Non-singular cosmology in a model of non-relativistic gravity
aa r X i v : . [ h e p - t h ] O c t Non-singular cosmology in a model of non-relativistic gravity
Yi-Fu Cai ∗ and Emmanuel N. Saridakis † Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, P.R. China Department of Physics, University of Athens, GR-15771 Athens, Greece
We present a model of non-relativistic gravitational theory which is power-counting renormalizablein 3+1 dimensional spacetime. When applied to cosmology, the relativity-violation terms lead toa dark radiation component, which can give rise to a bounce if dark radiation possesses negativeenergy density. Additionally, we investigate a cyclic extension of the non-singular cosmology in whichthe universe undergoes contractions and expansions periodically. In both scenarios the backgroundtheory is well defined at the quantum level.
PACS numbers: 04.60.-m, 98.80.-k, 04.60.Bc
I. INTRODUCTION
Relativity is commonly assumed to be the foundationin constructing models of particle physics and gravity.Yet, theories based on relativity often suffer from sometheoretical defects. Namely, a quantum field theory ofgeneral relativity cannot be well established since it isunable to be renormalized. Various attempts to break-ing relativity have been intensively discussed in the lit-erature. Recently, motivated by Hoˇrava [1, 2], models ofnon-relativistic quantum field theory were studied, notonly theoretically [3, 4, 5], but also in experimental de-tections [6, 7, 8] (see also [9]).In the present work we are interested in constructinga model of non-relativistic gravity and study its cosmo-logical implications. This model is power-counting renor-malizable in 3+1 dimensions and hence ultraviolet (UV)complete. Moreover, its action can recover the exactEinstein-Hilbert form in the infrared (IR) limit, and sogeneral relativity and Lorentz symmetry in local framedo emerge at low energy scales. A generic feature ofthis model is an existence of dark radiation for whichthe energy density can be either positive or negative. Inthe frame of a Friedmann-Robertson-Walker (FRW) uni-verse, dark radiation with negative energy density cangive rise to a bouncing solution [10], since it breaks en-ergy conditions if accompanied by normal matter com-ponents [11]. This is the so-called “quintom scenario”[12, 13, 14, 15] in which the equation-of-state of the uni-verse crosses the cosmological constant boundary. A re-markable point of this model is that the violation of en-ergy condition does not bring the quantum instabilities[16, 17] which often exist in usual quintom scenario. Thescenario of bouncing cosmology has been investigatedin models motivated by various approaches to quantumgravity [18, 19, 20, 21, 22, 23], and analyzed using ef-fective field techniques by introducing energy-condition-violating matters [11, 24, 25, 26], while the generation of ∗ Electronic address: [email protected] † Electronic address: [email protected] perturbation was studied in [27, 28, 29, 30, 31, 32, 33, 34](and we refer to [35] for a comprehensive review). Onemodel of bouncing cosmology with a matter-dominatedcontraction was found to be able to provide a scale-invariant spectrum [36, 37, 38, 39, 40] and sizable non-Gaussianities [41, 42], which may be responsible for thecurrent cosmological observations.To extend, we find that the evolution of a universe ina model of non-relativistic gravity might be free of anysingularities. One example of this scenario is an oscillat-ing universe in which the universe experiences a sequenceof contractions and expansions [43, 44, 45]. As shown in[44], for a pivot process (bounce or turnaround) to oc-cur, one must require that the Hubble parameter vanisheswhile its time derivative is non-vanishing at that point.Thus, it needs to be satisfied that the equation-of-state w of the universe is much less than − II. A MODEL OF NON-RELATIVISTICQUANTUM GRAVITY
We start with a discussion on the main obstacle againstthe usual approach to quantum gravity. In the contextof quantum field theory all successes are established ona solid construction of a perturbatively renormalizablemodel, namely the SU(2) Standard Model, in 3+1 di-mensions. However, this procedure does not work in thegravity sector, since the gravitational coupling constant G possesses a negative dimension ([ G ] = − S g = 116 πG Z dtd x √ gN (cid:18) K ij K ij − K + R (cid:19) , (1)where K ij = 12 N ( ˙ g ij − ∇ i N j − ∇ j N i ) , (2)is the extrinsic curvature and R is the three-dimensionalRicci scalar. The dynamical variables are the lapse andshift functions, N and N i respectively, and the spatialmetric g ij (roman letters indicate spatial indices), interms of the ADM metric ds = − N dt + g ij ( dx i + N i dt )( dx j + N j dt ) , (3)where indices are raised and lowered by g ij .Attempts on solving the UV incompletion have beenintensively studied in the literature. Motivated by a Lee-Wick model [46, 47] which shows an improved UV behav-ior [48], gravity involving higher-derivative terms may beapplied to provide a UV completion. However, this ap-proach suffers from the existence of unbounded from be-low energy state, and thus its quantization becomes un-reliable. Another path to quantum gravity is to build anon-local theory [49], using infinite high-derivative terms[50, 51, 52, 53], or string theory [54, 55], attempts whichare still in proceeding.Motivated by a recent work [1], one realizes that amodel of power-counting renormalizable gravity may beachieved just by adding higher-order spatial derivativeterms. One important peculiarity in this type of modelsis that Lorentz symmetry has to be given up but it mayappear as an emergent one at low energy scales. The orig-inal model, which is the so-called Hoˇrava gravity, sinceis required to satisfy a ‘detailed-balance’ condition refer-enced from condense matter physics, it still suffers fromproblems such as the over-constraining in UV region, be-ing not compatible with current observations even in theIR limit.Concerning to above issues, the logic of effective fieldtheory suggests that a complete action of gravity couldinclude all possible terms consistent with the imposedsymmetries, and the dimensions of these terms ought tobe bounded due to renormalization. In the frame of 3+1 dimensional spacetime, a renormalizable term may allow6th-order spatial derivatives at most, as pointed out by[1]. As a consequence, one could add a modified actionwhich involves all the permitted terms:∆ S g = 116 πG Z dtd x √ gN (cid:18) α R ij R ij + α R + α ∇ i R jk ∇ i R jk + α ∇ i R jk ∇ j R ki + α ∇ i R ∇ i R (cid:19) . (4)Adding these terms into action (1) and requiring the signsof the 6th-order spatial derivatives to be negative, weobtain a non-relativistic gravity theory which is power-counting renormalizable and the dispersion relation isbounded from below. Note that in the following we areinterested in the cosmological applications of this gravi-tational background, that is in its IR limit, and thus wedo not present the full “running” formalism but we re-strict ourselves to the IR expressions. Furthermore, with-out loss of generality, we will focus our investigation onthe quadratic terms, which preserves parity and Poincaresymmetries, although ones can straightforwardly exam-ine a model with all the coefficients included.We can insert a matter component in the construction,attributed to a canonical scalar field φ described by S m = Z dtd x √ gN [ 12 ∂ µ φ∂ µ φ − V ( φ )] . (5)In addition, we focus on the cosmological frame with anFRW metric, N = 1 , g ij = a ( t ) γ ij , N i = 0 , (6)with γ ij dx i dx j = dr − kr + r d Ω , (7)where k = − , , k ] = 2 in massunits). Both the scalar field and the metric are assumedto be homogenous, with their backgrounds being func-tions of cosmic time t . By varying N and g ij , we obtainthe Friedmann equations as follows H = 8 πG h ρ m + ρ k + ρ dr i , (8)˙ H + 32 H = − πG (cid:20) p m − ρ k + 13 ρ dr (cid:21) , (9)where H ≡ ˙ aa is the Hubble parameter, and the matterpressure and energy densities are expressed as p m ≡ ˙ φ − V ( φ ) , (10) ρ m ≡ ˙ φ V ( φ ) , (11) ρ k ≡ − k πGa , (12) ρ dr ≡ − α + 3 α ) k πGa , (13)respectively. In addition, the energy density for mat-ter component satisfies the continuity equation ˙ ρ m +3 H ( ρ m + p m ) = 0 following from the action (5), whichleads to the equation of motion for the scalar field¨ φ + 3 H ˙ φ + V ,φ = 0 , (14)where the subscript ‘ ,φ ’ denotes the derivative with re-spect to φ .In summary, in Friedmann equation (8) we have ob-tained the energy density for usual matter component(with equation-of-state parameter − ≤ w m ≤
1, where w m ≡ p m /ρ m ) and the usual spatial curvature term (ofwhich the equation-of-state parameter is w k = − / a − andthus its equation-of-state parameter is w dr = 1 / w m < w dr and α + 3 α > III. COSMOLOGICAL EVOLUTIONS
In this section we will study in detail the backgroundevolution in the context of non-relativistic gravity. Inparticular, we study a non-flat universe with k = ± ρ cr ≡ α +3 α ) k πG which needs to be larger than zero. Furthermore, wenormalize a = 1 at the bounce point. A. A bounce solution
Let us take a first look at how it is possible to obtain abouncing cosmology in this model. We assume that theuniverse starts in a contracting phase and that the con-tribution of the normal matter dominates over that of thedark radiation. This will typically be the case at low en-ergy densities and curvatures. As the universe contractsand the energy density increases, the relative importanceof dark radiation compared to the normal matter willgrow. From (8) it follows that there will be a time when H = 0. This is a necessary condition for the realizationof the bounce. By making use of the continuity equationsit follows that at the bounce point ˙ H >
0. Hence, we canacquire a transition from a contracting to an expandingphase, which is a cosmological bounce.Now, as a specific example, for the scalar field respon-sible for matter we consider a mass potential V ( φ ) = m φ . We begin the evolution during the contractingphase with a sufficiently large scale factor, so that we ex-pect the contribution of the dark radiation to the totalenergy density to be small. Besides we require the con-tribution of the spatial curvature term to be negligibletoo (we leave the discussion on this component for the next section). For these initial conditions the scalar fieldwill be oscillating around its vacuum, and the equation ofstate will hence be that of a matter dominated universewith its Hubble parameter being h H i ≃ t (15)on average. In fact, as follows from the Klein-Gordonequation (14), which in this case reduces to¨ φ + 3 H ˙ φ + m φ = 0 , (16)one well-known approximated solution is φ ( t ) ≃ √ πG sin mtmt . (17)When t is smaller than m − , the scalar field φ is oscil-lating and the universe behaves as a matter-dominatedone. Once t ∼ m − , the amplitude of φ reaches thePlanck scale and the scalar enters a “slow-climb” phase.During this period, the energy density of the scalar variesvery slowly but the scale factor decreases rapidly. There-fore, the dark radiation will become similar to the scalarfield density very soon. Once there is ρ m = ρ cr , wewill obtain a big bounce. Since in this case the uni-verse has experienced a matter-dominated contraction,this scenario is the so-called “matter-bounce”. B. Cyclic scenario
Having investigated the realization of bouncing cos-mology in the present non-relativistic gravitationalframework, we now study a sub-class of cosmological evo-lution without any singularities, that is a realistic andphysical cyclic scenario.A cyclic scenario could be straightforwardly obtainedif we consider a negative dark radiation term and a neg-ative curvature one. According to (12) and (13), thisis achieved if α + 3 α > k = 1, that is for aclosed universe. For simplicity we consider the mattercomponent of the universe to be dust, that is to pos-sess w m = 0. During the expansion, the energy densitiesof all components are decreasing. However, the energydensity of the curvature term decreases much slower rel-atively to others. Thus, its contribution will counterbal-ance that of dark matter, triggering a turnaround with H = 0 and ˙ H <
0, after which the universe enters in thecontracting phase. As described in the previous subsec-tion, after a contraction to sufficiently small scale factorsthe dark radiation term will lead the universe to experi-ence a bounce. Therefore, the universe in such a modelindeed presents a cyclic behavior, with a bounce and aturnaround at each cycle.A more careful analysis reveals that the negative cur-vature term, that is a closed universe with k = 1, is not anecessary condition. Indeed, its role on the competitionof the positive dark matter density can be equivalentlyfulfilled by a small negative cosmological constant addedto the potential V ( φ ). In other words, a slightly negative V ( φ ) can trigger the turnaround at large scale factors,even if the universe is open, i.e with k = − − and . Thus, havingextracted its general features, in the following we willexplicitly construct the class of models that allow forcyclicity, and in particular we wish to appropriately re-construct the corresponding scalar potential.Let us first start from the desired result, that is to im-pose a known scale factor a ( t ) possessing an oscillatorybehavior. In this case both H ( t ) and ˙ H ( t ) are straight-forwardly known. Therefore, we can use the the Fried-mann equations (8),(9) together with (10),(11), in orderto extract the relations for φ ( t ) (through ˙ φ ( t )) and V ( t ),acquiring: φ ( t ) = ± Z t dt ′ s − ˙ H ( t ′ )4 πG − ρ k ( t ′ ) − ρ dr ( t ′ ) , (18) V ( t ) = ˙ H ( t )8 πG + 3 H ( t )8 πG − ρ k ( t ) − ρ dr ( t ) . (19)Note that the a ( t )-form or the parameter-choices mustlead to a positive ˙ φ ( t ). Finally, eliminating time be-tween these two expressions we extract the explicit formof the potential V ( φ ). Thus, performing the procedureinversely we conclude that this particular V ( φ ) generatesthe desired a ( t )-form.We now proceed to a specific, simple, but quite generalexample. We assume a cyclic universe with an oscillatoryscale factor of the form a ( t ) = A sin( ωt ) + a c , (20)where we have shifted t in order to eliminate a possibleadditional parameter standing for the phase. Further-more, the non-zero constant a c is inserted in order toeliminate any possible singularities from the model. Insuch a scenario t varies between −∞ and + ∞ , and t = 0is just a specific moment without any particular phys-ical meaning. Finally, note that the bounce occurs at a B ( t ) = a c − A , which can be set to 1. Straightforwardlywe find: H ( t ) = Aω cos( ωt ) A sin( ωt ) + a c (21)˙ H ( t ) = − Aω [ A + a c sin( ωt )][ A sin( ωt ) + a c ] , (22)and thus substitution into (12),(13), (18) and (19) givesthe corresponding expressions for φ ( t ) and V ( t ).In order to provide a more transparent picture of theobtained cosmological behavior, in Fig. 1 we present the evolution of the scale factor (20) and of the Hubble pa-rameter (21) with A = 10, ω = 0 . a c = 11, whereall quantities are measured in units with 8 πG = 1. Thischoice is consistent with our setting a = 1 at the bounce.In Fig. 2 we depict the corresponding behavior of φ ( t ) a H t
FIG. 1:
The evolution of the scale factor a ( t ) and of the Hub-ble parameter H ( t ) of the ansatz (20), with A = 10 , ω = 0 . and a c = 11 . All quantities are measured in units where πG = 1 . and V ( t ) for the scale factor of Fig. 1, in the case of aclosed universe ( k = 1), and for model parameters α = 1and α = 1 (in units where 8 πG = 1). Finally, eliminat- V t
FIG. 2: φ ( t ) and V ( t ) for the cosmological evolution of Fig. 1,in the case of a closed universe ( k = 1 ) with α = 1 and α = 1 . All quantities are measured in units where πG = 1 . ing time between φ ( t ) and V ( t ) allows us to re-constructthe corresponding relation for V ( φ ), shown in Fig. 3. V FIG. 3: V ( φ ) for the cosmological evolution of Figs. 1 and 2.All quantities are measured in units where πG = 1 . From these figures we observe that an oscillating andsingularity-free scale factor, can be generated by an oscil-latory form of the scalar potential V ( φ ) (although of nota simple function as that of a ( t ), as can be seen by theslightly different form of V ( φ ) in its minima and its max-ima). This V ( φ )-form was more or less theoretically ex-pected, since a non-oscillatory V ( φ ) would be physicallyimpossible to generate an infinitely oscillating scale fac-tor and a universe with a form of time-symmetry. Finally,we stress that although we have presented the above spe-cific simple example, we can straightforwardly performthe described procedure imposing an arbitrary oscillat-ing ansatz for the scale factor.The aforementioned bottom to top approach was en-lightening about the form of the scalar potential thatleads to a cyclic cosmological behavior. Therefore, onecan perform the above procedure the other way around,starting from a specific oscillatory V ( φ ) and resulting toan oscillatory a ( t ). In particular, (18) is written in acompact form as ˙ φ ( t ) = Q ( a, ˙ a, ¨ a ) and similarly (19) as V ( t ) = Q ( a, ˙ a, ¨ a ). Thus, we can invert the known formof V ( φ ) ≡ V ( φ ( t )) obtaining φ ( t ) = V {− } ( Q ( a, ˙ a, ¨ a )).Therefore, ˙ φ ( t ) = (cid:8) ddt (cid:2) V {− } ( Q ( a, ˙ a, ¨ a )) (cid:3)(cid:9) . In con-clusion, the scale factor arises as a solution of the differ-ential equation Q ( a, ˙ a, ¨ a ) = (cid:26) ddt h V {− } ( Q ( a, ˙ a, ¨ a )) i(cid:27) . (23)As a specific example we consider the simple case V ( φ ) = V sin( ω V φ ) + V c . (24) In this case φ = ω V sin − (cid:16) V ( φ ( t )) − V c V (cid:17) , where V ( φ ( t )) ≡ V ( t ) = Q ( a, ˙ a, ¨ a ) with Q ( a, ˙ a, ¨ a ) the right hand side ofexpression (19). Therefore, differentiation leads to:˙ φ ( t ) = 1 V ω V r − h Q ( a, ˙ a, ¨ a ) − V c V i ddt [ Q ( a, ˙ a, ¨ a )] (25)and thus we obtain Q ( a, ˙ a, ¨ a ) = V ω V r − h Q ( a, ˙ a, ¨ a ) − V c V i ddt [ Q ( a, ˙ a, ¨ a )] , (26)where as we have mentioned, Q ( a, ˙ a, ¨ a ) is the right handside of expression (18).Differential equation (26) cannot be handled analyti-cally, but it can be easily solved numerically. In Fig. 4 wedepict the corresponding solution for a ( t ) (and thus for H ( t )) under the ansatz (24) with V = 5 . ω V = 0 . V c = 5 .
25, with k = 1, α = 1 and α = 1 (in unitswhere 8 πG = 1). The potential parameters have beenchosen in order to acquire a cyclic universe with a ( t ) ≈ -200 -100 0 100 200-0.4-0.20.00.20.405101520 H t a FIG. 4:
The evolution of the scale factor a ( t ) and of the Hub-ble parameter H ( t ) , for a scalar potential of the ansatz (24)with V = 5 . , ω V = 0 . and V c = 5 . , with k = 1 , α = 1 and α = 1 . All quantities are measured in unitswhere πG = 1 . Let us now present the corresponding simple cyclic ex-ample in the case of an open universe ( k = − a ( t ) and thusof the Hubble parameter H ( t ), under the ansatz (24) with V = 3 . ω V = 0 .
25 and V c = 3 .
13, with k = − α = 1and α = 1 (in units where 8 πG = 1). As described a H t
FIG. 5:
The evolution of the scale factor a ( t ) and of the Hub-ble parameter H ( t ) , for a scalar potential of the ansatz (24)with V = 3 . , ω V = 0 . and V c = 3 . , with k = − , α = 1 and α = 1 . All quantities are measured in unitswhere πG = 1 . above, in the case of an open universe one needs the scalarpotential to be negative for field values corresponding tolarge scale factors, in order for the turnaround to be trig-gered. However, since at that regime the curvature termis very small, even very small negative potential valuescan fulfill this condition, as can be seen by the specificexample of Fig. 5, where V = 3 .
15 and V c = 3 .
13 leadingthe minimal value of the potential to be − . a ( t ) one re-sults in the corresponding periodic scalar potential V ( φ ).Similarly, imposing any periodic potential V ( φ ) one cansolve the differential equation (23) and extract the result-ing periodic a ( t ).We close this section by mentioning that, although thebounce solutions arise owing to the presence of a darkradiation component with negative energy density, theycan also be obtained if ordinary radiation with positiveenergy density is present. When ordinary radiation isinvolved, it has to be generated from the reheating pro-cess of a primordial field namely inflaton or φ appearedin our model. Therefore its domination only takes placeafter reheating of which the energy scale is much lowerthan the bounce scale. Moreover, in the late time evo-lution normal radiation would be erased during matterdominated period, and hence will not affect the bouncesolution in next cycle. We will address on the details of this issue in future studies. IV. FLUCTUATIONS THROUGH THEBOUNCE
A model of non-relativistic gravity is usually able to re-cover Einstein’s general relativity as an emergent theoryat low energy scales. Therefore, the cosmological fluctu-ations generated in this model should be consistent withthose obtained in standard perturbation theory in the IRlimit [10]. This result has been intensively discussed inthe literature (see e.g. [34]). In particular, the pertur-bation spectrum presents a scale-invariant profile if theuniverse has undergone a matter-dominated contractingphase [26, 40, 42]. However, the non-relativistic correc-tions in the action (4) could lead to a modification of thedispersion relations of perturbations. This issue has beenaddressed in [56] , which shows that the spectrum in theUV regime may have a red tilt in a bouncing universe.Moreover, the perturbation modes cannot even enter theUV regime in the scenario of matter-bounce. So the anal-ysis of the cosmological perturbations in the IR regimeis quite reliable.Things become complicated but more interesting in acyclic scenario. Usually, a particular perturbation modein the contracting phase is dominated by its growing ten-dency, but in the expanding stage it becomes nearly con-stant on super-Hubble scales. Therefore, the metric per-turbation is amplified on super-Hubble scales cycle bycycle [59], and also the slope of its spectral index is vary-ing [60]. However, it is known that the contribution offluctuations has to be much less than the backgroundenergy. This prohibits the metric perturbations to enterthe next cycle if δρ/ρ ∼ O (1), unless the universe can beseparated into many parts independent of one another,each of which corresponding to a new universe and evolv-ing up to next cycle, then separate again and so on. Inthis case, the model of cyclic universe may be viewed asa realization of the multiverse scenario [59, 61, 62]. V. CONCLUSIONS
In this work, we have studied the possibility of con-structing a model of non-relativistic quantum gravity in3+1 dimensional spacetime. The novel features of thegravitational sector are reflected in new terms that arepresent in the IR, that is in the cosmologically interestingregime. Our results show that this model can give rise toa non-singular cosmology, for which the initial singular-ity is replaced by a big bounce if we require that the dark we refer to Refs. [57, 58] and references therein for the perturba-tions of a pure expanding universe in Hoˇrava-Lifshitz cosmology. radiation term is negative. Specifically, we have consid-ered an example in which the normal matter componentis a free scalar field. In this case we have obtained abouncing universe with a matter-dominated contractingphase, and so it may be responsible for the formationof a scale-invariant primordial spectrum. To extend, wehave also investigated the realization of a cyclic scenario,by re-constructing the potential of the scalar field whichleads to a universe with an oscillatory scale factor.Recently, the scenario of oscillating universe (originallyproposed by [63] and awaked later as ekpyrotic/cyclic by[64, 65, 66]), in which the universe experiences a sequenceof contractions and expansions, has been widely studiedin the literature, for instance in the context of loop quan-tum gravity [43, 67, 68, 69], including matter componentsviolating energy conditions [70, 71, 72, 73, 74, 75], in theframe of string cosmology [64, 76] and within the brane-world [22, 77, 78] (see Refs. [79, 80, 81, 82, 83, 84, 85,86, 87] for recent developments on oscillating universes).The main peculiarity of the current work is that the os-cillation depends on the dark radiation term, which ispresent only for not spatially flat geometry. However,the main advantage is that the background theory is welldefined at the quantum level.In conclusion, we see that the present model of power-counting renormalizable, non-relativistic gravitationaltheory in 3+1 dimensional spacetime, can naturally leadto a bounce and to cyclicity as particular sub-classes of itspossible induced cosmological behaviors. The fact thatthe theory is UV complete and that instabilities do notarise at the quantum level, makes future investigationson its cosmological implications quite interesting.As an end, we would like to comment on a principaldifference between our model and the intensely studiedHoˇrava-Lifshitz cosmology[3, 4]. In a model of Hoˇrava-Lifshitz cosmology (with or without detailed balance con- ditions), it is claimed by [88] that, there is one extrascalar modes in its perturbation theory which becomesstrongly coupled as the parameters approach a desired IRfixed point. To understand this point, we would like torecall that such a strong coupling problem only exists in asystem of which the original scalar modes have obtainedeffective mass terms and so lead to the generation of anextra degree of freedom, as what we have understoodin quantum field theory very well. As a consequence,modifications on general relativity often suffers from thisproblem, namely, in the theory of Pauli-Fierz massivegravity [89] the longitudinal scalar becomes strongly cou-pled when the mass approaches zero [90], leading to thefamous vDVZ discontinuity [91, 92]. Hoˇrava-Lifshitzcosmology also suffers from the strong coupling prob-lem, since the theory manifestly contains parity-violatingterms[93, 94] which bring effective masses for gravitons.However, this does not happen in our model since thespatial curvature terms and its spatial derivatives intro-duced phenomenologically only contribute to the disper-sion relations of two polarizations of gravitons withoutbringing effective mass terms. In this case, there is no ex-tra degree of freedom in our model and the longitudinalpart of metric perturbations can be fixed by Hamiltonianconstraint when combined with matter component. Acknowledgments
We would like to thank Xinmin Zhang for discussions.The research of Y.F.C. is supported in part by the Na-tional Science Foundation of China under Grants No.10533010 and 10675136, and by the Chinese Academyof Sciences under Grant No. KJCX3-SYW-N2. [1] P. Horava, Phys. Rev. D , 084008 (2009)[arXiv:0901.3775 [hep-th]].[2] P. Horava, JHEP , 020 (2009) [arXiv:0812.4287[hep-th]].[3] G. Calcagni, arXiv:0904.0829 [hep-th].[4] E. Kiritsis and G. Kofinas, arXiv:0904.1334 [hep-th].[5] M. Li and Y. Pang, arXiv:0905.2751 [hep-th].[6] L. Maccione, A. M. Taylor, D. M. Mattingly andS. Liberati, JCAP , 022 (2009) [arXiv:0902.1756[astro-ph.HE]].[7] D. Blas, D. Comelli, F. Nesti and L. Pilo,arXiv:0905.1699 [hep-th].[8] X. Gao, Y. Wang, R. Brandenberger and A. Riotto,arXiv:0905.3821 [hep-th].[9] D. Mattingly, Living Rev. Rel. , 5 (2005)[arXiv:gr-qc/0502097].[10] R. Brandenberger, arXiv:0904.2835 [hep-th].[11] Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang, JHEP , 071 (2007) [arXiv:0704.1090 [gr-qc]].[12] B. Feng, X. L. Wang and X. M. Zhang, Phys. Lett. B , 35 (2005) [arXiv:astro-ph/0404224].[13] M. Z. Li, B. Feng and X. M. Zhang, JCAP , 002(2005) [arXiv:hep-ph/0503268].[14] Y. F. Cai, H. Li, Y. S. Piao and X. M. Zhang, Phys. Lett.B , 141 (2007) [arXiv:gr-qc/0609039].[15] M. R. Setare and E. N. Saridakis, JCAP , 026 (2008)[arXiv:0809.0114 [hep-th]].[16] S. M. Carroll, M. Hoffman and M. Trodden, Phys. Rev.D , 023509 (2003) [arXiv:astro-ph/0301273].[17] J. M. Cline, S. Jeon and G. D. Moore, Phys. Rev. D ,043543 (2004) [arXiv:hep-ph/0311312].[18] G. Veneziano, Phys. Lett. B , 287 (1991);[19] M. Gasperini and G. Veneziano, Astropart. Phys. , 317(1993) [arXiv:hep-th/9211021].[20] R. Brustein and R. Madden, Phys. Rev. D , 712 (1998)[arXiv:hep-th/9708046].[21] M. Bojowald, Phys. Rev. Lett. , 5227 (2001)[arXiv:gr-qc/0102069].[22] Y. Shtanov and V. Sahni, Phys. Lett. B , 1 (2003).[23] P. S. Apostolopoulos, N. Brouzakis, E. N. Saridakis and N. Tetradis, Phys. Rev. D , 044013 (2005)[arXiv:hep-th/0502115].[24] L. R. Abramo and P. Peter, JCAP , 001 (2007)[arXiv:0705.2893 [astro-ph]].[25] Y. F. Cai, T. T. Qiu, J. Q. Xia and X. Zhang, Phys. Rev.D , 021303 (2009) [arXiv:0808.0819 [astro-ph]].[26] Y. F. Cai and X. Zhang, JCAP , 003 (2009)[arXiv:0808.2551 [astro-ph]].[27] V. F. Mukhanov and G. V. Chibisov, JETP Lett. (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. (1981) 549].[28] J. c. Hwang and H. Noh, Phys. Rev. D , 124010 (2002)[arXiv:astro-ph/0112079].[29] P. Peter and N. Pinto-Neto, Phys. Rev. D , 063509(2002) [arXiv:hep-th/0203013].[30] M. Gasperini, M. Giovannini and G. Veneziano, Phys.Lett. B , 113 (2003) [arXiv:hep-th/0306113].[31] J. Martin and P. Peter, Phys. Rev. D , 103517 (2003)[arXiv:hep-th/0307077].[32] Y. S. Piao, B. Feng and X. m. Zhang, Phys. Rev. D ,103520 (2004) [arXiv:hep-th/0310206].[33] L. E. Allen and D. Wands, Phys. Rev. D , 063515(2004) [arXiv:astro-ph/0404441].[34] Y. F. Cai, T. Qiu, R. Brandenberger, Y. S. Piao andX. Zhang, JCAP , 013 (2008) [arXiv:0711.2187 [hep-th]].[35] M. Novello and S. E. P. Bergliaffa, Phys. Rept. , 127(2008) [arXiv:0802.1634 [astro-ph]].[36] A. A. Starobinsky, JETP Lett. (1979) 682 [Pisma Zh.Eksp. Teor. Fiz. (1979) 719].[37] D. Wands, Phys. Rev. D , 023507 (1999)[arXiv:gr-qc/9809062].[38] F. Finelli and R. Brandenberger, Phys. Rev. D ,103522 (2002) [arXiv:hep-th/0112249].[39] P. Peter, E. J. C. Pinho and N. Pinto-Neto, Phys. Rev.D , 023516 (2007) [arXiv:hep-th/0610205].[40] Y. F. Cai, T. T. Qiu, R. Brandenberger and X. M. Zhang,Phys. Rev. D , 023511 (2009) [arXiv:0810.4677 [hep-th]].[41] Y. F. Cai, W. Xue, R. Brandenberger and X. Zhang,JCAP , 011 (2009) [arXiv:0903.0631 [astro-ph.CO]].[42] Y. F. Cai, W. Xue, R. Brandenberger and X. M. Zhang,JCAP , 037 (2009) [arXiv:0903.4938 [hep-th]].[43] H. H. Xiong, T. Qiu, Y. F. Cai and X. Zhang, Mod. Phys.Lett. A , 1237 (2009) [arXiv:0711.4469 [hep-th]].[44] H. H. Xiong, Y. F. Cai, T. Qiu, Y. S. Piao and X. Zhang,Phys. Lett. B , 212 (2008) [arXiv:0805.0413 [astro-ph]].[45] E. N. Saridakis, arXiv:0905.3532 [hep-th].[46] T. D. Lee and G. C. Wick, Nucl. Phys. B (1969) 209.[47] T. D. Lee and G. C. Wick, Phys. Rev. D , 1033 (1970).[48] B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev.D , 025012 (2008) [arXiv:0704.1845 [hep-ph]].[49] R. H. Brandenberger and C. Vafa, Nucl. Phys. B ,391 (1989).[50] T. Biswas, A. Mazumdar and W. Siegel, JCAP ,009 (2006) [arXiv:hep-th/0508194].[51] T. Biswas, R. Brandenberger, A. Mazum-dar and W. Siegel, JCAP , 011 (2007)[arXiv:hep-th/0610274].[52] N. Barnaby and N. Kamran, JHEP , 022 (2008)[arXiv:0809.4513 [hep-th]].[53] G. Calcagni and G. Nardelli, arXiv:0904.4245 [hep-th].[54] P. G. O. Freund and M. Olson, Phys. Lett. B , 186(1987). [55] P. G. O. Freund and E. Witten, Phys. Lett. B , 191(1987).[56] Y. F. Cai and X. Zhang, arXiv:0906.3341 [astro-ph.CO].[57] S. Mukohyama, arXiv:0904.2190 [hep-th].[58] Y. S. Piao, arXiv:0904.4117 [hep-th].[59] Y. S. Piao, Phys. Lett. B , 1 (2009) [arXiv:0901.2644[gr-qc]].[60] R. H. Brandenberger, arXiv:0905.1514 [hep-th].[61] J. K. Erickson, S. Gratton, P. J. Steinhardt and N. Turok,Phys. Rev. D , 123507 (2007) [arXiv:hep-th/0607164].[62] J. L. Lehners and P. J. Steinhardt, arXiv:0812.3388 [hep-th].[63] R. C. Tolman, Relativity, Thermodynamics and Cosmol-ogy, (Oxford U. Press, Clarendon Press, 1934).[64] P. J. Steinhardt and N. Turok, Science (2002) 1436.[65] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Stein-hardt and N. Turok, Phys. Rev. D , 086007 (2002)[arXiv:hep-th/0108187].[66] S. Tsujikawa, R. Brandenberger and F. Finelli, Phys.Rev. D , 083513 (2002) [arXiv:hep-th/0207228].[67] M. Bojowald, R. Maartens and P. Singh, Phys. Rev. D , 083517 (2004) [arXiv:hep-th/0407115].[68] J. E. Lidsey, D. J. Mulryne, N. J. Nunes and R. Tavakol,Phys. Rev. D , 063521 (2004) [arXiv:gr-qc/0406042].[69] P. Singh, K. Vandersloot and G. V. Vereshchagin, Phys.Rev. D , 043510 (2006) [arXiv:gr-qc/0606032].[70] M. G. Brown, K. Freese and W. H. Kinney, JCAP ,002 (2008) [arXiv:astro-ph/0405353].[71] B. Feng, M. Li, Y. S. Piao and X. Zhang, Phys. Lett. B , 101 (2006) [arXiv:astro-ph/0407432].[72] M. P. Dabrowski and T. Stachowiak, Annals Phys. ,771 (2006) [arXiv:hep-th/0411199].[73] L. Baum and P. H. Frampton, Phys. Rev. Lett. ,071301 (2007) [arXiv:hep-th/0610213].[74] T. Clifton and J. D. Barrow, Phys. Rev. D , 043515(2007) [arXiv:gr-qc/0701070].[75] K. Freese, M. G. Brown and W. H. Kinney,arXiv:0802.2583 [astro-ph].[76] P. J. Steinhardt and N. Turok, Phys. Rev. D , 126003(2002) [arXiv:hep-th/0111098].[77] N. Kanekar, V. Sahni and Y. Shtanov, Phys. Rev. D ,083520 (2001) [arXiv:astro-ph/0101448].[78] E. N. Saridakis, Nucl. Phys. B , 224 (2009)[arXiv:0710.5269 [hep-th]].[79] Y. S. Piao, Phys. Rev. D , 101302 (2004)[arXiv:hep-th/0407258].[80] X. Zhang, Eur. Phys. J. C , 661 (2009)[arXiv:0708.1408 [gr-qc]].[81] T. Biswas and S. Alexander, arXiv:0812.3182 [hep-th].[82] T. Biswas and A. Mazumdar, arXiv:0901.4930 [hep-th].[83] J. Liu, H. Li, J. Xia and X. Zhang, arXiv:0901.2033[astro-ph.CO].[84] K. Nozari, M. R. Setare, T. Azizi and S. Akhshabi,arXiv:0901.0090 [hep-th].[85] X. Zhang, Eur. Phys. J. C , 755 (2009).[86] X. Zhang, J. f. Zhang, J. l. Cui and L. Zhang,arXiv:0902.0928 [gr-qc].[87] A. S. Koshelev and S. Y. Vernov, arXiv:0903.5176 [hep-th].[88] C. Charmousis, G. Niz, A. Padilla and P. M. Saffin,arXiv:0905.2579 [hep-th].[89] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A (1939) 211.[90] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys. , 96 (2003) [arXiv:hep-th/0210184].[91] H. van Dam and M. J. G. Veltman, Nucl. Phys. B ,397 (1970).[92] V. I. Zakharov, JETP Lett. (1970) 312 [Pisma Zh.Eksp. Teor. Fiz.12