# Periodic Cosmological Evolutions of Equation of State for Dark Energy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Brief Review of the CG Type Models

## 3. Periodical and Quasi-periodical GCG Type Models

#### 3.1. Periodical Generalizations

#### 3.1.1. MG-XXI Model

#### 3.1.2. MG-XXII Model

#### 3.1.3. MG-XXIII Model

#### 3.1.4. MG-XXIV Model

#### 3.2. Quasi-periodical Generalizations

#### 3.2.1. MG-XXV Model

#### 3.2.2. MG-XXVI Model

#### 3.2.3. MG-XXVII Model

## 4. Other Two Periodical FLRW Models

#### 4.1. MG-XII Model

**Figure 1.**The EoS ω in Equation (29) as a function of t for $\wp \left(t,1,1\right)$, i.e., the model parameters of the Weierstrass invariants of ${g}_{2}=1$ and ${g}_{3}=1$. The line of $\omega =-1$ is also plotted.

#### 4.2. MG-XIII Model

#### 4.3. MG-XIV Model

#### 4.4. MG-XV Model

#### 4.5. MG-XVI Model

#### 4.6. MG-XVII Model

#### 4.7. MG-XVIII Model

#### 4.8. MG-XIX Model

#### 4.9. MG-XX Model

#### 4.10. MG-XXXIII Model

## 5. Conclusions

## Acknowledgments

## References

- Spergel, D.N.; Verde, L.; Peiris, H.V.; Komatsu, E.; Nolta, M.R.; Bennett, C.L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; et al. First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophys. J. Suppl.
**2003**, 148, 175–194. [Google Scholar] [CrossRef] - Spergel, D.N.; Bean, R.; Dore, O.; Nolta, M.R.; Bennett, C.L.; Hinshaw, G.; Jarosik, N.; Komatsu, E.; Page, L.; Peiris, H.V.; et al. Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology. Astrophys. J. Suppl.
**2007**, 170, 377–408. [Google Scholar] [CrossRef] - Komatsu, E.; Dunkley, J.; Nolta, M.R.; Bennett, C.L.; Gold, B.; Hinshaw, G.; Jarosik, N.; Larson, D.; Limon, M.; Page, L.; et al. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. Astrophys. J. Suppl.
**2009**, 180, 330–376. [Google Scholar] [CrossRef] - Komatsu, E.; Smith, K.M.; Dunkley, J.; Bennett, C.L.; Gold, B.; Hinshaw, G.; Jarosik, N.; Larson, D.; Nolta, M.R.; Page, L.; et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. Astrophys. J. Suppl.
**2011**, 192, 18. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiattia, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron. J.
**1998**, 116, 1009–1038. [Google Scholar] [CrossRef] - Tegmark, M.; Strauss, M.A.; Blanton, M.R.; Abazajian, K.N.; Dodelson, S.; Sandvik, H.; Wang, X.; Weinberg, D.H.; Zehavi, I.; Bahcall, N.A.; et al. Cosmological parameters from SDSS and WMAP. Phys. Rev. D
**2004**. [Google Scholar] [CrossRef] - Seljak, U.; Makarov, A.; McDonald, P.; Anderson, S.; Bahcall, N.; Brinkmann, J.; Burles, S.; Cen, R.; Doi, M.; Gunn, J.; Ivezic, Z.; et al. Cosmological parameter analysis including SDSS Ly-alpha forest and galaxy bias: Constraints on the primordial spectrum of fluctuations, neutrino mass, and dark energy. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Eisenstein, D.J.; Zehavi, I.; Hogg, D.W.; Scoccimarro, R.; Blanton, M.R.; Nichol, R.C.; Scranton, R.; Seo, H.; Tegmark, M.; Zheng, Z.; et al. Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies. Astrophys. J.
**2005**, 633, 560–574. [Google Scholar] [CrossRef] - Jain, B.; Taylor, A. Cross-correlation Tomography: Measuring Dark Energy Evolution with Weak Lensing. Phys. Rev. Lett.
**2003**, 91, 141302. [Google Scholar] [CrossRef] [PubMed] - Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753–1936. [Google Scholar] [CrossRef] - Durrer, R.; Maartens, R. Dark Energy and Dark Gravity. Gen. Rel. Grav.
**2008**, 40, 301–328. [Google Scholar] [CrossRef] [Green Version] - Durrer, R.; Maartens, R. Dark Energy and Modified Gravity. In Dark Energy: Observational Theoretical Approaches; Ruiz-Lapuente, P., Ed.; Cambridge University press: Cambridge, UK, 2010; pp. 48–91. [Google Scholar]
- Cai, Y.F.; Saridakis, E.N.; Setare, M.R.; Xia, J.Q. Quintom Cosmology: Theoretical implications and observations. Phys. Rept.
**2010**, 493, 1–60. [Google Scholar] [CrossRef] - Tsujikawa, S. Dark energy: investigation and modeling. 2010; arXiv:1004.1493 [astro-ph.CO]. [Google Scholar]
- Amendola, L.; Tsujikawa, S. Dark Energy; Cambridge University press: Cambridge, United Kingdom, 2010; pp. 1–491. [Google Scholar]
- Li, M.; Li, X.D.; Wang, S.; Wang, Y. Dark Energy. Commun. Theor. Phys.
**2011**, 56, 525–604. [Google Scholar] [CrossRef] - Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci.
**2012**, 342, 155–228. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models. Phys. Rept.
**2011**, 505, 59–144. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Meth. Mod. Phys.
**2007**, 4, 115–146. [Google Scholar] [CrossRef] - Sotiriou, T.P.; Faraoni, V. f(R) Theories Of Gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef] - Capozziello, S.; Faraoni, V. Beyond Einstein Gravity; Springer: London, UK, 2010; pp. 1–424. [Google Scholar]
- Capozziello, S.; de Laurentis, M. Extended Theories of Gravity. Phys. Rept.
**2011**, 509, 167–321. [Google Scholar] [CrossRef] - De Felice, A.; Tsujikawa, S. f(R) theories. Living Rev. Rel.
**2010**, 13, 3. [Google Scholar] [CrossRef] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. Phys. Rept.
**2012**, 513, 1–189. [Google Scholar] [CrossRef] - Capozziello, S.; de Laurentis, M.; Odintsov, S.D. Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology. Eur. Phys. J. C
**2012**, 72, 2068. [Google Scholar] [CrossRef] - Caldwell, R.R.; Kamionkowski, M.; Weinberg, N.N. Phantom Energy and Cosmic Doomsday. Phys. Rev. Lett.
**2003**. [Google Scholar] [CrossRef] - McInnes, B. The dS/CFT correspondence and the big smash. J. High Energy Phys.
**2002**, 0208, 029. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Quantum deSitter cosmology and phantom matter. Phys. Lett. B
**2003**, 562, 147–152. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Effective equation of state and energy conditions in phantom / tachyon inflationary cosmology perturbed by quantum effects. Phys. Lett. B
**2003**, 571, 1–10. [Google Scholar] [CrossRef] - Faraoni, V. Superquintessence. Int. J. Mod. Phys. D
**2002**, 11, 471–482. [Google Scholar] [CrossRef] - Gonzalez-Diaz, P.F. K-essential phantom energy: Doomsday around the corner? Phys. Lett. B
**2004**, 586, 1–4. [Google Scholar] [CrossRef] - Elizalde, E.; Nojiri, S.; Odintsov, S.D. Late-time cosmology in (phantom) scalar-tensor theory: Dark energy and the cosmic speed-up. Phys. Rev. D
**2004**. [Google Scholar] [CrossRef] - Singh, P.; Sami, M.; Dadhich, N. Cosmological dynamics of phantom field. Phys. Rev. D
**2003**. [Google Scholar] [CrossRef] - Csaki, C.; Kaloper, N.; Terning, J. Exorcising w < −1. Annals Phys.
**2005**, 317, 410–422. [Google Scholar] - Wu, P.X.; Yu, H.W. Avoidance of Big Rip In Phantom Cosmology by Gravitational Back Reaction. Nucl. Phys. B
**2005**, 727, 355–367. [Google Scholar] [CrossRef] - Nesseris, S.; Perivolaropoulos, L. The fate of bound systems in phantom and quintessence cosmologies. Phys. Rev. D
**2004**. [Google Scholar] [CrossRef] - Sami, M.; Toporensky, A. Phantom Field and the Fate of Universe. Mod. Phys. Lett. A
**2004**, 19, 1509–1517. [Google Scholar] [CrossRef] - Stefancic, H. Generalized phantom energy. Phys. Lett. B
**2004**, 586, 5–10. [Google Scholar] [CrossRef] - Chimento, L.P.; Lazkoz, R. On the link between phantom and standard cosmologies. Phys. Rev. Lett.
**2003**. [Google Scholar] [CrossRef] - Hao, J.G.; Li, X.Z. Generalized quartessence cosmic dynamics: Phantom or quintessence with de Sitter attractor. Phys. Lett. B
**2005**, 606, 7–11. [Google Scholar] [CrossRef] - Elizalde, E.; Nojiri, S.; Odintsov, S. D.; Wang, P. Dark energy: Vacuum fluctuations, the effective phantom phase, and holography. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Dabrowski, M.P.; Stachowiak, T. Phantom Friedmann cosmologies and higher-order characteristics of expansion. Annals Phys.
**2006**, 321, 771–812. [Google Scholar] [CrossRef] - Lobo, F.S.N. Phantom energy traversable wormholes. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Cai, R.G.; Zhang, H.S.; Wang, A. Crossing w = -1 in Gauss-Bonnet brane world with induced gravity. Commun. Theor. Phys.
**2005**, 44, 948–954. [Google Scholar] [CrossRef] - Aref’eva, I.Y.; Koshelev, A.S.; Vernov, S.Y. Crossing of the w=-1 Barrier by D3-brane Dark Energy Model. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Lu, H. Q.; Huang, Z.G.; Fang, W. Quantum and classical cosmology with Born-Infeld scalar field.
**2005**. [Google Scholar] - Godlowski, W.; Szydlowski, M. Which cosmological model with dark energy - phantom or LambdaCDM. Phys. Lett. B
**2005**, 623, 10–16. [Google Scholar] [CrossRef] - Sola, J.; Stefancic, H. Effective equation of state for dark energy: mimicking quintessence and phantom energy through a variable Lambda. Phys. Lett. B
**2005**, 624, 147–157. [Google Scholar] [CrossRef] - Guberina, B.; Horvat, R.; Nikolic, H. Generalized holographic dark energy and the IR cutoff problem. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Shtanov, Y.; Sahni, V. Unusual cosmological singularities in braneworld models. Class. Quant. Grav.
**2002**, 19, L101–L107. [Google Scholar] [CrossRef] - Barrow, J.D. Sudden Future Singularities. Class. Quant. Grav.
**2004**, 21, L79–L82. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Quantum escape of sudden future singularity. Phys. Lett. B
**2004**, 595, 1–8. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. The final state and thermodynamics of dark energy universe. Phys. Rev. D
**2004**. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Inhomogeneous equation of state of the universe: Phantom era, future singularity and crossing the phantom barrier. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Cotsakis, S.; Klaoudatou, I. Future singularities of isotropic cosmologies. J. Geom. Phys.
**2005**, 55, 306–315. [Google Scholar] [CrossRef] - Dabrowski, M.P. Inhomogenized sudden future singularities. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Fernandez-Jambrina, L.; Lazkoz, R. Geodesic behaviour of sudden future singularities. Phys. Rev. D
**2004**. [Google Scholar] [CrossRef] - Fernandez-Jambrina, L.; Lazkoz, R. Singular fate of the universe in modified theories of gravity. Phys. Lett. B
**2009**, 670, 254–258. [Google Scholar] [CrossRef] - Barrow, J.D.; Tsagas, C.G. New Isotropic and Anisotropic Sudden Singularities. Class. Quant. Grav.
**2005**, 22, 1563–1571. [Google Scholar] [CrossRef] - Stefancic, H. ’Expansion’ around the vacuum equation of state: Sudden future singularities and asymptotic behavior. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Cattoen, C.; Visser, M. Necessary and sufficient conditions for big bangs, bounces, crunches, rips, sudden singularities, and extremality events. Class. Quant. Grav.
**2005**, 22, 4913–4930. [Google Scholar] [CrossRef] - Tretyakov, P.; Toporensky, A.; Shtanov, Y.; Sahni, V. Quantum effects, soft singularities and the fate of the universe in a braneworld cosmology. Class. Quant. Grav.
**2006**, 23, 3259–3274. [Google Scholar] [CrossRef] - Balcerzak, A.; Dabrowski, M.P. Strings at future singularities. Phys. Rev. D
**2006**. [Google Scholar] [CrossRef] - Sami, M.; Singh, P.; Tsujikawa, S. Avoidance of future singularities in loop quantum cosmology. Phys. Rev. D
**2006**. [Google Scholar] [CrossRef] - Bouhmadi-Lopez, M.; Gonzalez-Diaz, P.F.; Martin-Moruno, P. Worse than a big rip? Phys. Lett. B
**2008**, 659, 1–5. [Google Scholar] [CrossRef] - Yurov, A.V.; Astashenok, A.V.; Gonzalez-Diaz, P.F. Astronomical bounds on future big freeze singularity. Grav. Cosmol.
**2008**, 14, 205–212. [Google Scholar] [CrossRef] - Koivisto, T. Dynamics of Nonlocal Cosmology. Phys. Rev. D
**2008**. [Google Scholar] [CrossRef] - Barrow, J.D.; Lip, S.Z.W. The Classical Stability of Sudden and Big Rip Singularities. Phys. Rev. D
**2009**. [Google Scholar] [CrossRef] - Bouhmadi-Lopez, M.; Tavakoli, Y.; Moniz, P.V. Appeasing the Phantom Menace? JCAP
**2010**, 1004, 016. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Tsujikawa, S. Properties of singularities in (phantom) dark energy universe. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. The future evolution and finite-time singularities in F(R)-gravity unifying the inflation and cosmic acceleration. Phys. Rev. D
**2008**. [Google Scholar] [CrossRef] - Bamba, K.; Nojiri, S.; Odintsov, S.D. The Universe future in modified gravity theories: Approaching the finite-time future singularity. JCAP
**2008**, 0810, 045. [Google Scholar] [CrossRef] - Bamba, K.; Odintsov, S.D.; Sebastiani, L.; Zerbini, S. Finite-time future singularities in modified Gauss-Bonnet and F(R,G) gravity and singularity avoidance. Eur. Phys. J. C
**2010**, 67, 295–310. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turok, N. Cosmic evolution in a cyclic universe. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turok, N. A cyclic model of the universe. Science
**2002**, 296, 1436–1439. [Google Scholar] [CrossRef] [PubMed] - Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. Density perturbations in the ekpyrotic scenario. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turok, N. The cyclic universe: An informal introduction. Nucl. Phys. Proc. Suppl.
**2003**, 124, 38–49. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turok, N. A cyclic model of the universe. Science
**2002**, 296, 1436–1439. [Google Scholar] [CrossRef] [PubMed] - Khoury, J.; Steinhardt, P.J.; Turok, N. Designing Cyclic Universe Models. Phys. Rev. Lett.
**2004**. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turok, N. Why the cosmological constant is small and positive. Science
**2006**, 312, 1180–1182. [Google Scholar] [CrossRef] [PubMed] - Saaidi, K.; Sheikhahmadi, H.; Mohammadi, A.H. Interacting New Agegraphic Dark Energy in a Cyclic Universe. Astrophys. Space Sci.
**2012**, 338, 355–361. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Saez-Gomez, D. Cyclic, ekpyrotic and little rip universe in modified gravity. AIP Conf. Proc.
**2011**, 1458, 207–221. [Google Scholar] - Cai, Y.F.; Saridakis, E.N. Non-singular Cyclic Cosmology without Phantom Menace. J. Cosmol.
**2011**, 17, 7238–7254. [Google Scholar] - Sahni, V.; Toporensky, A. Cosmological Hysteresis and the Cyclic Universe. Phys. Rev. D
**2012**. [Google Scholar] [CrossRef] - Chung, D.Y. The Cyclic universe. 2001; arXiv:physics/0105064. [Google Scholar]
- Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. The ekpyrotic universe: Colliding branes and the origin of the hot big bang. Phys. Rev. D
**2001**. [Google Scholar] [CrossRef] - Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. A brief comment on ’The pyrotechnic universe’. 2001; arXiv:hep-th/0105212. [Google Scholar]
- Donagi, R.Y.; Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. Visible branes with negative tension in heterotic M-theory. JHEP
**2001**, 0111, 041. [Google Scholar] [CrossRef] - Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. Density perturbations in the ekpyrotic scenario. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Page, D. N. A Fractal Set of Perpetually Bouncing Universes? Class. Quant. Grav.
**1984**, 1, 417–427. [Google Scholar] [CrossRef] - Peter, P.; Pinto-Neto, N. Has the Universe always expanded? Phys. Rev. D
**2001**. [Google Scholar] [CrossRef] - Peter, P.; Pinto-Neto, N. Primordial perturbations in a non singular bouncing universe model. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Shtanov, Y.; Sahni, V. Bouncing braneworlds. Phys. Lett. B
**2003**, 557, 1–6. [Google Scholar] [CrossRef] - Biswas, T.; Mazumdar, A.; Siegel, W. Bouncing universes in string-inspired gravity. J. Cosmol. Astropart. Phys.
**2006**, 0603, 009. [Google Scholar] [CrossRef] - Cai, Y.F.; Qiu, T.; Piao, Y.S.; Li, M.; Zhang, X. Bouncing Universe with Quintom Matter. J. High Energ. Phys.
**2007**, 0710, 071. [Google Scholar] [CrossRef] - Creminelli, P.; Senatore, L. A smooth bouncing cosmology with scale invariant spectrum. J. Cosmol. Astropart. Phys.
**2007**. [Google Scholar] [CrossRef] - Piao, Y.S. Proliferation in Cycle. Phys. Lett. B
**2009**, 677, 1–5. [Google Scholar] [CrossRef] - Zhang, J.; Liu, Z.G.; Piao, Y.S. Amplification of curvature perturbations in cyclic cosmology. Phys. Rev. D
**2010**. [Google Scholar] [CrossRef] - Piao, Y.S. Design of a Cyclic Multiverse. Phys. Lett. B
**2010**, 691, 225–229. [Google Scholar] [CrossRef] - Liu, Z.G.; Piao, Y.S. Scalar Perturbations Through Cycles. Phys. Rev. D
**2012**. [Google Scholar] [CrossRef] - Novello, M.; Bergliaffa, S.E.P. Bouncing Cosmologies. Phys. Rept.
**2008**, 463, 127–213. [Google Scholar] [CrossRef] - Myrzakulov, R. F(T) gravity and k-essence. To appear in Gen. Relativ. Gravit.
**2012**. [Google Scholar] [CrossRef] - Myrzakulov, R. Knot Universes in Bianchi Type I Cosmology. Advances in High Energy Physics
**2012**, 2012, 868203. [Google Scholar] [CrossRef] - Esmakhanova, K.; Myrzakulov, Y.; Nugmanova, G.; Myrzakulov, R. A note on the relationship between solutions of Einstein, Ramanujan and Chazy equations. Int. J. Theor. Phys.
**2012**, 51, 1204–1210. [Google Scholar] [CrossRef] - Yesmakhanova, K.R.; Myrzakulov, N.A.; Yerzhanov, K.K.; Nugmanova, G.N.; Serikbayaev, N.S.; Myrzakulov, R. Some Models of Cyclic and Knot Universes. 2012; arXiv:1201.4360 [physics.gen-ph]. [Google Scholar]
- Gibbons, G.W.; Vyska, M. The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics. Class. Quant. Grav.
**2012**, 29, 065016. [Google Scholar] [CrossRef] - Bochicchio, I.; Capozziello, S.; Laserra, E. The Weierstrass Criterion and the Lemaitre-Tolman-Bondi Models with Cosmological Constant Λ. Int. J. Geom. Meth. Mod. Phys.
**2011**, 8, 1653–1666. [Google Scholar] [CrossRef] - Dimitrov, B.G. Cubic algebraic equations in gravity theory, parametrization with the Weierstrass function and non-arithmetic theory of algebraic equations. J. Math. Phys.
**2003**, 44, 2542–2578. [Google Scholar] [CrossRef] [Green Version] - D’Ambroise, J. Applications of Elliptic and Theta Functions to Friedmann-Robertson-Lemaitre-Walker Cosmology with Cosmological Constant. 2009; arXiv:0908.2481 [gr-qc]. [Google Scholar]
- Bouhmadi-Lopez, M.; Garay, L.J.; Gonzalez-Diaz, P.F. Quantum behavior of FRW radiation-filled universes. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Bamba, K.; Yesmakhanova, K.; Yerzhanov, K.; Myrzakulov, R. Reconstruction of the equation of state for the cyclic universes in homogeneous and isotropic cosmology. 2012; arXiv:1203.3401 [gr-qc]. [Google Scholar] [CrossRef]
- Nojiri, S.; Odintsov, S.D. Inhomogeneous equation of state of the universe: Phantom era, future singularity and crossing the phantom barrier. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Stefancic, H. ’Expansion’ around the vacuum equation of state: Sudden future singularities and asymptotic behavior. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Kamenshchik, A.Y.; Moschella, U.; Pasquier, V. An Alternative to quintessence. Phys. Lett. B
**2001**, 511, 265–268. [Google Scholar] [CrossRef] - Bento, M.C.; Bertolami, O.; Sen, A.A. Generalized Chaplygin gas, accelerated expansion and dark energy matter unification. Phys. Rev. D
**2002**. [Google Scholar] [CrossRef] - Benaoum, H.B. Accelerated universe from modified Chaplygin gas and tachyonic fluid. 2002; arXiv:hep-th/0205140. [Google Scholar]
- Saez-Gomez, D. Oscillating Universe from inhomogeneous EoS and coupled dark energy. Grav. Cosmol.
**2009**, 15, 134–140. [Google Scholar] [CrossRef] - Sharif, M.; Yesmakhanova, K.R.; Rani, S.; Myrzakulov, R. Solvable K-essence Cosmologies and Modified Chaplygin Gas Unified Models of Dark Energy and Dark Matter. Eur. Phys. J. C
**2012**, 72, 2067. [Google Scholar] [CrossRef] - Bamba, K.; Nojiri, S.; Odintsov, S.D.; Sasaki, M. Screening of cosmological constant for De Sitter Universe in non-local gravity, phantom-divide crossing and finite-time future singularities. Gen. Relativ. Gravit.
**2012**, 44, 1321–1356. [Google Scholar] [CrossRef] - Alam, U.; Sahni, V.; Starobinsky, A.A. The case for dynamical dark energy revisited. J. Cosmol. Astropart. Phys.
**2004**, 0406, 008. [Google Scholar] [CrossRef] - Alam, U.; Sahni, V.; Starobinsky, A.A. Exploring the Properties of Dark Energy Using Type Ia Supernovae and Other Datasets. J. Cosmol. Astropart. Phys.
**2007**, 0702, 011. [Google Scholar] [CrossRef] - Nesseris, S.; Perivolaropoulos, L. Crossing the Phantom Divide: Theoretical Implications and Observational Status. J. Cosmol. Astropart. Phys.
**2007**, 0701, 018. [Google Scholar] [CrossRef] - Wu, P.U.; Yu, H.W. Constraints on a variable dark energy model with recent observations. Phys. Lett. B
**2006**, 643, 315–318. [Google Scholar] [CrossRef] - Jassal, H.K.; Bagla, J.S.; Padmanabhan, T. Understanding the origin of CMB constraints on Dark Energy. Mon. Not. Roy. Astron. Soc.
**2010**, 405, 2639–2650. [Google Scholar] [CrossRef] - Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Abramowitz, M.; Stegun, l.A. (Eds.) Dover Publications: New York, NY, USA, 1965; pp. 1–1046.
- Zhao, G.B.; Xia, J.Q.; Li, M.; Feng, B.; Zhang, X. Perturbations of the quintom models of dark energy and the effects on observations. Phys. Rev. D
**2005**. [Google Scholar] [CrossRef] - Chiba, T.; Okabe, T.; Yamaguchi, M. Kinetically driven quintessence. Phys. Rev. D
**2000**. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Damour, T.; Mukhanov, V.F. k-Inflation. Phys. Lett. B
**1999**, 458, 209–218. [Google Scholar] [CrossRef] - Garriga, J.; Mukhanov, V.F. Perturbations in k-inflation. Phys. Lett. B
**1999**, 458, 219–225. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Mukhanov, V.F.; Steinhardt, P.J. A dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration. Phys. Rev. Lett.
**2000**, 85, 4438–4441. [Google Scholar] [CrossRef] [PubMed] - Armendariz-Picon, C.; Mukhanov, V.F.; Steinhardt, P.J. Essentials of k-essence. Phys. Rev. D
**2001**. [Google Scholar] [CrossRef] - de Putter, R.; Linder, E.V. Kinetic k-essence and Quintessence. Astropart. Phys.
**2007**, 28, 263–272. [Google Scholar] [CrossRef] - Nicolis, A.; Rattazzi, R.; Trincherini, E. The Galileon as a local modification of gravity. Phys. Rev. D
**2009**. [Google Scholar] [CrossRef] - Deffayet, C.; Esposito-Farese, G.; Vikman, A. Covariant Galileon. Phys. Rev. D
**2009**. [Google Scholar] [CrossRef] - Deffayet, C.; Deser, S.; Esposito-Farese, G. Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors. Phys. Rev. D
**2009**, 80. [Google Scholar] [CrossRef] - Deffayet, C.; Deser, S.; Esposito-Farese, G. Arbitrary p-form Galileons. Phys. Rev. D
**2010**, 82, 061501-1–061501-5. [Google Scholar] [CrossRef] - Shirai, N.; Bamba, K.; Kumekawa, S.; Matsumoto, J.; Nojiri, S. Generalized Galileon model: Cosmological reconstruction and the Vainshtein mechanism. Phys. Rev. D
**2012**. [Google Scholar] [CrossRef] - Planck Science Team Home. Available online: http://www.sciops.esa.int/index.php?project=PL-ANCK (accessed on 17th November, 2012).

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**MDPI and ACS Style**

Bamba, K.; Debnath, U.; Yesmakhanova, K.; Tsyba, P.; Nugmanova, G.; Myrzakulov, R.
Periodic Cosmological Evolutions of Equation of State for Dark Energy . *Entropy* **2012**, *14*, 2351-2374.
https://doi.org/10.3390/e14112351

**AMA Style**

Bamba K, Debnath U, Yesmakhanova K, Tsyba P, Nugmanova G, Myrzakulov R.
Periodic Cosmological Evolutions of Equation of State for Dark Energy . *Entropy*. 2012; 14(11):2351-2374.
https://doi.org/10.3390/e14112351

**Chicago/Turabian Style**

Bamba, Kazuharu, Ujjal Debnath, Kuralay Yesmakhanova, Petr Tsyba, Gulgasyl Nugmanova, and Ratbay Myrzakulov.
2012. "Periodic Cosmological Evolutions of Equation of State for Dark Energy " *Entropy* 14, no. 11: 2351-2374.
https://doi.org/10.3390/e14112351