Emergent Universe Scenario Via Quintom Matter

Physics Letters B 718 (2012) 248–254

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Physics Letters B

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Emergent universe scenario via Quintom matter ∗ Yi-Fu Cai a,b, , Mingzhe Li c,d, Xinmin Zhang e a Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada b Department of Physics, Arizona State University, Tempe, AZ 85287, USA c Department of Physics, Nanjing University, Nanjing 210093, China d Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing University, Purple Mountain Observatory, Nanjing 210093, China e Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China article info abstract

Article history: The emergent universe scenario provides a possible alternative to bouncing cosmology to avoid the Big Received 3 October 2012 Bang singularity problem. In this Letter we study the realization of the emergent universe scenario by Received in revised form 23 October 2012 making use of Quintom matter with an equation of state across the cosmological constant boundary. We Accepted 23 October 2012 will show explicitly the analytic and numerical solutions of emergent universe in two Quintom models, Available online 26 October 2012 which are a phenomenological ﬂuid and a nonconventional spinor ﬁeld, respectively. Editor: M. Trodden © 2012 Elsevier B.V. All rights reserved.

1. Introduction universe has to cross over the cosmological constant boundary w =−1 twice. This type of bounce model was extensively studied Inflation is considered as the most successful model of describ- in the literature in recent years (for example see [15–18] and ref- ing physics of very early universe, which has explained concep- erences therein, and see [19] for a review on Quintom cosmology). tual issues of the Big Bang cosmology [1–3] (see [4–6] for early A bouncing model combining the Matter Bounce scenario [20,21] works). Among these remarkable achievements, inflation has pre- and Quintom scenario [14] was achieved in the frame of Lee–Wick dicted a nearly scale-invariant primordial power spectrum which cosmology [22], which showed that such kind of bounces can give was later verified in high precision by Cosmic Microwave Back- rise to a scale-invariant power spectrum for primordial curvature ground (CMB) observations [7]. The success of inflation is mainly perturbation. Later, the cosmological perturbation theory of bounc- based on a series of assumptions including an enough long period ing cosmology was established to a complete frame which includes of quasi-exponential expansion and the applicability of perturba- primordial non-gaussianities [23,24], entropy fluctuations and cur- tion theory during this phase. However, these assumptions often vaton scenario [25], and related preheating process [26].Arecent bring troubles to inflation models, such as the fine-tuning problem nonsingular bouncing model [12] which combined the benefit of of the potential parameters. Moreover, it was pointed out that in- Matter Bounce [20,21] and Ekpyrotic cosmology [27,28] was pro- flation models suffer from the initial singularity problem inherited posed and the w crossing −1 scenario can be realized without from the Big Bang cosmology [8]. pathologies through the Galileon-like Lagrangian (see [29] for the Recently, there are increasing interests in alternatives to infla- Galileon model and [30,31] for extensions). tionary cosmology which cannot only be the same successful as Besides the bouncing, there is another interesting cosmolog- inflation in explaining early universe physics but also avoid the ini- ical scenario which could be a strong competitor of inflation, tial spacetime singularity (for example see [9] for a recent review). which is the so-called emergent universe scenario [32–34].This One class of the alternative scenario is the bouncing cosmology, scenario suggests that our universe was initially emergent from which suggests the expansion of our universe was preceded by a non-vanishing minimal radius and experienced a enough long an initial contraction and then a non-vanishing bouncing point quasi-Minkowski phase and then entered the normal thermal ex- happened to connect the contraction and the expansion [10–12]. pansion. It was obtained in the String Gas cosmology in which It was pointed out in Ref. [13] that a realistic cosmological model the emergent universe was achieved in the Hagedorn phase of realizing a nonsingular bounce requires a matter field with Quin- a thermal system composed of a gas of superstrings [32,35,36]. tom behavior [14] of which the equation of state (EoS) of the It can be implemented by a model dubbed as Galilean Genesis as well [37]. Phenomenological study on the causal generation of primordial field fluctuations in this scenario was performed Corresponding author at: Department of Physics, Arizona State University, * via the so-called Conformal Cosmology [38–43], and the Pseudo- Tempe, AZ 85287, USA. E-mail addresses: [email protected] (Y.-F. Cai), [email protected] (M. Li), Conformal Cosmology [44,45]. Later, the issue of successfully trans- [email protected] (X. Zhang). ferring scale-invariant primordial field fluctuations to curvature

0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.10.065 Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254 249 perturbations was discussed in [46] and [47], respectively. More- The emergent universe is a scenario of non-singular cosmology. over, there is a modified version of emergent universe in which the Differing from the bouncing or cyclic modes where the expand- universe has experienced a process of slow contraction [27,48–50] ing universe was preceded by a contracting phase, the picture of or slow expansion [51–56]. Recently, it has been also shown that emergent universe requires the universe expands forever beginning emergent-type universes, which start out with a very small Hub- with a finite scale factor in the infinite past. This implies the Hub- ble rate, are quite generally preferred over inflationary models in a ble rate H cannot be negative. landscape, as described in [57,58]. To be non-singular requires the spacetime of the universe is In this Letter, we are going to show that within the frame of the geodesically complete, i.e., the affine parameters of geodesics are 4-dimensional Friedmann–Robertson–Walker (FRW) universe gov- divergent in the limit of infinite past [59]. The null geodesic obeys erned by standard Einstein gravity, the Quintom-like matter field the equations is needed to realize a realistic model of emergent universe, which dkν drives the universe from a quasi-Minkowski phase to a normal + ν μ ρ = Γμρ k k 0, (4) thermal expanding history. We start with a brief examination on dλ this necessary condition. and Consider a universe initially emergences from a nonzero mini- μ ν mal size and experiences enough long period of quasi-Minkowski gμνk k = 0, (5) phase. In this phase, the scale factor a(t) is almost a constant and where λ is the affine parameter and kμ = dxμ/dλ is the vector ˙ its time derivative a(t) is nearly zero. In order to make the universe tangential to the geodesic. In the spatially flat FRW universe with exit this phase gradually, we need its second order time derivative the line element (2), one can show that a¨(t) to be a very small positive value. Thus, if we use the Hubble ≡ ˙ 0 parameter H ( a/a) to characterize dynamics of this phase, we dk 0 → + ˙ + Hk = 0. (6) find H 0 and H > 0. Having assumed Einstein gravity is still dt available to describe the gravity sector during this period, one can By making use of H = d ln a/dt, the above equation yields immediately read there is an effective EoS of the universe which satisfies, dt 1 k0 = ∝ , dλ ∝ a(t) dt (7) dλ a 2H˙ w =−1 − −1. (1) and hence, the requirement of the completion of null geodesics can 2 3H be expressed as After exiting the quasi-Minkowski evolution gracefully, the uni- t =t a(t =t) verse needs to enter into the normal thermal expanding phase = −1 =∞ which requires the EoS of the whole universe to be roughly equal a t dt H da . (8) to 1/3, 0 and −1 along with the observable history, respectively. t =−∞ a(t =−∞) Therefore, this requires a transit of the background EoS from Because the scale factor a is finite and non-negative, the diver- w < −1tow > −1, which is exactly the Quintom behavior. − gence of the second integral needs H 1 should be singular at In this Letter we study particular realizations of the emergent − certain moment. For example, in bouncing cosmology, H 1 =∞ universe picture by using several explicit Quintom models. The at the bouncing point which is located at some intermediate time. Letter is organized as follows. In Section 2, we study the general However, for the emergent universe scenario, it is natural to con- requirements of the emergent universe picture. Then in Section 3, − clude that H 1 →∞when t →−∞, which means, the spacetime we present the analytic and numerical solution of the emergent of the universe approaches Minkowskian in the infinite past. universe from a toy model of a phenomenological Quintom-like For time-like geodesics, we can think that they are worldlines fluid with a parameterized EoS across the cosmological constant of some free particles with masses. Consider a wordline of a free boundary. For a much concrete model building, we consider a non- particle with mass m,thepropertimes itself is the affine param- conventional spinor field [60] in Section 3. Particularly we make eter of the wordline. The four-momentum is defined as use of an ansatz of the EoS and reconstruct the potential of the μ spinor Quintom analytically and numerically. Section 5 is devoted dx P μ = m , (9) to a summary of the Letter. ds μ ν 2 which obeys gμν P P = m . The condition for the geodesic com- 2. The requirements of the emergent universe pletion is obtained from the geodesic equation, which is

a(t =t) To start, we consider a spatially flat FRW universe, of which the −1/2 − C p metric is given by H 1 a2 + da =∞, (10) m2 a(t =−∞) ds2 = dt2 − a2(t) dx · dx. (2) where C p = P i P i is a non-negative constant. This also requires the Without modifying General Relativity, it is well known that the universe starts at infinite past from a Minkowski spacetime where − background dynamics in this frame follow the following two equa- H 1 =∞ or H = 0. The evolution of a Minkowski spacetime to tions of motion, an expanding universe requires the H increases for some time during which H˙ > 0, this implies the equation of state w < −1 ρ ρ + P H2 = , H˙ =− , (3) and the null energy condition was violated. At later time the uni- 2 2 3M p 2M p verse should enter into the radiation dominated phase in which √ w = 1/3. So w crosses −1 at some intermediate time. That is to in which M p ≡ 1/ 8π G is the reduced Planck mass. In addition, say, the matter dominating the universe has Quintom behavior. ρ and P are the energy density and the pressure of the matter Furthermore, in the emergent universe the Hubble rate reached fields filled in the universe, respectively. amaximumHmax at the crossing point. It defines a mass scale 250 Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254

the universe cannot go beyond. If Hmax is much smaller than the Planck mass, the validity of the general relativity as a low energy effective theory is guaranteed and the corrections of quantum gravity are suppressed.

3. Emergent universe with a Quintom-like ﬂuid

As a ﬁrst example, we illustrate the possibility of obtaining the emergent universe solution in a phenomenological Quintom ﬂuid described by the following EoS:

1 2α − w(t) = − e αM pt , (11) 3 3 where α is a positive-valued dimensionless parameter. In this particular parametrization, we have assumed the universe can auto- matically enters the radiation dominated period after the quasi- Minkowski expansion. Thus, one can see the EoS approaches 1/3 when the cosmic time t goes to positive inﬁnity. Moreover, when t approaches to a far past moment, the EoS would have fallen down to negative inﬁnity very soon due to the expression of the exponential term. To substitute the parametrization of the EoS (11) into the Fried- mann equations, one can solve out the explicit solution to the Hubble parameter as follows,

αM pt M pe H(t) = , (12) αM t 1 + (C + 2M pt)e p where C is a constant which can determine the energy scale of the Hubble parameter at the occurrence of emergent universe. Further, one obtains the energy density and the pressure of the Quintom fluid governing the universe as well. As a consequence, the evolu- Fig. 1. Plot of the evolutions of the scale factor a, the Hubble parameter H and the tion of the scale factor in this model can be numerically integrated background EoS w as a function of cosmic time in the case of Quintom fluid. In the = = out as shown in Fig. 1. From the figure, one can explicitly read that numerical calculation, we take α 3andC 10. All dimensional parameters are of Planck units. the scale factor a approaches to a non-vanishing minimal value in the limit of far past, and connects the thermal expansion when μ where e denotes the vierbein, gμν is the spacetime metric, and t →+∞. a η is the Minkowski metric with η = diag(+1, −1, −1, −1). Notice that, the occurrence moment t of the emergent uni- ab ab E Note that the Latin indices represents the local inertial frame and verse scenario can be characterized by the moment of w cross- the Greek indices represents the spacetime frame. ing −1. Thus, by solving w(t ) =−1, one gets E We choose the Dirac–Pauli representation as = 1 α tE log , (13) 0 = 10 i = 0 σi αM p 2 γ , γ , (16) 0 −1 −σi 0 which in our explicit example takes the value of 0.14 approxi- a where σi is Pauli matrices. One can see that the 4 × 4 γ satisfy mately. At this moment, the Hubble parameter arrives at the max- μ the Clifford algebra {γ a, γ b}=2ηab.Theγ a and e provide the imal value a definition of a new set of Gamma matrices M p H = , (14) μ E + 2 − 2 Γ μ = e γ a, (17) C α (1 log( α )) a { μ ν}= which is around 0.1M p . Therefore, we are able to trust Einstein which satisfy the algebra Γ ,Γ 2gμν . The generators of gravity in this case. the Spinor representation of the Lorentz group can be written as ab = 1 [ a b] Σ 4 γ , γ . So the covariant derivative are given by 4. Emergent universe and spinor Quintom Dμψ = (∂μ + Ωμ)ψ, (18) In this section, we consider a class of Quintom model described ¯ ¯ ¯ Dμψ = ∂μψ − ψΩμ, (19) by a nonconventional spinor field. To begin with, we simply review + the background dynamics of a spinor field which is minimally cou- where the Dirac adjoint is defined as ψ¯ ≡ ψ γ 0.The4× 4matrix = 1 ab = ν∇ pled to Einstein’s gravity (see Refs. [61,62] for detailed introduction Ωμ 2 ωμabΣ is the spin connection, where ωμab ea μeνb and see [60,63,64] for recent phenomenological study in cosmol- are Ricci spin coefficients. ogy). By the aid of the above algebra we can write down the following Dirac action in a curved spacetime background 4.1. Algebra of a cosmological spinor 4 i ¯ μ ¯ μ ¯ Sψ = d xe ψΓ Dμψ − DμψΓ ψ − V (ψψ) . (20) Following the general covariance principle, a connection be- 2 tween the metric gμν and the vierbein is given by a Here, e is the determinant of the vierbein eμ and V stands for μ ν = ¯ gμνea eb ηab, (15) the potential of the spinor field ψ and its adjoint ψ.Duetothe Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254 251 requirement of covariance, the potential V only depends on the Note that, a realistic cosmological evolution requires a thermal scalar bilinear ψψ¯ and “pseudo-scalar” term ψ¯ γ 5ψ. For simplicity expansion after the primordial period. After radiation dominated we drop the latter term and only assume V = V (ψψ)¯ . phase, the universe enters into a matter dominated era. However, μ Varying the action with respect to the vierbein ea ,weobtain we usually do not expect that a fermion field could be responsible the energy–momentum tensor, for a large amount of radiation.1 Therefore, for a simple and natural choice, we parameterize the form of EoS for the spinor field as e δS = μa ψ follows, Tμν ν e δea i ¯ ¯ ¯ ¯ 2α = [ψΓ D ψ + ψΓ D ψ − D ψΓ ψ − D ψΓ ψ] −αM pt ν μ μ ν μ ν ν μ wψ =− e . (29) 4 3 − gμνLψ . (21) To observe this form, one can see that the EoS falls into nega- On the other hand, varying the action with respect to the field ψ¯ , tive infinity when t −1, but approaches 0 at t →+∞.This ψ respectively yields the following equations of motion, parametrization can give rise to a emergent universe solution with μ a dust-like expansion following the quasi-Minkowski phase with- iΓ Dμψ − V ψ = 0, (22) out the radiation domination. ¯ μ ¯ iDμψΓ + V ψ = 0, (23) Substituting Eq. (29) into the Friedmann equations, one can solve out the Hubble parameter as a function of cosmic time: where V ≡ ∂ V /∂(ψψ)¯ denotes the derivative of the spinor poten- ¯ tial with respect to ψψ. αM pt M pe We deal with the homogeneous and isotropic FRW metric. Cor- H(t) = , (30) + + 3 αM pt respondingly, the vierbein are given by 1 (C 2 M pt)e

μ μ μ 1 μ e = δ , e = δ . (24) and thus the potential of the spinor evolves as 0 0 i a i 4 2αM pt Assuming the spinor field is space-independent, the equation of 12M pe 0 ˙ 3 V = . (31) motion reads iγ (ψ + Hψ) − V ψ = 0, where a dot denotes a αM t 2 2 (2 + 2(C + 3M pt)e p ) derivative with respect to the cosmic time and H is the Hubble parameter. Taking a further derivative, we can obtain: Similar to the case of Quintom fluid, the coefficient C is a inte- N gral constant which is used to determine the energy scale of the ψψ¯ = , (25) a3 occurrence of emergent universe scenario. By numerically solving the Friedmann equations, one can solve where N is a positive time-independent constant. out the evolution of the scale factor along cosmic time. We show From the expression of the energy–momentum tensor in the evolutions of the scale factor a, the Hubble parameter H and Eq. (21), one can obtain the expressions of the energy density and the EoS w in Fig. 2. the pressure of the spinor field as follows, After having known the evolution of the scale factor, one can ¯ 0 numerically obtain that of the scalar bilinear ψψ due to the rela- ρ = T = V , (26) − ψ 0 tion that ψψ¯ ∝ a 3. In addition, the energy density of the spinor

=− i = ¯ − field only depends on the potential V and thus one can derive V pψ Ti V ψψ V . (27) evolving along cosmic time. They are shown in Fig. 3.Fromtheup- As a consequence, the EoS of the spinor field is given by per panel of this figure, one can see that ψψ¯ is a monotonically ¯ decreasing function and approaches a constant in the emergent pψ V ψψ wψ ≡ =−1 + . (28) universe period. Note that, we use log scale to show the wide V ρψ range of scales for V in the longitudinal coordinate of the lower panel of Fig. 3, where it can be seen that V evolves as an expo- 4.2. Reconstruction of spinor Quintom realizing emergent universe nential form in the phase of quasi-Minkowski. scenario Since the evolutions of ψψ¯ and V were already obtained in above numerics, we can further derive V as a function of ψψ¯ . The above formulae show that a cosmological spinor field might This is achievable as ψψ¯ is decreasing monotonically along the realize its EoS to cross the cosmological constant boundary when ¯ cosmic time and then can lead to a inverse function t = t(ψψ).To the sign of V changes. For a conventional spinor with its potential combine t = t(ψψ)¯ and V = V (t), we then numerically solve out taking the form of mψψ¯ , its EoS is exactly zero which coincides V = V (ψψ)¯ as shown in Fig. 4. with that of normal non-relativistic dust matter. Thus, one has to Interestingly, from Fig. 4 one can find that V is a linear func- consider a nonconventional form of the potential for the cosmo- tion of ψψ¯ at the regime of small ψψ¯ values. This implies that logical spinor to achieve the Quintom scenario. the spinor Quintom recovers the conventional form of a massive Interestingly, observing the form of the Friedmann equations fermion with its potential V ∼ mψψ¯ at low energy limit. More- and the formulae of the spinor field, we can see that a model over, in the UV limit, one can derive the asymptotical form of the of spinor Quintom can be reconstructed to fulfill the scenario of scale factor as follows, emergent universe. Suppose a fixed form of the EoS such as what we have illustrated in the case of Quintom fluid. We then read the 1 ˙ αM pt αC expression of the Hubble parameter H and its time derivative H, a(t) aE 1 + Ce , (32) and thus can further derive the evolutions of V and V as func- − tions of cosmic time. By making use of the relation that ψψ¯ ∝ a 3, the form of V can be derived out explicitly. We will do the recon- 1 During radiation dominated phase, the main contribution comes from the gauge struction in the following context. photons, and only a few part is from nearly massless hot neutrinos. 252 Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254

Fig. 4. Plot of the potential V as a function of ψψ¯ in the case of spinor Quintom. In the numerical calculation, we take the values of α, C and N the same as in Fig. 2. All dimensional parameters are of Planck units.

and thus the asymptotical forms of ψψ¯ and V . To take the inverse function of ψψ(¯ t), eventually we can get the approximate expression of the potential in the phase of quasi-Minkowski: 4 ¯ αC 2 3M p ψψ 3 V (ψψ)¯ 1 − aαC , (33) C 2 E N

where aE being the minimal value of the scale factor in the emergent universe scenario. Note that, the potential for the spinor Quintom in low energy Fig. 2. Plot of the evolutions of the scale factor a, the Hubble parameter H and the limit is very normal in quantum field theory, which is merely a background EoS w as functions of cosmic time in the case of spinor Quintom. In the mass term. However, the potential becomes very nontrivial at the = = numerical calculation, we take α 3andC 10whicharethesameasinthecase high energy scale, which takes the form (33). Although this form of Quintom fluid (Fig. 1), and N = 100 for the scalar bilinear ψψ¯ . All dimensional is purely phenomenologically constructed, one can find that it im- parameters are of Planck units. plies a condensate of the tachyonic spinor field in UV regime and thus is expected to be related to the spinor formalism in open string field theory [65,66].

5. Conclusion

As well known, the conception of Quintom scenario was orig- inally from the study of dark energy physics, which shows the EoS of dark energy might cross the cosmological constant boundary [14]. This scenario is mildly favored by a group of cosmological observations [67,68], and its dynamics were extensively analyzed in a number of works (for example see [69–76] for phenomenological study and see [19] for a comprehensive review). However, it was soon found that the Quintom behavior has many significant implications to early universe physics. Namely, it can give rise to nonsingular bouncing and cyclic [77–79] solutions when applied to high energy scale. To conclude, we in the current Letter have studied the realization of emergent universe scenarios in the presence of Quintom matters. In the literature there have been a lot of efforts in building the models of emergent universe to avoid the big bang singularity. Also there are efforts on investigating the perturbation analysis in the phase of emergent universe assuming the gravity sector is still described by standard Einstein gravity. However, we in the present Letter point out that a model giving rise to the emergent universe scenario has to be of Quintom behavior within the standard 4-dimensional FRW framework under the assumption of Einstein gravity. Fig. 3. Plot of the evolutions of the scalar bilinear ψψ¯ and the potential V as func- In explicit realizations, we first considered the Quintom fluid tions of cosmic time in the case of spinor Quintom. In the numerical calculation, we take the values of α, C and N thesameasinFig. 2. All dimensional parameters with a parameterized form of EoS to illustrate its possibility. Af- are of Planck units. ter that, we have studied in detail the model of spinor Quintom. Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254 253 ˙ 3 As the model of spinor Quintom possesses a very nice algebra rela- i ψ¯ + Hψ¯ γ 0 + V ψ¯ = 0. (35) tion in the curved spacetime, we are able to reconstruct a suitably 2 chosen form of the potential and force it to give rise to the emer- According to the expression of the energy–momentum tensor gent universe scenario. Moreover, this type of model is able to given in Eq. (21), we calculate the expression of the energy density exit the phase of quasi-Minkowski expansion gracefully and enter of the spinor field: a normal expanding phase dominated by dust matter. The scenario proposed in the current Letter and its implemen- = 0 ρψ T0 tation in the detailed model has some remarkable properties in phenomenological applications. i ¯ 0 ¯ 0 ¯ 0 0 ¯ = ψΓ D0ψ + ψΓ0 D ψ − D0ψΓ ψ − D ψΓ0ψ First of all, it is interesting to realize the Quintom scenario in 4 other effective field models, such as the Galileon-type fields. In the i ¯ μ ¯ μ ¯ − ψΓ Dμψ − DμψΓ ψ − V (ψψ) original model of Galilean Genesis, it was found that the universe 2 could hardly exit the emergent state smoothly since its equation − i ¯ i ¯ i ¯ of state was unable to cross 1 from below to above. According to = ψγ ∂iψ − ∂iψγ ψ + V (ψψ) our analysis, one can expect a much improved model of Galilean 2a Genesis which can take the advantage of Quintom scenario to exit = V (ψψ),¯ (36) the primordial era smoothly. where in the last equality we used the assumption of scale inde- Additionally, a crucial issue in the cosmology of emergent uni- pendence ∂ ψ = 0. Furthermore, we can compute the pressure of verse is the processing of cosmic perturbations throughout the i the spinor field: quasi-Minkowski expanding phase. In the literature, there have been extensive studies on the generation of primordial power p =−T i spectrum. As a first step, we in the present Letter only consid- ψ i ered the background evolution but requires the energy scale of i ¯ 0 ¯ 0 i ¯ i ¯ i = ψγ D0ψ − D0ψγ ψ − ψγ ∂iψ − ∂iψγ ψ the emergent universe to be much lower than the Planck scale, so 2 2a that the assumption of Einstein gravity is trustable in our model. − V (ψψ)¯ However, we should be aware of that since a cosmic spinor was i ¯ 0 ¯ 0 ¯ introduced to realize the scenario of emergent universe, it may = ψγ D0ψ − D0ψγ ψ − V (ψψ) seed some unwanted fluctuations modes such as vector modes 2 ¯ through gauge interactions. Such a complete perturbation analy- = V ψψ − V , (37) sis of the emergent universe Quintom model lies beyond the scope where we have used the assumption ∂ ψ = 0 in the second line of the present work and it is left for future investigation. i and applied Eqs. (34) and (35) in the last line. Acknowledgements A.2. The stability issue We thank Robert Brandenberger and Yun-Song Piao for useful discussions. The work of C.Y.F. is supported in part by Depart- Regarding the stability issue of a cosmological model, there are ment of Physics in McGill University and the Cosmology Initia- mainly two possible concerns. One is to check whether or not the tive in Arizona State University. The research of M.L. is supported model leads to a ghost degree of freedom. Obviously, this issue in part by National Science Foundation of China under Grants does not exist in our emergent universe model which is realized Nos. 11075074 and 11065004, by the Specialized Research Fund by a spinor Quintom. As one can see from the action, the kinetic for the Doctoral Program of Higher Education (SRFDP) under Grant term for the spinor field is very standard without any modifica- No. 20090091120054 and by SRF for ROCS, SEM. X.Z. is supported tion. Thus, there does not exist any extra degree of freedom in our in part by the National Science Foundation of China under Grants model. The other is to study the evolution of cosmological pertur- Nos. 10975142 and 11033005, and by the Chinese Academy of Sci- bations seeded by the spinor field and make sure the backreaction ences under Grant No. KJCX3-SYW-N2. One of the author (C.Y.F.) of perturbations to the background is controllable. is grateful to the other two (M.L. and X.Z.) and Yun-Song Piao for Here we would like to show the perturbation theory of spinor hospitality during his visit in Nanjing University and the Institute Quintom crudely. In order to simplify the derivative, we would like ≡ 3 of High Energy Physics at CAS and the Graduate University of CAS to redefine the spinor as ψN a 2 ψ, and perturb it as while this work was initiated. i ψN → ψN (t) + δψN t, x , (38) Appendix A around the homogeneous background. Then perturbing the equation of motion of the spinor field, one can obtain the perturbation In the first part of this appendix, we derive the explicit expres- equation as follows, sions for the energy density and pressure provided in Eqs. (26) and (27), respectively. In the second part of the appendix, we ad- 2 d 2 2 2 0 ¯ dress on the stability of our model throughout the cosmological δψN −∇ δψN + a V + iγ HV − 3HV ψψ δψN dτ 2 evolution. 2 ¯ μ ¯ =−2a V V δ(ψψ)ψN − iγ ∂μ aV δ(ψψ) ψN , (39) A.1. Energy density and pressure where τ is the conformal time defined by dτ ≡ dt/a.Sincethe − right-hand side of the equation decays proportional to a 3 or even We assume the spinor field is space-independent but evolves faster, we can neglect those terms throughout the evolution of the along cosmic time. Then the equations of motion given in (22) universe for simplicity. and (23) are simplified as follows, From the perturbation equation above, we can read that 3 the sound speed is equal to 1 which eliminates the instability iγ 0 ψ˙ + Hψ − V ψ = 0, (34) 2 of the system in short wavelength. Moreover, when the equation 254 Y.-F. Cai et al. / Physics Letters B 718 (2012) 248–254

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