toosmall for inflation to solve the various problems of stan-dard cosmology which inflation was invented to solve [22](see also [23]) (such as the horizon and flatness problems).Let us add some comments on the effects of allowinga coupling between the two scalar fields. We expect thatthis will lead to a gradual flow of energy between theregular scalar and the Lee-Wick scalar such that at anenergy density corresponding to the scale of the Lee-Wickscalar, the energy density in the Lee-Wick scalar will be-gin to dominate. Thus, allowing for λ = 0 will lead to ashorter deflationary phase and may completely eliminatethe period of deflation. Complete elimination of the de-flationary phase will occur if the energy density in φ islarger than M at φ = (12 π ) − / m pl , where G ≡ m − pl .This is the case if (making use of (11) M < (cid:0) (12 π ) − / mm pl (cid:1) / . (17)The approximate analytical analysis summarizedabove is supported by exact numerical results. We have solved the coupled equations of motion for the scale fac-tor and the two scalar fields φ and ˜ φ numerically. Figure1 presents the results in the base of a non-interactingmodel. We plot the time evolution of the scalar field φ (denoted φ in the figure), its Lee-Wick partner ˜ φ (denoted by φ ) and the equation of state parameter w . As is evident, there is a non-singular cosmologicalbounce, there is no deflationary phase, but the equationof state parameter w crosses the phantom divide aroundthe bounce point. Note that the scalar field φ stops os-cillating near the bounce, whereas the Lee-Wick scalarcontinues to oscillate and therefore increases in magni-tude by a large factor during the latter stages of the con-tracting phase (which is why we have plotted the timeevolution of ˜ φ on two different scales).Note that in the non-interacting model, the bounce issymmetric. In Figure 2 we present the corresponding fig-ure in the case of an interacting model with the value of λ chosen to be λ = 1 . × − . In this case, the bounceis clearly asymmetric. As a second major difference com-pared to the simulation of Figure 1, the ratio of masseswas chosen to be almost 100 in this case as opposed toonly 2 in the first simulation. Because of the large ratio of the masses (and the corresponding initial conditionsfor which the energy in φ greatly dominates over than in˜ φ ), the background evolution enters a brief deflationaryphase at the end of the contraction phase. However, dueto the presence of interactions, the energy density in φ does not come to dominate again right after the bounceand hence the period of inflation which would be the timereversal of the phase of delation is absent.Finally, in Figure 3 we plot the number of e-foldingsof the deflationary phase as a function of the ratio of ρ φ to ρ ˜ φ , in the model without interactions between the twoscalar fields. As predicted by our analytical approxima-tions, the scaling of the period of deflation as a functionof the ratio is roughly logarithmic. IV. COSMOLOGICAL FLUCTUATIONSA. General considerations
It is useful to first consider the space-time sketch (4)of our non-singular bouncing cosmology. We choose thebounce time to correspond to t = 0. Long before thebounce, the equation of state is that of matter. Duringthis period, the Hubble radius is decreasing linearly and˙ H <
1. At a time denoted − t R (in analogy with thenotation in inflationary cosmology) there is a transitionto a period of deflation during which the Hubble radius | H | − is constant. However, as argued in the previoussection, this period will be of short duration and endsat a time − t i when a brief bouncing phase covering thetime interval [ − t i < t < t i begins. During this period˙ H >
0. After the bouncing phase there is a short periodof inflation lasting from t i to t R , after which the universe enters a matter-dominated expansion phase with ˙ H < k ( k standing for the co-moving wavenumber) which wewant to follow. The wavelength begins in the matter-dominated phase of contraction on sub-Hubble scale, ex-its the Hubble radius during this phase at a time whichwe denote − t H ( k ), and re-enters the Hubble radius dur-ing the matter-dominated phase at the time t H ( k ).Note that if the energy density at the bounce point isgiven by the scale η of Grand Unification ( η ∼ GeV),then the physical wavelength of a perturbation mode cor-responding to the current Hubble radius is of the order of1mm, i.e. in the far infrared. In this sense, the evolutionof fluctuations in this bouncing cosmology is free of thetrans-Planckian problem [24, 25] which effects the evo-lution of fluctuations in all inflationary models in whichthe period of inflation lasts more than about 70 e-foldings(this number assumes that the scale of inflation is of theorder of Grand Unification).Since our bounce is non-singular, the computation ofthe evolution of fluctuations is free of the matching con-dition ambiguities which affect the study of fluctuationsin singular bouncing cosmologies such as the Ekpyroticscenario (see [26, 27, 28, 29, 30, 31] for some early pa-pers on the problem of matching fluctuations through the -20 -15 -10 -5 0 5 10 15 20-4-3-2-101-2.0x10 -1x10 -150-100-50050100150 t w ~ ~ FIG. 1: Evolution of the background fields φ , ˜ φ and of the background equation of state parameter w in a non-interacting modelas a function of cosmic time (horizontal axis). The background fields are plotted in dimensionless units by normalizing by themass M rec = 10 − m pl . Similarly, the time axis is displayed in units of M − rec . The mass parameters m and M were chosen tobe m = 5 M rec and M = 10 M rec . The initial conditions were φ i = 1 . × M rec , ˙ φ i = 1 . × M rec , ˜ φ i = 8 . M rec , ˙˜ φ i = − . M rec . bounce in Ekpyrotic cosmology).There is another important difference in the study ofcosmological fluctuations between non-singular bouncingcosmologies and the inflationary scenario. It is usuallyargued that the exponential expansion of space duringinflation red-shifts any pre-existing matter and the re-lated matter fluctuations, leaving behind a vacuum mat-ter state. Thus, perturbations in this setup are quantumvacuum fluctuations [71]. On the other hand, in a bounc-ing cosmology the fluctuations are set up at low densitiesand temperatures in the contracting phase. There is nomechanism that red-shifts initial classical fluctuations.Thus, there is no reason to prefer vacuum over thermal initial perturbations. In the following, we will considerboth choices. B. Equations for cosmological perturbations
We begin by writing the metric including cosmologicalfluctuations in longitudinal gauge, assuming that thereis no anisotropic stress (see [32] for a comprehensive dis-cussion of the theory of cosmological perturbations and[33] for a briefer survey) ds = a ( η ) (cid:2) (1 + 2Φ) dη − (1 − d x (cid:3) , (18) -5.0x10 -4.0x10 -2.0x10 -500 -400 -300 -200 -100 0 100 200 300 400 500-4-3-2-101 ~ w t FIG. 2: Evolution of the background fields φ , ˜ φ and of the background equation of state parameter w in a non-interacting modelas a function of cosmic time (horizontal axis). The background fields are plotted in dimensionless units by normalizing by themass M rec = 10 − m pl . Similarly, the time axis is displayed in units of M − rec . The mass parameters m and M were chosento be m = 1 . × − M rec and M = 10 M rec . The initial conditions were φ i = − . × M rec , ˙ φ i = 5 . × M rec , ˜ φ i =2 . × − M rec , ˙˜ φ i = − . × − M rec . N | | FIG. 3: Plot of the duration of the deflationary phase as afunction of the ratio of energy densities of φ and ˜ φ (horizontalaxis). The duration (vertical axis) is shown in terms of thee-folding number of deflation. | | H - Xx Τ τ T τ B+ T τ B T τ B- Λ λ = FIG. 4: A sketch of the evolution of scales in a bouncinguniverse. The horizontal axis is co-moving spatial coordinate,the vertical axis is conformal time. Plotted are the Hubbleradius |H| −∞ and the wavelength λ of a fluctuations withcomoving wavenumber k . where Φ( x , t ) is the generalized Newtonian gravitationalpotential which represents the metric fluctuations. It isconvenient to write the equations in terms of conformaltime η defined via dt = a ( t ) dη .The Einstein equations linearly expanded in Φ lead tothe following equation of motion for the Fourier mode ofΦ with co-moving wave-number k :Φ ′′ + 2 (cid:0) H − φ ′′ φ ′ (cid:1) Φ ′ + 2 (cid:0) H ′ − H φ ′′ φ ′ (cid:1) Φ + k Φ= 8 πG (cid:0) H + φ ′′ φ ′ (cid:1) ˜ φ ′ δ ˜ φ (19)where the derivative with respect to conformal time isdenoted by a prime, H ≡ a ′ /a , and δ ˜ φ is the fluctuationin ˜ φ . In deriving this equation, we have assumed thatthe background is dominated by the field φ . This will bethe case except at the bounce.In inflationary cosmology, it has proven to be conve-nient to use the variable ζ , the curvature fluctuation inco-moving gauge, which in terms of Φ is given by ζ = Φ + HH − H ′ (cid:0) Φ ′ + H Φ (cid:1) . (20)In any eternally expanding universe in which 1 + w =0, the variable ζ is conserved on super-Hubble scales inthe absence of entropy fluctuations [34, 35, 36], since -neglecting terms of the order k - the equation of motion(19) for Φ is equivalent to(1 + w ) ˙ ζ = 0 . (21)When considering the quantum theory of cosmologicalperturbations, it is important to identify the fluctuationvariable which has canonical kinetic term. It is with re-spect to this variable, commonly denoted by v , that thecanonical commutation relations must be imposed (see[37, 38] for the quantum theory of cosmological pertur-bations). It turns out that the variable is simply relatedto ζ v = zζ , (22)where the background variable z is the following com-bination of the background metric and the backgroundmatter field φ (for simplicity we are assuming here onlyone matter field) z = aφ ′ H . (23)If the equation of state is constant in time, then z ( η ) isproportional to a ( η ).The equation of motion for v is v ′′ + (cid:2) k − z ′′ z (cid:3) v = 0 , (24)On sub-Hubble scales, it follows from (24) that v is per-forming harmonic oscillations as a function of conformal time. On the other hand, on super-Hubble scales v isfrozen in and v ( η ) ∼ z ( η ).In terms of the variable z , the relationship between themetric fluctuation Φ and the canonical field v takes theform [32] Φ = 4 πGk φ ′ H (cid:0) vz (cid:1) ′ . (25)The variable ζ has proven to be a convenient variable touse in inflationary cosmology. It was therefore taken forgranted that it would also be a useful variable in bouncingcosmologies, and that it would remain conserved betweenwhen the mode k exits the Hubble radius during the pe-riod of contraction at the time − t H ( k ) and the time t H ( k )of re-entry in the expanding phase. In the context ofsingular bouncing cosmologies, the Hwang-Vishniac [39](Deruelle-Mukhanov [40]) matching conditions for fluctu-ations across the singularity lead to the conclusion that ζ should be conserved. However, as pointed out in [31],the applicability of these matching conditions is ques-tionable since the matching conditions are not satisfiedby the background.Non-singular bouncing cosmologies do not require ad-hoc matching conditions - the fluctuations can be fol-lowed through the bounce (as long as their amplituderemains sufficiently small such that linear perturbationtheory does not break down). As has recently been shownin several examples of non-singular bounces, the equationof motion for ζ develops singularities around the bouncepoint [41, 42, 43], whereas the equation of motion for Φremains well defined. One of the reasons for the singular-ities in the equation of motion for ζ is that the co-movinggauge has a singularity at a cosmological bounce. Thus,in the following we will follow the evolution of the fluc-tuations in terms of Φ.If the initial fluctuations in the contracting phase aredue to thermal matter, then the initial values of Φ andits derivative follow from the perturbations in the energydensity of matter. If, on the other hand, we assume vac-uum initial fluctuations, then the initial inhomogeneitiesare given in terms of the canonical variable v and theinitial values of Φ and ˙Φ must be induced from v via therelation (25). C. General solutions
Let us briefly review the general solution of the equa-tion of motion (19) for Φ on super-Hubble scales. Wewill keep the discussion quite general in this subsectionand assume that the equation of state parameter is givenby some w . In this case, a ( t ) scales as a ( t ) ∼ t p (26)with p = 23(1 + w ) . (27)From the definition of conformal time η it then followsthat η ∼ t − p . (28)The condition for the Hubble radius crossing time t H ( k ) for a mode with co-moving wave-number k is a ( t H ( k )) k − = H − ( t H ( k )) ∼ t H ( k ) . (29)Hence η H ( k ) ∼ k − . (30)As is well known, one of the two modes of Φ on super-Hubble scales is constant, whereas the other is decayingin an expanding universe and growing in a contractingone. Specifically we have (see e.g. [11])Φ( k, η ) = D ( k ) + S ( k ) η − ν , (31)where 2 ν = 5 + 3 w w , (32)and where D ( k ) and S ( k ) are independent of time andcarry the information about the spectra of the two modes.In the following, we will determine the spectra of thesetwo modes for various thermal and vacuum initial condi-tions. D. Thermal initial conditions
Here we assume that the initial fluctuations are givenby thermal matter perturbations. As was done in thecase of string gas cosmology [44, 45] (see [46] for a recentcomprehensive review), we follow the matter perturba-tions up to Hubble radius crossing and then convert tometric fluctuations by making use of the perturbed Ein-stein constraint equation (the time-time component ofthe perturbed Einstein equations) which reads − H (cid:16) H Φ + Φ ′ (cid:17) + ∇ Φ = 4 πGa δT . (33)In the above, δT is the fluctuation in the energy den-sity, and ∇ is the co-moving spatial gradient. At Hubbleradius crossing all three terms on the left-hand side ofthe above equation are of the same order of magnitude.Hence, modulo a constant of the order 1, the Fourierspace correlation function of Φ becomes < | Φ( k ) | > = 16 π G k − a < | δT ( k ) | > . (34)where the pointed brackets indicate ensemble averaging.The energy density fluctuations are determined bythermodynamics. First, we express the momentum spaceenergy density correlation function for co-moving wave-number k in terms of the r.m.s. position space massfluctuation δM ( R ) = R < | δT ( k ) | > , (35) where R = ak − is the physical radius of the region cor-responding to the wave-number k . The mass fluctuationsare determined by the specific heat capacity C V δM ( R ) = T C V ( R ) , (36)where T is the temperature of the system. For a gas ofpoint particles, the heat capacity is proportional to R ,i.e. C V ( R ) = c V T R , (37)where c V is a constant. Note that this result is in agree-ment with the intuition that on scales larger than thethermal correlation length T − , the heat capacity scalesas a random walk.Inserting (35), (36) and (37) into (34) we obtain thefollowing power spectrum for Φ k < | Φ( k ) | > = 16 π G k − T c V . (38)We need to evaluate this expression at Hubble radiuscrossing t H ( k ) since we will be using the correspondingvalue as the initial condition for the evolution of Φ onsuper-Hubble scales: k < | Φ( k ) | > ( t H ( k )) = 16 π G k − T ( t H ( k )) c V . (39)We now use (39) to infer the power spectra of the S and D modes in the case of thermal matter initial conditions.We are assuming that the initial value of Φ at Hubbleradius crossing gets distributed equally among the twomodes The power spectrum P D ( k ) of the constant mode D is the same as that of Φ at Hubble radius crossing P D ( k ) ∼ k − p − p (40)where the second exponent comes from making use of T ( t H ( k )) ∼ a − ( t H ( k )) ∼ t H ( k ) − p ∼ k p − p . (41)The power spectrum of S is the spectrum of Φ at Hub-ble radius crossing modulated by the factor η H ( k ) ν : P S ( k ) ∼ k − p − p − ν . (42)In the example we are interested in, fluctuations leavethe Hubble radius is the matter epoch and hence p = 2 / w = 0. Thus, from the above we see that the spectraof D and S scale as P D ( k ) ∼ k P S ( k ) ∼ k − . (43)The spectrum of the D mode is extremely blue. The bluetilt is due to the thermal suppression of the spectrumat large wavelengths. It is also easy to understand whythe spectrum of the S mode is less blue than that ofthe D mode: the S mode grows on super-Hubble scales,and large wavelength modes experience the growth fora longer period of time. Our calculation shows that thedifference in growth on super-Hubble scales dominatesover the thermal suppression of long wavelength modes. E. Vacuum initial conditions
Vacuum initial conditions are given in terms of thecanonically normalized variable v k being in its quantumvacuum state [32] v k ( η ) ∼ k − / e iηk . (44)Inserting this into (25) and making use of the fact that onsub-Hubble scales the derivative of the oscillating factordominates over the derivative of other terms leads to thefollowing initial conditions in terms of Φ:Φ k ( η ) ∼ i πGk / φ ′ z H , (45)i.e. a spectrum which is proportional to k − / . The sameconclusion can be reached [27] by starting with vacuumfluctuations in the matter field φ and inserting the resultinto the equations expressing Φ and ˙Φ in terms of thematter field. Making use of (23) to eliminate z and of thebackground Friedmann equation to eliminate ˙ φ in favorof H we find the following result for the power spectrumof Φ: P Φ ( k, η ) ≡ π k | Φ k ( η ) | (46) ≃ π (cid:0) H ( η ) m pl (cid:1) . Making use of the definition of z from (23) it followsthat the time dependent terms in (45) scale as H − .Thus, Φ k ( t ) ∼ k − / t − , (47)which allows us to evaluate the result at Hubble radiuscrossing Φ k ( t H ( k )) ∼ k − + − p . (48)The above result (48) allows us to compute the spectraof both D and S modes of Φ on super-Hubble scales,assuming - as we did in the previous subsection - thatΦ k ( t H ( k )) sources both modes equally:Φ D ( k ) ∼ k − + − p (49)Φ S ( k ) ∼ k − + − p − ν . (50)In the case we are interested in p = 2 / w = 0 and2 ν = 5 we obtain Φ D ( k ) ∼ k / (51)Φ S ( k ) ∼ k − / . (52)Note that, as pointed out in [11], the S − mode leadsto a scale-invariant spectrum of fluctuations of ζ in thecontracting phase. This compares to the results obtained for Ekpyrotictype contraction [27] where p = 0, w = ∞ and 2 ν = 1and therefore Φ D ( k ) ∼ k − / (53)Φ S ( k ) ∼ k − / , (54)which leads to a scale-invariant spectrum for the S modewhich is growing in the phase of contraction.As we expect from the Hwang-Vishniac (Deruelle-Mukhanov) matching conditions, the S − mode will cou-ple with a k suppression to the dominant mode in theexpanding phase. If this is realized, we will obtain ascale-invariant spectrum of curvature fluctuations in theexpanding phase. In the following subsection we willevolve the fluctuations through the non-singular bounceand infer the spectrum at late times. We indeed find alate-time scale-invariant spectrum. F. Evolution of the fluctuations through thebounce
Let us step back and write down the equation of motionfor Φ in a slightly modified form (which is equivalent to(19) except that we allow for a general speed of sound c s which is equal to 1 in our scalar field model)Φ ′′ k + 2 σ H Φ ′ k + k c s Φ k = 0 , (55)where σ ≡ − ¨ H H ˙ H . (56)Contracting phaseIn the contracting phase the equation (55) takes theform Φ ′′ k + 1 + 2 ν c η − ˜ η B − Φ ′ k + k c s Φ k = 0 , (57)with ν c ≡ w c w c ) , (58)where the subscript “c” indicates that we are discussingthe contracting phase. The general analytical solution isΦ k = ( η − ˜ η B − ) − ν c (cid:26) k − ν c D − J ν c [ c s k ( η − ˜ η B − )]+ k ν c S − J − ν c [ c s k ( η − ˜ η B − )] (cid:27) , (59)where the coefficients D − and S − can be determined bythe initial condition of the gravitational potential as de-scribed in the two previous subsections for different setsof initial conditions. In the above, η B − is a fixed timethat corresponds to when the singular bounce would oc-cur if the universe were to remain matter-dominated.0Note that, when the wavelength of the perturbation islarger than Hubble radius with k ≪ |H| , the asymptoti-cal form of Φ k can be written asΦ ck = ¯ D − + ¯ S − ( η − ˜ η B − ) ν c , (60)where we define¯ D − ≡ c ν c s D − ν c Γ(1 + ν c ) , ¯ S − ≡ ν c S − c ν c s Γ(1 − ν c ) , (61)As discussed in previous subsections, the ¯ D − mode isconstant and the ¯ S − mode is growing in a contractinguniverse. Note from the definition of ζ k we have to lead-ing order in k ζ ck = 5 + 3 w c w c ) ¯ D − . (62)Thus, to this order, in the contracting phase ζ k is de-termined by the constant mode of Φ k which is sub-dominant. As discussed in detail in [27] the S mode doeseffect ζ when k corrections to the solutions are takeninto account.If we match the asymptotic form for Φ (60) to theinitial power spectrum of Φ (see (46)) at Hubble radiuscrossing and assume that the initial power is equally dis-tributed into the two modes, we obtain12 π k | S − ( k ) | ≃ π m pl t H ( k ) − η H ( k ) ν c , (63)where the subscript H stands for the time of Hubble ra-dius crossing. Bouncing phaseAs we have shown in the section on the background dy-namics, the contribution of the higher derivative termsin the Lee-Wick model becomes more and more impor-tant as the universe contracts and will lead to a non-singular bounce. Thus, the universe will exit from thephase of matter-dominated contraction at some time t B − ,and then the EoS of the universe will cross − η B . After the bounce, the Lee-Wickfield will recover its normal state with the higher deriva-tive terms rapidly decreasing in importance.It is rather complicated to solve the perturbation equa-tion directly from Eq. (55). In order to solve the equa-tion analytically, we need to make some approximationsto simplify it. Our approximation consists of choosing aconvenient modelling of the Hubble parameter near thebounce of the form H = αt (64)with some positive constant α which has dimensions of k and whose magnitude is set by the microphysics ofthe bounce, in our case by the mass M of the Lee-Wickscalar. The time of the bounce was chosen to be t = 0. In this case, we can obtain an analytical form for thecomoving Hubble parameter in the bouncing phase: H = y ( η − η B )1 − y ( η − η B ) ,y = 12 π αa B , (65)where a B denotes the value of the scale factor at thebounce point η B .Since the above parametrization should be valid onlyin the neighborhood of the bounce point, the quadraticand higher order terms of | η − η B | can be neglected. Con-sequently, the perturbation equation takes the followingformΦ ′′ k + 2 y ( η − η B )Φ ′ k + ( c s k + 23 y )Φ k = 0 . (66)The solution of this equation can be written asΦ k = (cid:26) E k H l [ √ y ( η − η B )] + F k F [ − l , , y ( η − η B ) ] (cid:27) × exp[ − y ( η − η B ) ] , (67)which is constructed from the l -th Hermite polynomialand a confluent hypergeometric function with l ≡ −
23 + c s k y (68)and two undetermined coefficients E k , F k . These twofunctions are linearly independent, and their asymptot-ical behaviors are mainly determined by the parameter l . When c s k ≫ y , i.e. the wave-number of the modeis larger than the mass scale of the bounce, then bothfunctions are oscillating. This case was already studiedin Ref. [43].However, in the current paper we are interested in theopposite limit, the limit in which the wavelength is muchlarger than the inverse mass scale of the bounce, i.e. thelimit when c s k ≪ y . (69)As we have argued in the section of our paper on thebackground evolution, the bounce takes place very fastand thus the condition (69) will be satisfied for all wave-lengths we are interested in. In this case, we can expandthe solution of the perturbation equation in a power se-ries in terms of √ y ( η − η B ). Then the solution is givenbyΦ bk = ˆ F k + ˆ E k √ y ( η − η B ) + ( − − l ) ˆ F k y ( η − η B ) + O ( y ( η − η B ) ) , (70)with ˆ E k ≡ − l √ π Γ( − l ) E k , (71)ˆ F k ≡ l √ π Γ( − l ) E k + F k , (72)1and the subscript “b” represents the bouncing phase. Inthis case, we have ζ bk ≃ ˆ F k [1 + c s k η − η B ) ] . (73)Therefore, the conservation of ζ k is realized by the modeˆ F k when the bounce is fast enough.Now we study how to establish the coefficients ˆ E k andˆ F k . We need to use the Hwang-Vishniac [39] (Deruelle-Mukhanov [40]) matching condition to link the fluctu-ations in contracting phase with those in the bounc-ing phase at the momentum η B − . Note that since weare matching two contracting universes across a non-singular surface, the background satisfies the matchingconditions, unlike the situation in the Ekpyrotic scenariowith a singular bounce, Thus, it is justified to apply thematching conditions [31].The matching conditions say that both Φ k andˆ ζ k ≡ ζ k + c s k k H − H ′ (74)are continuous on the matching surface of constant en-ergy density. Taking use of matching conditions in thesolutions (60) and (70), we can obtain the following re-lations:ˆ E k √ y ( η B − − η B ) = − ( + 2 l )Φ ck − ˆ ζ ck | B − , ˆ F k = ( + 2 l )Φ ck + ˆ ζ ck | B − . (75)These relations show that the constant and growingmodes of gravitational potential get mixed during thebounce. However, if we consider large wavelengths com-pared to the duration of the bounce, the second relationshows that ζ k is indeed conserved across the bounce.Expanding phaseAfter the bounce, the higher derivative terms of Lee-Wick field rapidly decay. Therefore, a phase of matter-dominated expansion starts at the time η B + . In the ab-sence of interactions between the two scalar fields, thebackground cosmology will be time-symmetric about thebounce point. During the period after η B + the back-ground evolution is the time reverse of the contractingphase. In the case λ = 0 an asymmetric bounce is pos-sible. To render our analysis more general, we assumethat the equation of state in the expanding phase is w e which could be different from that in the contractingstage which is w c .The equation of motion for the gravitational potentialis similar as Eq. (57) but with the indexes ν e ≡ w e w e ) (76)and “ B +” instead of ν c and “ B − ”. Then the solutionon super-Hubble scales take the formΦ ek = ¯ D + + ¯ S + ( η − ˜ η B + ) ν e , (77)with ˜ η B + ≡ η B + −
21 + 3 w e H B + . (78)The ¯ D + mode of the gravitational potential is constantin time, as is the ¯ D − mode in contracting phase. How-ever, the role of the S mode is very different. In the ex-panding phase ¯ S + is the sub-dominant decreasing mode,whereas in the contracting phase ¯ S − is the dominant ex-panding mode. Therefore, the dominant mode of thecurvature perturbation in the period of expansion is ¯ D + .As we will shown in the following, it inherits contribu-tions from both ¯ D − and ¯ S − since these modes mix duringthe bounce.To determine the coefficients of the two modes in theexpanding phase, we need to apply the matching condi-tion again, this time at the surface η B + . A straightfor-ward calculation yieldsˆ E k √ y ( η B + − η B ) = − ( + 2 l )Φ ek − ˆ ζ ek | B + , ˆ F k = ( + 2 l )Φ ek + ˆ ζ ek | B + . (79)Note again that it is justified to apply the matching con-ditions since the universe is expanding on both sides ofthe matching surface and thus the background also sat-isfies the matching conditions.By combining Eqs. (75) and (79) , we can establish therelation between the gravitational potentials in contract-ing and expanding phases. Since ¯ S + is a decaying mode,we will not write down its expression and focus our at-tention instead on the dominant mode ¯ D + . In terms ofthe modes in the contracting phase, it is given by¯ D + = (5 + 3 w c )(1 + w e )(1 + w c )(5 + 3 w e ) ¯ D − + 3(1 + w e )(5 + 3 w e ) c s k × (cid:26) η B + − η B η B − − η B M + ( 2 ¯ D − w c ) − ¯ S − ( η B − − ˜ η B − ) ν c ) − w c w c ) M + ¯ D − + M − ( ¯ D − + ¯ S − ( η B − − ˜ η B − ) ν c ) (cid:27) + O ( k ) , (80)where we defined the parameters M ± ≡ H B ± (1 + w ec ) + 1 y , (81) which are independent of k .2From the above result we see that both the con-stant and growing modes of gravitational potential inthe contracting phase effect the dominant mode afterthe bounce. However, the growing mode is suppressedby k on large scales whereas the constant one transfersthrough the bounce without a change in the spectral in-dex. These results agree with what is obtained using thematching conditions at a singular hypersurface betweenthe contracting and the expanding phase, as shown in[28].Inspecting our result (80), we see that there are twoways to obtain a scale-invariant spectrum of cosmologicalperturbations after the bounce. The first is to considera model in which the D − mode in the contracting phasehas a scale invariant spectrum, i.e. D − ( k ) ∼ k − / , theother is to take a scenario where S − ( k ) ∼ k − / . Asfollows from (49, the first possibility is realized if p = ∞ ,i.e. in an inflationary contracting phase. The second wayis realized in the case of a matter-dominated contraction,a possibility already pointed out in [11] (see also [47]).The Lee-Wick model yields a natural realization of thisway.Let us now come back to our Lee-Wick background,and assume quantum vacuum fluctuations. We insertthe values w c = w e = 0, c s = 1 into (80) and assume asymmetric fast bounce. Thus¯ D + = ¯ D − + (cid:20) −
45 ¯ D − + 35 H B − ¯ S − (cid:21) k H B − . (82)As discussed in the subsection on vacuum initial condi-tions, then if the initial conditions are imposed at a time η sufficiently early compared to the transition point B − ,we have Φ inik ∝ k − and therefore obtain ¯ D − ∝ k and¯ S − ∝ k − . Substituting these relations into Eq. (82),one can see that whereas the contribution of ¯ D − to thefinal spectrum of D + vanishes on large scales, the con-tribution of ¯ S − which starts out deep red is blue-tiltedby exactly the right amount to yield a final spectrumproportional to k − which is the scale-invariant form.To compute the amplitude of the spectrum, we insertthe values (81) and the expression (63) for the value of S − into (82) and use the background Friedmann equationto replace the Hubble parameter by the energy density.This yields ¯ D + = − √ ρ B − √ M p k − . (83)Therefore, the power spectrum of the gravitational po-tential in this case can be expressed as P Φ ≡ k π | ¯ D + | = ρ B − (20 π ) M p . (84)Note that as long as M ≪ M pl , the power spectrumof metric fluctuations remains much smaller than 1 andthus linear perturbation theory is applicable throughoutthe bouncing phase. G. Numerical analysis
Our analytical calculations involve approximations.Specifically, in the contracting phase the scalar fields andhence the equation of state are oscillating. But in ouranalytical analysis we have replaced the time-dependentequation of state parameter by its temporal average. Itis thus important to confirm the results by numerical in-tegration of the full equations, namely Eq. (19) coupledto the equation for the scalar matter field fluctuation.At first sight, it appears that the equation (19) con-tains a singularity at all turnaround points of φ . Suchsingularities are known from the study of the evolutionof Φ during reheating taking into account the oscillatorynature of the inflaton field [48, 49]. However, this singu-larity is actually not present. Let us consider in additionto the dynamical perturbed Einstein equation (19) theperturbed Einstein constraint equationΦ ′ + H Φ = 4 πG (cid:0) φ ′ δφ − ˜ φ ′ δ ˜ φ (cid:1) . (85)Inserting (85) into (19) yieldsΦ ′′ + 6 H Φ ′ + 2 (cid:0) H ′ + 2 H (cid:1) Φ + k Φ= 8 πG (cid:0) H + φ ′′ φ ′ (cid:1) φ ′ δφ , (86)from which it is clear that the singularity has disap-peared. Thus, we numerically solve (86) coupled to theperturbed φ equation δφ ′′ + 2 H δφ ′ + (cid:0) k + a V φφ (cid:1) δφ = 4 φ ′ Φ ′ − a V φ Φ , (87)where the subscripts on V indicate the variables withrespect to which the potential is differentiated.Figures 5 and 6 show the results of our numerical in-tegration. The first figure shows the evolution in time ofthe metric fluctuation Φ as a function of physical time(left side) and conformal time (right side) for differentvalues of the comoving wavenumber k . We have chosenthe bounce point to correspond to physical and confor-mal time 0. The initial conditions for Φ were set at theinitial time of the simulation according to the vacuuminitial condition prescription discussed earlier. We seefrom this figure that before the bounce the perturbationsare dominated by the growing mode. When the universeenters the bouncing phase, we see that the amplitudeapproaches a constant and passes smoothly through thebounce. The numerical evolution agrees well with theanalytical solution (70) in the bouncing phase. After thebounce, the perturbations are dominated by the constantmode. The numerical evolution demonstrates that thismode can be inherited from the growing mode in con-tracting phase.Figure 6 shows the power spectrum of Φ (lower panel)and the spectral index n s (upper panel) as a function ofcomoving wavenumber k . On large scales (small values of k ), the power spectrum tends to a constant. The rise ofthe spectrum for large values of k is on scales which are3comparable to maximal value of the Hubble rate, i.e. formodes which have not spent time outside of the Hubbleradius. -0.2 -0.1 0.0 0.1 0.2-15 -10 -5 0 510 -20 -16 -12 -8 -4 k=10 -3 k=1 k=10 k=80 k=180 P t
FIG. 5: Result of the numerical evolution of the curvatureperturbations with different comoving wavenumbers k in theLee-Wick bounce. The horizontal axis in the left panel iscosmic time, and in the right panel it is comoving time. Theinitial values of the background parameters are the same as inFigure 1. The units of the time axis are M − rec , the comovingwavenumber k is unity for k = M rec , as in Figure 2. -10 -9 -9 -9 P k n s FIG. 6: Plot of the power spectrum of the curvature per-turbation Φ (lower panel) and of the spectral index (upperpanel) as functions of comoving wavenumbers k in the Lee-Wick bounce. The initial values of the background parametersare the same as in Figure 1. V. GRAVITATIONAL WAVES
Now we turn to consider the evolution of gravitationalwaves (tensor perturbations) in our background, assum-ing they start out in the vacuum state on sub-Hubblescales in the contracting phase. Since at the level of lin-ear perturbation theory scalar metric fluctuations andgravitational waves decouple, we can focus on a metriccontaining only gravitational waves propagating in thebackground. The standard form of this metric in a spa-tially flat FRW background is ds = a ( η ) [ − dη + ( δ ij + ¯ h ij ) dx i dx j ] , (88)where the Latin indexes run over the spatial coordinates,and the tensor perturbation ¯ h ij is real, transverse andtraceless, i.e.¯ h ij = ¯ h ji ; ¯ h ii = 0 ; ¯ h ij,j = 0 . (89)Due to these constraints, we only have two degrees offreedom in ¯ h ij which correspond to two polarizations ofgravitational waves. For each polarization state (labelledby r in the following), we can write ¯ h ij ( η, x ) as a scalarfield h r ( η, x ) multiplied by a polarization tensor e rij whichis constant in space and time.If matter contains an anisotropic stress tensor σ ij ,there is a non-vanishing source term in the equation ofmotion for tensor perturbations, namely¯ h ′′ ij + 2 a ′ a ¯ h ′ ij − ∇ ¯ h ij = 16 πGa σ ij . (90)If matter consists of a set of canonically normalized scalarfields or a set of perfect fluids, there is no anisotropicstress and thus no source term at linear order in pertur-bation theory for gravitational waves.As usual, we go to Fourier space The Fourier trans-formations of the tensor perturbations and anisotropicstress tensor are give by,¯ h ij ( η, x ) = X r =1 Z d k (2 π ) h r ( η, k ) e rij e i kx . (91)In order to canonically quantize the gravitational waves,it is important to identify the variable in terms of whichthe action has canonical kinetic term. This variable turnsout to be (see [50] for a derivation) v rk = s ( e r ) ij ( e r ) ji πG ah rk (92)(where h rk is a short hand notation for h r ( η, k )) in termsof which the Einstein action expanded to second order in v r becomes S = r X r =1 Z (cid:0) | ( v r ) ′ k | − ( k − a ′′ a ) | v rk | dηd k . (93)4The resulting equation of motion for v r is( v r ) ′′ + ( k − a ′′ a ) v r = 0 . (94)We are interested in computing the power spectrumof the tensor modes. Making use of (92), it is relatedto the power spectrum of v (which is the same for eachpolarization state) via P Tk ( h ) = a − πG X r =1 P k ( v r )= a − πG k π | v k | . (95)The tensor spectral index n T is defined by n T ≡ d ln P T d ln k . (96)The evolution of tensor perturbations is very similarto that of scalar perturbations. Initially the perturba-tions are inside the Hubble radius in the far past. Sincethe Hubble radius shrinks in the contracting phase, themodes with small comoving wave number exit the Hubbleradius. After that the universe bounces to an expandingphase, so these Fourier modes will return into the Hubbleradius.In the current paper we focus on a mode with small k so that it exits the Hubble radius in the contracting phase(rather than the bounce phase), then passes through thebounce and finally re-enters the Hubble radius during theexpanding phase.We divide the time interval into three periods like wedid for the analysis of scalar metric fluctuations. Dur-ing the phase when the universe is contracting with anequation of state oscillating around w = 0, we have v = ( η − ˜ η B − ) (cid:26) A Tk J − [ k ( η − ˜ η B − )]+ B Tk J [ k ( η − ˜ η B − )] (cid:27) , (97)where ˜ η B − = η B − − / H B − . Here, the parameters A Tk and B Tk can be determined by the initial condition forgravitational waves, which is taken as the Bunch-Daviesvacuum v ∼ e − ikη / √ k . (98)So we have A Tk = i √ π B Tk = − √ π . (99)Therefore, the asymptotic form of the solution to thetensor perturbation in the contracting phase is v ( k, η ) = − i √ k − ( η − ˜ η B − ) − , outside Hubble radius; √ k e − ik ( η − ˜ η B − ) , inside Hubble radius . (100)During the bouncing phase, we have the approximaterelation a ′′ a ≃ π αa B = y . (101)Solving Eq. (94), we have v ( k, η ) = C Tk cos[ l ( η − η B )] + D Tk sin[ l ( η − η B )] , k ≥ y3 ; C Tk e l ( η − η B ) + D Tk e − l ( η − η B ) , k < y3 , (102)where we define l = | k − y | . Since the Hubble pa-rameter approaches zero when the universe is transitingfrom the contracting to the expanding phase, all fluctu-ation modes return to the sub-Hubble region, but onlyfor a very brief time. However, from the above solutionwe interestingly find that k ph ( ∼ k /a B ) and ˙ H ( ∼ α ) arecomparable.After the bounce, an expanding phase with its EoS w = 0 takes place. So the solution to the gravitationalwaves is given by v = ( η − ˜ η B + ) × (cid:26) E Tk J − [ k ( η − ˜ η B + )]+ F Tk J [ k ( η − ˜ η B + )] (cid:27) , (103)where ˜ τ B + = τ B + − / H B + . This solution takes on theasymptotic form, v ≃ r π F Tk k ( η − ˜ η B + ) , (104)when the modes are outside the Hubble radius.Having obtained the solutions of the tensor pertur-bations in the different phases, now we need to matchthese solutions and determine the coefficients C Tk , D Tk , E Tk and F Tk respectively. This procedure is analogous tothe matching process of scalar perturbations performedin the previous section. For a non-singular bounce sce-nario such as the bounce we are considering, the continu-ity of the background evolution implies that both v and v ′ are able to pass through the bounce smoothly. So wematch v and v ′ in (100) and (102) on the surface τ B − ,and those in (102) and (103) on the surface τ B + . Withthese matching conditions, we can determine all the co-efficients and finally get the solution for v at late times.Since in the specific model we considered in this paper,the evolution of the universe is symmetric with respectiveto the bounce point, we can simply take H B − ≃ −H B + .In addition, we have shown that the bounce takes placevery fast on the time scale set by k − , so we have l ( τ B + − τ B − ) ≫
1. Therefore, we eventually obtain theapproximate result F Tk ≃ i √ π H B + k (105)5and the asymptotical form of v in the final stage can beexpressed as v f → i √ H B + k ( η − ˜ η B + ) . (106)Now we are able to derive the power spectrum of pri-mordial gravitational waves. From the definition of Eq.(95), the primordial power spectrum is given by P T ( k ) = G k π | v f a | = 2 ρ B + π M p . (107)From Eq. (107), we can read that the spectrum are scale-invariant on large scales (which is consistent with theresult in Ref. [51]).Comparing our result of the tensor power spectrumwith the result (84) for the power spectrum of scalar met-ric fluctuations, we obtain a tensor to scalar ratio of theorder of 30, which is in excess of the current observationalbounds. The exact value of the ratio, however, will de-pend on the detailed modelling of the bounce phase [72]However, the conclusion that the tensor to scalar ratiowill be rather large will be robust, and also agrees withthe analysis of [47] done in a different context. VI. CONCLUSIONS AND DISCUSSION
Recently, the Lee-Wick Standard Model has been sug-gested as an extension of the Standard Model of parti-cle physics providing an alternative to supersymmetry interms of addressing the hierarchy problems.In this paper, we have considered the cosmology of theHiggs sector of the Lee-Wick Standard Model, an alterna-tive to supersymmetry to solving the hierarchy problem.We have found that homogeneous and isotropic solutionsare non-singular. Thus, the Lee-Wick model provides apossible solution of the cosmological singularity problem.We then considered the spectrum of cosmological per-turbations and find that quantum vacuum fluctuationsestablished in the contracting phase evolve into a scale-invariant spectrum in the expanding phase. Note thatthese results emerge without having to introduce anyadditional features into the model, unlike the situationin inflationary cosmology where the existence of a newscalar field satisfying slow-roll conditions must be as-sumed, or the situation in Ekpyrotic models where onceagain a scalar field with special features must be assumed.Tuning the amplitude of the spectrum of scalar metricfluctuations to agree with the amplitude inferred fromCMB observations [52], we can determine the scale M ofthe new physics which is present in the Lee-Wick model.The required value of M turns out to be about 10 GeV.We have also computed the spectrum of gravitationalwaves and also find a scale-invariant spectrum assumingthat the fluctuations are quantum vacuum in nature. The tensor to scalar ratio may be in excess of the currentobservational bounces, but the exact value will dependon the detailed modelling of the bounce phase.One of the main successes of cosmological inflation isthe solution of the horizon, homogeneity, size and flatnessproblems of Standard Big Bang cosmology which it pro-vides. How does a bouncing cosmology such as our Lee-Wick model measure up against these successes? Firstof all, if the universe starts out large and cold, there areno horizon and size problems. If the spatial curvature attemperatures in the contracting phase comparable to thecurrent temperature is not larger than the current spatialcurvature, then there will be no flatness problem eitherbecause the deviation of Ω K from 0 decreases in the con-tracting phase at the same rate that it increases in theexpanding phase. The key challenge for any bouncingcosmology is to control the magnitude of the inhomo-geneities and to provide a mechanism for preventing theuniverse to collapse into a gas of black holes at the endof the phase of contraction. For an attempt to addressthis issue in the case of string gas cosmology see [53].We would like to conclude this paper by putting ourwork in the context of previous work on perturbations inbouncing cosmologies. The issue of the mixing of the S and D modes at a cosmological bounce has been hotlydebated in the literature since the Ekpyrotic scenariowas proposed. In the case of the Ekpyrotic scenario,for vacuum initial conditions the S mode of Φ inheritsa scale-invariant spectrum, whereas the D mode obtainsa blue spectrum with index n = 3 [26, 27, 28, 29] (seealso the more recent analysis of [54] and the recent re-view of [55]). This is also the spectrum of ζ . Accord-ing to the Hwang-Vishniac [39] (Deruelle-Mukhanov [40])matching conditions applied at a hypersurface on whichwe glue the expanding to the contracting universe, themixing between the S − mode and the D + mode is sup-pressed by a power of k (see e.g. [28] for a discussionof this point). Hence, the spectrum of metric fluctua-tions after the bounce is not scale-invariant. The Pre-Big-Bang scenario faces a similar problem [56]. Theseconclusions were confirmed in some specific models inwhich the bounce was smoothed out by making use ofhigher derivative gravity terms (see [57] in the case ofthe Pre-Big-Bang model and [58, 59] in the case of theEkpyrotic scenario). However, the use of the matchingconditions was challenged in [31] where it was pointedout that if the background solution does not satisfy thematching conditions at the bounce, there is no reasonto expect the fluctuations to do so. In fact, in the caseof the Ekpyrotic scenario (which is intrinsically a higher-dimensional cosmology), computations done in the higherdimensional framework yielded a successful transfer ofthe scale-invariant spectrum of metric fluctuations fromthe contracting to the expanding phase [60], a conclusionwhich was confirmed in [61] and, in a slightly differentsetting, in [62] [73].Calculations have also been done in some other non-singular bouncing models [63]. For example, studies done6in models in which the bounce is induced by a negativeenergy density scalar field found no unsuppressed match-ing between the growing perturbation mode in the con-tracting phase and the constant mode in the expandingphase [64, 65], in contrast to what was obtained in someinitial work [7, 8]. Both studies in models in which abounce was generated by a curvature term in the Einsteinaction [66] and analyses in some other bouncing models[67, 68] yielded un-suppressed matching of the dominantmodes of the contracting and expanding phases.The upshot of these analyses is that the transfer offluctuations through a cosmological bounce can dependquite sensitively on the physics of the bounce.It was realized that the equation of motion for ζ hassingularities in the case of a non-singular bounce, thuscasting doubt on the belief that in all cases ζ is con-served at a bounce. It was shown that the Φ equationis free of such singularities and is thus a safer equationto use [41, 42]. In our previous work [43] it was shownin the case of the quintom bounce model that there isunsuppressed mixing between the D + and S − modes onlength scales which are small compared to the durationof the bouncing period, whereas on longer length scalesthe mixing is suppressed (but not completely absent). Inthe present work, the bounce is short compared to thelength scales we are interested in.Our work shows that the evolution of fluctuationsthrough the non-singular bounce in the Lee-Wick modelis rather standard. There is no un-suppressed couplingbetween the dominant modes of the contracting and ex-panding phases, and ζ is conserved at the bounce.In the current paper we have not considered radiation.Since the energy density in radiation increases faster thanthat in matter, radiation would dominate at early times.However, in the Lee-Wick standard model there is a Lee-Wick partner to each field. In particular, there is a Lee-Wick photon partner ˜ γ of the radiation field γ . At highenergy densities, then as a consequence of interactions between γ and ˜ γ , we expect that energy will flow from γ into ˜ γ , like it flows from φ to ˜ φ in our scalar field model.Then, a cosmological bounce would occur in a mannersimilar to how it occurs in our model. Adding inter-mediate phases of radiation between the bouncing phaseand the contracting and expanding matter phases willnot change our results concerning the spectrum of fluc-tuations for modes which exit the Hubble radius duringthe phase of matter domination, which are the modes weare interested in when trying to explain the large-scalestructure of the universe and the CMB anisotropies. Astudy of these issues is left to a followup paper.It would also be interesting to consider entropy fluctu-ations and non-Gaussian signatures of our scenario. Weleave these topics for future research.Note added: while this paper was being prepared forsubmission, a preprint appeared [69] pointing out thatthe Lee-Wick model provides a realization of the quin-tom scenario and could be applied to study the currentacceleration of the universe. We find it more naturalto consider the corrections to the cosmological evolutionwhich are obtained in the very early universe. Acknowledgments
We wish to thank Andy Cohen, Ben Grinstein, HongLi, Jie Liu and Mark Wise for useful discussions. RBwishes to thank the Theory Division of the Institute ofHigh Energy Physics (IHEP) for their wonderful hospi-tality and financial support. RB is also supported by anNSERC Discovery Grant and by the Canada ResearchChairs Program. The research of X.Z., Y.C. and T.Q.is supported in part by the National Science Foundationof China under Grants No. 10533010 and 10675136, bythe 973 program No. 2007CB815401, and by the ChineseAcademy of Sciences under Grant No. KJCX3-SYW-N2 [1] T. D. Lee and G. C. Wick, “Negative Metric and theUnitarity of the S Matrix,” Nucl. Phys. B , 209 (1969);T. D. Lee and G. C. Wick, “Finite Theory of QuantumElectrodynamics,” Phys. Rev. D , 1033 (1970).[2] B. Grinstein, D. O’Connell and M. B. Wise, “The Lee-Wick standard model,” Phys. Rev. D , 025012 (2008)[arXiv:0704.1845 [hep-ph]].[3] R. R. Caldwell, “A Phantom Menace?,” Phys. Lett. B , 23 (2002) [arXiv:astro-ph/9908168].[4] J. M. Cline, S. Jeon and G. D. Moore, “The phantommenaced: Constraints on low-energy effective ghosts,”Phys. Rev. D , 043543 (2004) [arXiv:hep-ph/0311312].[5] B. Feng, X. L. Wang and X. M. Zhang, “Dark EnergyConstraints from the Cosmic Age and Supernova,” Phys.Lett. B , 35 (2005) [arXiv:astro-ph/0404224];B. Feng, M. Li, Y. S. Piao and X. Zhang, “Oscillatingquintom and the recurrent universe,” Phys. Lett. B ,101 (2006) [arXiv:astro-ph/0407432]; Z. K. Guo, Y. S. Piao, X. M. Zhang and Y. Z. Zhang,“Cosmological evolution of a quintom model ofdark energy,” Phys. Lett. B , 177 (2005)[arXiv:astro-ph/0410654].[6] Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang,“Bouncing Universe with Quintom Matter,” JHEP ,071 (2007) [arXiv:0704.1090 [gr-qc]].[7] F. Finelli, “Study of a class of four dimensional non-singular cosmological bounces,” JCAP , 011 (2003)[arXiv:hep-th/0307068].[8] J. Martin, P. Peter, N. Pinto Neto and D. J. Schwarz,“Passing through the bounce in the ekpyrotic models,”Phys. Rev. D , 123513 (2002) [arXiv:hep-th/0112128];P. Peter and N. Pinto-Neto, “Primordial perturbationsin a non singular bouncing universe model,” Phys. Rev.D , 063509 (2002) [arXiv:hep-th/0203013].[9] M. Gasperini and G. Veneziano, “Pre - big bangin string cosmology,” Astropart. Phys. , 317 (1993) [arXiv:hep-th/9211021].[10] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,“The ekpyrotic universe: Colliding branes and the originof the hot big bang,” Phys. Rev. D , 123522 (2001)[arXiv:hep-th/0103239].[11] F. Finelli and R. Brandenberger, “On the generation of ascale-invariant spectrum of adiabatic fluctuations in cos-mological models with a contracting phase,” Phys. Rev.D , 103522 (2002) [arXiv:hep-th/0112249].[12] D. Wands, “Duality invariance of cosmological per-turbation spectra,” Phys. Rev. D , 023507 (1999)[arXiv:gr-qc/9809062].[13] P. Peter and N. Pinto-Neto, “Cosmology without infla-tion,” arXiv:0809.2022 [gr-qc].[14] A. A. Starobinsky, “Spectrum of relict gravitational radi-ation and the early state of the universe,” JETP Lett. ,682 (1979) [Pisma Zh. Eksp. Teor. Fiz. , 719 (1979)].[15] D. G. Boulware and D. J. Gross, “Lee-Wick IndefiniteMetric Quantization: A Functional Integral Approach,”Nucl. Phys. B , 1 (1984).[16] N. Nakanishi, “Lorentz noninvariance of the complex-ghost relativistic field theory,” Phys. Rev. D , 811(1971);N. Nakanishi, “Remarks on the complex-ghost relativis-tic field theory,” Phys. Rev. D , 3235 (1971).[17] A. M. Gleeson, R. J. Moore, H. Rechenberg andE. C. G. Sudarshan, “Analyticity, covariance, and uni-tarity in indefinite-metric quantum field theories,” Phys.Rev. D , 2242 (1971).[18] R. E. Cutkosky, P. V. Landshoff, D. I. Olive andJ. C. Polkinghorne, “A non-analytic S matrix,” Nucl.Phys. B , 281 (1969).[19] A. van Tonder, “Unitarity, Lorentz invariance and causal-ity in Lee-Wick theories: An asymptotically safe comple-tion of QED,” arXiv:0810.1928 [hep-th].[20] A. D. Linde, “Chaotic Inflation,” Phys. Lett. B , 177(1983).[21] T. Biswas, A. Mazumdar and W. Siegel, “Bouncinguniverses in string-inspired gravity,” JCAP , 009(2006) [arXiv:hep-th/0508194];T. Biswas, R. Brandenberger, A. Mazumdar andW. Siegel, “Non-perturbative gravity, Hagedornbounce and CMB,” JCAP , 011 (2007)[arXiv:hep-th/0610274].[22] A. H. Guth, “The Inflationary Universe: A Possible So-lution To The Horizon And Flatness Problems,” Phys.Rev. D , 347 (1981).[23] K. Sato, “First Order Phase Transition Of A VacuumAnd Expansion Of The Universe,” Mon. Not. Roy. As-tron. Soc. , 467 (1981).[24] R. H. Brandenberger, “Inflationary cosmology: Progressand problems,” arXiv:hep-ph/9910410.[25] J. Martin and R. H. Brandenberger, “The trans-Planckian problem of inflationary cosmology,” Phys.Rev. D , 123501 (2001) [arXiv:hep-th/0005209];R. H. Brandenberger and J. Martin, “The robustness ofinflation to changes in super-Planck-scale physics,” Mod.Phys. Lett. A , 999 (2001) [arXiv:astro-ph/0005432].[26] D. H. Lyth, “The primordial curvature perturbation inthe ekpyrotic universe,” Phys. Lett. B , 1 (2002)[arXiv:hep-ph/0106153];D. H. Lyth, “The failure of cosmological perturbationtheory in the new ekpyrotic scenario,” Phys. Lett. B ,173 (2002) [arXiv:hep-ph/0110007]. [27] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,“Density perturbations in the ekpyrotic scenario,” Phys.Rev. D , 046005 (2002) [arXiv:hep-th/0109050].[28] R. Brandenberger and F. Finelli, “On the spec-trum of fluctuations in an effective field theory ofthe ekpyrotic universe,” JHEP , 056 (2001)[arXiv:hep-th/0109004].[29] J. c. Hwang, “Cosmological structure problem in theekpyrotic scenario,” Phys. Rev. D , 063514 (2002)[arXiv:astro-ph/0109045].[30] J. Martin, P. Peter, N. Pinto Neto and D. J. Schwarz,“Passing through the bounce in the ekpyrotic models,”Phys. Rev. D , 123513 (2002) [arXiv:hep-th/0112128].[31] R. Durrer and F. Vernizzi, “Adiabatic perturbationsin pre big bang models: Matching conditions andscale invariance,” Phys. Rev. D , 083503 (2002)[arXiv:hep-ph/0203275].[32] V. F. Mukhanov, H. A. Feldman and R. H. Branden-berger, “Theory of cosmological perturbations. Part 1.Classical perturbations. Part 2. Quantum theory of per-turbations. Part 3. Extensions,” Phys. Rept. , 203(1992).[33] R. H. Brandenberger, “Lectures on the theory of cosmo-logical perturbations,” Lect. Notes Phys. , 127 (2004)[arXiv:hep-th/0306071].[34] J. M. Bardeen, “Gauge Invariant Cosmological Pertur-bations,” Phys. Rev. D , 1882 (1980).[35] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, “Spon-taneous Creation Of Almost Scale - Free Density Pertur-bations In An Inflationary Universe,” Phys. Rev. D ,679 (1983).[36] R. H. Brandenberger and R. Kahn, “Cosmological Per-turbations In Inflationary Universe Models,” Phys. Rev.D , 2172 (1984).[37] M. Sasaki, “Large Scale Quantum Fluctuations in the In-flationary Universe,” Prog. Theor. Phys. , 1036 (1986).[38] V. F. Mukhanov, “Quantum Theory of Gauge InvariantCosmological Perturbations,” Sov. Phys. JETP , 1297(1988) [Zh. Eksp. Teor. Fiz. , 1 (1988)].[39] J. c. Hwang and E. T. Vishniac, “Gauge-invariant joiningconditions for cosmological perturbations,” Astrophys. J. , 363 (1991).[40] N. Deruelle and V. F. Mukhanov, “On matching condi-tions for cosmological perturbations,” Phys. Rev. D ,5549 (1995) [arXiv:gr-qc/9503050].[41] R. Brandenberger, H. Firouzjahi and O. Saremi, “Cos-mological Perturbations on a Bouncing Brane,” JCAP , 028 (2007) [arXiv:0707.4181 [hep-th]].[42] S. Alexander, T. Biswas and R. H. Brandenberger, “Onthe Transfer of Adiabatic Fluctuations through a Nonsin-gular Cosmological Bounce,” arXiv:0707.4679 [hep-th].[43] Y. F. Cai, T. Qiu, R. Brandenberger, Y. S. Piao andX. Zhang, “On Perturbations of Quintom Bounce,”JCAP , 013 (2008) [arXiv:0711.2187 [hep-th]].[44] A. Nayeri, R. H. Brandenberger and C. Vafa, “Producinga scale-invariant spectrum of perturbations in a Hage-dorn phase of string cosmology,” Phys. Rev. Lett. ,021302 (2006) [arXiv:hep-th/0511140].[45] R. H. Brandenberger, A. Nayeri, S. P. Patil andC. Vafa, “String gas cosmology and structure formation,”arXiv:hep-th/0608121.[46] R. H. Brandenberger, “String Gas Cosmology,”arXiv:0808.0746 [hep-th].[47] L. E. Allen and D. Wands, “Cosmological perturbations through a simple bounce,” Phys. Rev. D , 063515(2004) [arXiv:astro-ph/0404441].[48] F. Finelli and R. H. Brandenberger, “Paramet-ric amplification of gravitational fluctuations dur-ing reheating,” Phys. Rev. Lett. , 1362 (1999)[arXiv:hep-ph/9809490];F. Finelli and R. H. Brandenberger, “Parametric am-plification of metric fluctuations during reheating intwo field models,” Phys. Rev. D , 083502 (2000)[arXiv:hep-ph/0003172].[49] W. B. Lin, X. H. Meng and X. M. Zhang, “Adiabaticgravitational perturbation during reheating,” Phys. Rev.D , 121301 (2000) [arXiv:hep-ph/9912510].[50] V. Mukhanov, “Physical foundations of cosmology,” Cambridge, UK: Univ. Pr. (2005) 421 p [51] Y. F. Cai and X. Zhang, “Evolution of Metric Perturba-tions in Quintom Bounce model,” arXiv:0808.2551 [astro-ph].[52] G. F. Smoot et al. , “Structure in the COBE differen-tial microwave radiometer first year maps,” Astrophys.J. , L1 (1992).[53] N. Lashkari and R. H. Brandenberger, “Speed of Soundin String Gas Cosmology,” arXiv:0806.4358 [hep-th].[54] P. Creminelli, A. Nicolis and M. Zaldarriaga, “Pertur-bations in bouncing cosmologies: Dynamical attractorvs scale invariance,” Phys. Rev. D , 063505 (2005)[arXiv:hep-th/0411270].[55] D. Wands, “Cosmological perturbations through the bigbang,” arXiv:0809.4556 [astro-ph].[56] R. Brustein, M. Gasperini, M. Giovannini,V. F. Mukhanov and G. Veneziano, “Metric per-turbations in dilaton driven inflation,” Phys. Rev. D ,6744 (1995) [arXiv:hep-th/9501066].[57] C. Cartier, J. c. Hwang and E. J. Copeland, “Evo-lution of cosmological perturbations in non-singularstring cosmologies,” Phys. Rev. D , 103504 (2001)[arXiv:astro-ph/0106197].[58] S. Tsujikawa, R. Brandenberger and F. Finelli, “On theconstruction of nonsingular pre-big-bang and ekpyroticcosmologies and the resulting density perturbations,”Phys. Rev. D , 083513 (2002) [arXiv:hep-th/0207228].[59] A. Cardoso and D. Wands, “Generalised perturbationequations in bouncing cosmologies,” Phys. Rev. D ,123538 (2008) [arXiv:0801.1667 [hep-th]].[60] A. J. Tolley, N. Turok and P. J. Steinhardt, “Cosmologi-cal perturbations in a big crunch / big bang space-time,”Phys. Rev. D , 106005 (2004) [arXiv:hep-th/0306109].[61] J. L. Lehners, P. McFadden, N. Turok and P. J. Stein-hardt, “Generating ekpyrotic curvature perturbations before the big bang,” arXiv:hep-th/0702153.[62] T. J. Battefeld, S. P. Patil and R. H. Brandenberger, “Onthe transfer of metric fluctuations when extra dimensionsbounce or stabilize,” Phys. Rev. D , 086002 (2006)[arXiv:hep-th/0509043].[63] J. Martin and P. Peter, “On the properties of the transi-tion matrix in bouncing cosmologies,” Phys. Rev. D ,107301 (2004) [arXiv:hep-th/0403173].[64] V. Bozza and G. Veneziano, “Scalar perturbations in reg-ular two-component bouncing cosmologies,” Phys. Lett.B , 177 (2005) [arXiv:hep-th/0502047].[65] F. Finelli, P. Peter and N. Pinto-Neto, “Spectraof primordial fluctuations in two-perfect-fluid reg-ular bounces,” Phys. Rev. D , 103508 (2008)[arXiv:0709.3074 [gr-qc]].[66] J. Martin and P. Peter, “Parametric amplification of met-ric fluctuations through a bouncing phase,” Phys. Rev.D , 103517 (2003) [arXiv:hep-th/0307077].[67] L. R. Abramo and P. Peter, “K-Bounce,” JCAP ,001 (2007) [arXiv:0705.2893 [astro-ph]].[68] P. Peter, N. Pinto-Neto and D. A. Gonzalez, “Adi-abatic and entropy perturbations propagation ina bouncing universe,” JCAP , 003 (2003)[arXiv:hep-th/0306005].[69] S. Lee, “Lee Wick Dark Energy,” arXiv:0810.1145 [astro-ph].[70] The use of a second scalar field to obtain a nonsingularbounce was already discussed in [7, 8][71] In light of the trans-Planckian problem for inflationaryfluctuations, one may view this prescription with somescepticism.[72] We have modelled the contracting and expanding phaseswith a background with a constant equation of state. Inthis case, the squeezing factor for scalar and tensor cos-mological perturbations is the same. If the equation ofstate changes, then the amplitude of the scalar spectrumcan be enhanced, as happens in inflationary cosmology atthe time of reheating [35, 36]. Since 1+ ww