Abstract
In this paper we present the cosmological dynamics of a perfect fluid and the Dark Energy component of the Universe, where our model of the dark energy is the string-theoritic Dirac-Born-Infeld (DBI) model. We assume that the potential of the scalar field and the warp factor of the warped throat region of the compact space in the extra dimension for the DBI model are both exponential in nature. In the background of spatially flat Friedman–Robertson–Walker–Lemaître Universe, the Einstein field equations for the DBI dark energy reduce to a system of autonomous dynamical system. We then perform a dynamical system analysis for this system. Our analysis is motivated by the invariant manifold approach of the mathematical dynamics. In this method, it is possible to reach a definite conclusion even when the critical points of a dynamical system are non-hyperbolic in nature. Since we find the complete set of critical points for this system, the center manifold analysis ensures that our investigation of this model leaves no stone unturned. We find some interesting results such as that for some critical points there are situations where scaling solutions exist. Finally we present various topologically different phase planes and stability diagrams and discuss the corresponding cosmological scenario.
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References
Reiss, A.J., Supernova Search Team, et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116(3), 1009–1038 (1998)
Perlmutter, S.J., Supernova Cosmology Project Collaboration, et al.: Measurements of Omega and Lambda from 42 high redshift supernovae. Astrophys. J. 517(2), 565–586 (1999)
Spergel, D.N., WMAP Collaboration, et al.: Three year Wilkinson microwave anisotropy probe (WMAP) observations: implications for cosmology. Astrophys. J. Suppl. Ser. 170(2), 377–408 (2007)
Komatsu, E., WMAP Collaboration, et al.: Five-year Wilkinson microwave anisotropy probe observations: cosmological interpretation. Astrophys. J. Suppl. Ser. 180(2), 330–376 (2009)
Percival, W.J., et al.: Measuring the Baryon acoustic oscillation scale using the Sloan digital sky survey and 2dF Galaxy Redshift Survey. Mon. Not. R. Astron. Soc. 381(3), 1053–1066 (2007)
Amendola, L., Tsujikawa, S.: Dark Energy: Theory and Observations. Cambridge University Press, Cambridge (2010)
Elijalde, E., Nojiri, S., Odintsov, S.D.: Late-time Cosmology in a (phantom) scalar-tensor theory: dark energy and the cosmic speed-up. Phys. Rev. D 70(4), 043539 (2004)
Silverstein, E., Tong, D.: Scalar speed limits and cosmology: acceleration from D-eceleration. Phys. Rev. D 70(10), 103505 (2004)
Guendelman, E., Singleton, D., Youngram, N.: A two measure model of dark energy and dark matter. JCAP 2004(07), 004 (2004)
Pavon, D., Wang, B.: Le Chatelier–Braun principle in cosmological physics. Gen. Rel. Grav. 41(1), 1–5 (2009)
Weinberg, S.: The cosmological constant problem. Rev. Mod. Phys. 61(1), 1–23 (1989)
Carroll, S.M.: Liv. Rev. Lett. 4, 1 (2001)
Armendariz-Picon, C., Mukhanov, V., Steinherdt, P.J.: Dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration. Phys. Rev. Lett. 85(21), 4438–4441 (2000)
Caldwell, R.R., Kamionkowski, M., Weinberg, N.N.: Causes a cosmic doomsday. Phys. Rev. Lett. 91(7), 071301 (2003)
Sen, A.: J. H. E. P 207, 65 (2002)
Copeland, E.J., Riddle, A.R., Wands, D.: Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57(8), 4686–4690 (1998)
Sahni, V., Starobinsky, A.: The case for a positive cosmological \(\Lambda \)- TERM. Int. J. Mod. Phys. D 09(04), 373–443 (2000)
Ratra, B., Peebles, P.J.E.: Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 37(12), 3406–3427 (1988)
Caldwell, R.R., Dave, R., Steinherdt, P.J.: Cosmological imprint of an energy component with general equation of state. Phys. Rev. Lett. 80(8), 1582–1585 (1998)
Zlatev, I., Wang, L., Steinhardt, P.J.: Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett. 82(5), 896–899 (1999)
Olivares, G., Atrio-Barandela, F., Pavon, D.: Observational constraints on interacting quintessence models. Phys. Rev. D 71(6), 063523 (2005)
Olivares, G., Atrio-Barandela, F., Pavon, D.: Metter density perturbations in interacting quintessence models. Phys. Rev. D 74(4), 043521 (2006)
Armendariz-Picon, C., Mukhanov, V., Steinherdt, P.J.: Essentials of k-essence. Phys. Rev. D 63(10), 103510 (2001)
Armendariz-Picon, C., Damour, T., Mukhanov, V.: K-Inflation. Phys. Lett. B 458(2–3), 209–218 (1999)
Caldwell, R.R.: A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Phys. Lett. B 545(1–2), 23–29 (2002)
Nojiri, S., Odintsov, S.D., Tsujikawa, S.: Properties of singularities in the (phantom) dark energy Universe. Phys. Rev. D 71(6), 063004 (2005)
Piazza, F., Tsujikawa, S.: Dilatonic ghost condensate as dark energy. JCAP 2004(07), 004 (2004)
Padmanabhan, T.: Accelerated expansion of the Universe driven by tachyonic matter. Phys. Rev. D 66(2), 021301 (2002)
Chen, X.: Multithroat brane inflation. Phys. Rev. D 71(6), 063506 (2005)
Chimento, Luis P., Lazkoz, Ruth, Richarte, Martin G.: Enhanced inflation in the Dirac-Born-Infeld framework. Phys. Rev. D 83(6), 063505 (2011)
Kaeonikhom, C., Singleton, D., Sushkov, S.V., Youngram, N.: Dynamics of Dirac-Born-Infeld dark energy interacting with dark matter. Phys. Rev. D 86(12), 124049 (2012)
Mahata, N., Chakraborty, S.: Dynamical system analysis for DBI dark energy interacting with dark matter. Mod. Phys. Lett. A 30(02), 1550009 (2015)
Copeland, E.J., Shuntaro, M., Shaeri, M.: Cosmological dynamics of a Dirac-Born-Infeld field. Phys. Rev. D 81(12), 123501 (2010)
Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1991)
Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, Berlin (2003)
Banerjee, A.S.: An Introduction to Center Manifold Theory
Hirsch, M., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York (1974)
Boehmer, C.G., Chan, N., Lazkoz, R.: Dynamics of dark energy models and centre manifolds. Phys. Lett. B 714, 11 (2012)
Copeland, E.J., Sami, M., Tsujikawa, S.: Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753 (2006)
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Pal, S., Chakraborty, S. Dynamical system analysis of a Dirac-Born-Infeld model: a center manifold perspective. Gen Relativ Gravit 51, 124 (2019). https://doi.org/10.1007/s10714-019-2608-0
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DOI: https://doi.org/10.1007/s10714-019-2608-0