Abstract
The bouncing evolution of an universe in Loop Quantum Cosmology can be described very well by a set of effective equations, involving a function sin x. Recently, we have generalised these effective equations to \((d + 1)\) dimensions and to any function f(x). Depending on f(x) in these models inspired by Loop Quantum Cosmology, a variety of cosmological evolutions are possible, singular as well as non singular. In this paper, we study them in detail. Among other things, we find that the scale factor \(a(t) \propto t^{ \frac{2 q}{(2 q - 1) (1 + w) d}}\) for \(f(x) = x^q\), and find explicit Kasner-type solutions if \(w = 2 q - 1 \) also. A result which we find particularly fascinating is that, for \(f(x) = \sqrt{x}\), the evolution is non singular and the scale factor a(t) grows exponentially at a rate set, not by a constant density, but by a quantum parameter related to the area quantum.
Similar content being viewed by others
References
Bojowald, M.: Absence of singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227 (2001). doi:10.1103/PhysRevLett.86.5227. [arXiv:gr-qc/0102069]
Bojowald, M.: The Inverse scale factor in isotropic quantum geometry. Phys. Rev. D 64, 084018 (2001). doi:10.1103/PhysRevD.64.084018. [arXiv:gr-qc/0105067]
Bojowald, M.: Isotropic loop quantum cosmology. Class. Quantum Gravity 19, 2717 (2002). doi:10.1088/0264-9381/19/10/313. [arXiv:gr-qc/0202077]
Bojowald, M.: Homogeneous loop quantum cosmology. Class. Quantum Gravity 20, 2595 (2003). doi:10.1088/0264-9381/20/13/310. [arXiv:gr-qc/0303073]
Ashtekar, A., Bojowald, M., Lewandowski, J.: Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys. 7, 233 (2003). doi:10.4310/ATMP.2003.v7.n2.a2. [arXiv:gr-qc/0304074]
Ashtekar, A., Pawlowski, T., Singh, P.: Quantum nature of the big bang. Phys. Rev. Lett. 96, 141301 (2006). doi:10.1103/PhysRevLett.96.141301. [arXiv:gr-qc/0602086]
Ashtekar, A., Pawlowski, T., Singh, P.: Quantum nature of the big bang: improved dynamics. Phys. Rev. D 74, 084003 (2006). doi:10.1103/PhysRevD.74.084003. [arXiv:gr-qc/0607039]
Ashtekar, A., Singh, P.: Loop quantum cosmology: a status report. Class. Quantum Gravity 28, 213001 (2011). doi:10.1088/0264-9381/28/21/213001. arXiv:1004.2952 [gr-qc]
Date, G.: Lectures on LQG/LQC. arXiv:1004.2952 [gr-qc]
Ashtekar, A.: New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244 (1986). doi:10.1103/PhysRevLett.57.2244
Ashtekar, A.: New Hamiltonian formulation of general relativity. Phys. Rev. D 36, 1587 (1987). doi:10.1103/PhysRevD.36.1587
Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53 (2004). doi:10.1088/0264-9381/21/15/R01. [arXiv:gr-qc/0404018]
Ashtekar, A.: Lectures on non-perturbative canonical gravity. Notes prepared in collaboration with R. S. Tate, World Scientific, Singapore (1991)
Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)
Thiemann, T.: Introduction to Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2005)
Rovelli, C., Vidotto, F.: Covariant Loop Quantum Gravity. Cambridge University Press, Cambridge (2014)
Date, G.: Absence of the Kasner singularity in the effective dynamics from loop quantum cosmology. Phys. Rev. D 71, 127502 (2005). doi:10.1103/PhysRevD.71.127502. [arXiv:gr-qc/0505002]
Ashtekar, A., Wilson-Ewing, E.: Loop quantum cosmology of Bianchi I models. Phys. Rev. D 79, 083535 (2009). doi:10.1103/PhysRevD.79.083535. [arXiv:0903.3397 [gr-qc]]
Linsefors, L., Barrau, A.: Modified Friedmann equation and survey of solutions in effective Bianchi-I loop quantum cosmology. Class. Quantum Gravity 31, 015018 (2014). doi:10.1088/0264-9381/31/1/015018. [arXiv:1305.4516 [gr-qc]]
Diener, P., Joe, A., Megevand, M., Singh, P.: Numerical simulations of loop quantum Bianchi-I spacetimes. Class. Quantum Gravity 34, 094004 (2017). doi:10.1088/1361-6382/aa68b5. [arXiv:1701.05824 [gr-qc]]
Kalyana Rama, S.: A class of LQC–inspired models for homogeneous, anisotropic cosmology in higher dimensional early universe. Gen. Relativ. Gravit. 48, 155 (2016). doi:10.1007/s10714-016-2150-2. arXiv:1608.03231 [gr-qc]
Bodendorfer, N., Thiemann, T., Thurn, A.: New variables for classical and quantum gravity in all dimensions I. Hamiltonian analysis. Class. Quantum Gravity 30, 045001 (2013). doi:10.1088/0264-9381/30/4/045001. [arXiv:1105.3703 [gr-qc]]
Bodendorfer, N., Thiemann, T., Thurn, A.: New variables for classical and quantum gravity in all dimensions II. Lagrangian analysis. Class. Quantum Gravity 30, 045002 (2013). doi:10.1088/0264-9381/30/4/045002. [arXiv:1105.3704 [gr-qc]]
Bodendorfer, N., Thiemann, T., Thurn, A.: New variables for classical and quantum gravity in all dimensions III. Quantum theory. Class. Quantum Gravity 30, 045003 (2013). doi:10.1088/0264-9381/30/4/045003. [arXiv:1105.3705 [gr-qc]]
Kalyana Rama, S., Priya Saha, A.: Unpublished notes
Zhang, X.: Higher dimensional loop quantum cosmology. Eur. Phys. J. C 76(7), 395 (2016). doi:10.1140/epjc/s10052-016-4249-8. [arXiv:1506.05597 [gr-qc]]
Mielczarek, J.: Multi-fluid potential in the loop cosmology. Phys. Lett. B 675, 273 (2009). doi:10.1016/j.physletb.2009.04.034. arXiv:0809.2469 [gr-qc]
Wilson-Ewing, E.: The matter bounce scenario in loop quantum cosmology. JCAP 03, 026 (2013). doi:10.1088/1475-7516/2013/03/026. arXiv:1211.6269 [gr-qc]
Bowick, M.J., Wijewardhana, L.C.R.: Superstrings at high temperature. Phys. Rev. Lett. 54, 2485 (1985). doi:10.1103/PhysRevLett.54.2485
Bowick, M.J., Wijewardhana, L.C.R.: Superstring gravity and the early universe. Gen. Relativ. Gravit. 18, 59 (1986). doi:10.1007/BF00843749
Brandenberger, R.H., Vafa, C.: Superstrings in the early universe. Nucl. Phys. B 316, 391 (1989). doi:10.1016/0550-3213(89)90037-0
Tseytlin, A.A., Vafa, C.: Elements of string cosmology. Nucl. Phys. B 372, 443 (1992). doi:10.1016/0550-3213(92)90327-8. [arXiv:hep-th/9109048]
Veneziano, G.: A model for the big bounce. JCAP 03, 004 (2004). doi:10.1088/1475-7516/2004/03/004. [arXiv:hep-th/0312182]
Nayeri, A., Brandenberger, R.H., Vafa, C.: Producing a scale-invariant spectrum of perturbations in a Hagedorn phase of string cosmology. Phys. Rev. Lett. 97, 021302 (2006). doi:10.1103/PhysRevLett.97.021302. [arXiv:hep-th/0511140]
Kalyana Rama, S.: A stringy correspondence principle in cosmology. Phys. Lett. B 638, 100 (2006). doi:10.1016/j.physletb.2006.05.047. [arXiv:hep-th/0603216]
Kalyana Rama, S.: A principle to determine the number (3 + 1) of large spacetime dimensions. Phys. Lett. B 645, 365 (2007). doi:10.1016/j.physletb.2006.11.077. [arXiv:hep-th/0610071]
Bodendorfer, N.: Black hole entropy from loop quantum gravity in higher dimensions. Phys. Lett. B 726, 887 (2013). doi:10.1016/j.physletb.2013.09.043. [arXiv:1307.5029 [gr-qc]]
Chowdhury, B.D., Mathur, S.D.: Fractional brane state in the early universe. Class. Quantum Gravity 24, 2689 (2007). doi:10.1088/0264-9381/24/10/014. [arXiv:hep-th/0611330]
Kalyana Rama, S.: Entropy of anisotropic universe and fractional branes. Gen. Relativ. Gravit. 39, 1773 (2007). doi:10.1007/s10714-007-0488-1. [arXiv:hep-th/0702202 [hep-th]]
Kalyana Rama, S.: Consequences of U dualities for intersecting branes in the Universe. Phys. Lett. B 656, 226 (2007). doi:10.1016/j.physletb.2007.09.069. [arXiv:0707.1421 [hep-th]]
Bhowmick, S., Kalyana Rama, S.: 10 + 1 to 3 + 1 in an early universe with mutually BPS intersecting branes. Phys. Rev. D 82, 083526 (2010). doi:10.1103/PhysRevD.82.083526. [arXiv:1007.0205 [hep-th]]
Olmo, G .J., Singh, P.: Effective action for loop quantum cosmology a la Palatini. JCAP 01, 030 (2009). doi:10.1088/1475-7516/2009/01/030. arXiv:0806.2783 [gr-qc]
Sotiriou, T .P.: Covariant effective action for loop quantum cosmology from order reduction. Phys. Rev. D 79, 044035 (2009). doi:10.1103/PhysRevD.79.044035. [arXiv:0811.1799 [gr-qc]]
Date, G., Sengupta, S.: Effective actions from loop quantum cosmology: correspondence with higher curvature gravity. Class. Quantum Gravity 26, 105002 (2009). doi:10.1088/0264-9381/26/10/105002. [arXiv:0811.4023 [gr-qc]]
Barragan, C., Olmo, G .J., Sanchis-Alepuz, H.: Bouncing cosmologies in Palatini f(R) gravity. Phys. Rev. D 80, 024016 (2009). doi:10.1103/PhysRevD.80.024016. arXiv:0907.0318 [gr-qc]
Helling, R.C.: Higher curvature counter terms cause the bounce in loop cosmology. arXiv:0912.3011 [gr-qc]
Barragan, C., Olmo, G .J.: Isotropic and anisotropic bouncing cosmologies in Palatini gravity. Phys. Rev. D 82, 084015 (2010). doi:10.1103/PhysRevD.82.084015. arXiv:1005.4136 [gr-qc]
Acknowledgements
We thank G. Date for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rama, S.K. Variety of \((d + 1)\) dimensional cosmological evolutions with and without bounce in a class of LQC-inspired models. Gen Relativ Gravit 49, 113 (2017). https://doi.org/10.1007/s10714-017-2277-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-017-2277-9