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Emergent Universe Scenario Via Quintom Matter Physics Letters B 718 (2012) 248–254 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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.
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