Abstract
We explore whether generalised Brans–Dicke theories, which have a scalar field \(\Phi \) and a function \(\omega (\Phi )\), can be the effective actions leading to the effective equations of motion of the LQC and the LQC-inspired models, which have a massless scalar field \(\sigma \) and a function f(m). We find that this is possible for isotropic cosmology. We relate the pairs \((\sigma , f)\) and \((\Phi , \omega )\) and, using examples, illustrate these relations. We find that near the bounce of the LQC evolutions for which \(f(m) = sin \; m\), the corresponding field \(\Phi \rightarrow 0\) and the function \(\omega (\Phi ) \propto \Phi ^2\). We also find that the class of generalised Brans–Dicke theories, which we had found earlier to lead to non singular isotropic evolutions, may be written as an LQC-inspired model. The relations found here in the isotropic cases do not apply to the anisotropic cases, which perhaps require more general effective actions.
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Notes
In the following, the convention of summing over repeated indices is not always applicable. Hence we will write explicitly the indices to be summed over.
See the review [13] for a complete description of the various LQG/C terms and concepts mentioned here and in the following.
In LQC, \((3 + 1)\) dimensional effective actions have been constructed in [19,20,21,22,23,24] by generalising Einstein’s term R to F(R) and finding an appropriate function F which will give the isotropic LQC evolution. Any F(R)-theory, including that for LQC, can be written as a scalar–tensor theory with the Brans–Dicke constant \(\omega = 0\) and with a potential that depends on F [16,17,18]. Similar approach may also work for the present \((d + 1)\) dimensional LQC-inspired models for any arbitrary function \(f \;\), but we will not pursue it in this paper.
References
Ashtekar, A.: New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244 (1986). https://doi.org/10.1103/PhysRevLett.57.2244
Ashtekar, A.: New Hamiltonian formulation of general relativity. Phys. Rev. D 36, 1587 (1987). https://doi.org/10.1103/PhysRevD.36.1587
Bojowald, M.: Absence of singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227 (2001). https://doi.org/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). https://doi.org/10.1103/PhysRevD.64.084018. arXiv:gr-qc/0105067
Bojowald, M.: Isotropic loop quantum cosmology. Class. Quant. Gravit. 19, 2717 (2002). https://doi.org/10.1088/0264-9381/19/10/313. arXiv:gr-qc/0202077
Bojowald, M.: Homogeneous loop quantum cosmology. Class. Quant. Gravit. 20, 2595 (2003). https://doi.org/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). https://doi.org/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). https://doi.org/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). https://doi.org/10.1103/PhysRevD.74.084003. arXiv:gr-qc/0607039
Varadarajan, M.: On the resolution of the big bang singularity in isotropic loop quantum cosmology. Class. Quant. Gravit. 26, 085006 (2009). https://doi.org/10.1088/0264-9381/26/8/085006. arXiv:0812.0272 [gr-qc]
Ashtekar, A., Wilson-Ewing, E.: Loop quantum cosmology of Bianchi I models. Phys. Rev. D 79, 083535 (2009). https://doi.org/10.1103/PhysRevD.79.083535. arXiv:0903.3397 [gr-qc]
Date, G.: Lectures on LQG/LQC. arXiv:1004.2952 [gr-qc]
Ashtekar, A., Singh, P.: Loop quantum cosmology: a status report. Class. Quant. Grav. 28, 213001 (2011). https://doi.org/10.1088/0264-9381/28/21/213001. arXiv:1108.0893 [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). https://doi.org/10.1007/s10714-016-2150-2. arXiv:1608.03231 [gr-qc]
Kalyana Rama, S.: 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. arXiv:1706.08220 [gr-qc]
Sotiriou, T.P., Faraoni, V.: \(f(R)\) theories of gravity. Rev. Mod. Phys. 82, 451 (2010). https://doi.org/10.1103/RevModPhys.82.451. arXiv:0805.1726 [gr-qc]
De Felice, A., Tsujikawa, S.: \(f(R)\) theories. Living Rev. Relativ. 13, 3 (2010). https://doi.org/10.12942/lrr-2010-3. arXiv:1002.4928 [gr-qc]
Nojiri, S., Odintsov, S.D., Oikonomou, V.K.: Modified gravity theories on a nutshell: inflation, bounce and late-time evolution. Phys. Rep. 692, 1 (2017). https://doi.org/10.1016/j.physrep.2017.06.001. arXiv:1705.11098 [gr-qc]
Olmo, G.J., Singh, P.: Effective action for loop quantum cosmology a la palatini. JCAP 01, 030 (2009). https://doi.org/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). https://doi.org/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. Quant. Gravit. 26, 105002 (2009). https://doi.org/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). https://doi.org/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). https://doi.org/10.1103/PhysRevD.82.084015. arXiv:1005.4136 [gr-qc]
Will, C.M.: Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge (1993)
Dicke, R.H.: Principle of equivalence and the weak interactions. Rev. Mod. Phys. 29, 363 (1957). https://doi.org/10.1103/RevModPhys.29.355
Dicke, R.H.: Gravitation without a principle of equivalence. Rev. Mod. Phys. 29, 363 (1957). https://doi.org/10.1103/RevModPhys.29.363
Dicke, R.H.: Mach’s principle and invariance under transformation of units. Phys. Rev. 125, 2163 (1962). https://doi.org/10.1103/PhysRev.125.2163
Brans, C., Dicke, R.H.: Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925 (1961). https://doi.org/10.1103/PhysRev.124.925
Damour, T., Polyakov, A.M.: The string dilaton and a least coupling principle. Nucl. Phys. B 423, 532 (1994). https://doi.org/10.1016/0550-3213(94)90143-0. arXiv:hep-th/9401069
Damour, T., Polyakov, A.M.: String theory and gravity. Gen. Relativ. Gravit. 26, 1171 (1994). https://doi.org/10.1007/BF02106709. arXiv:gr-qc/9411069
Husain, V., Pawlowski, T.: Time and a physical Hamiltonian for quantum gravity. Phys. Rev. Lett. 108, 141301 (2012). https://doi.org/10.1103/PhysRevLett.108.141301. arXiv:1108.1145 [gr-qc]
Husain, V., Pawlowski, T.: Dust reference frame in quantum cosmology. Class. Quant. Gravit. 28, 225014 (2011). https://doi.org/10.1088/0264-9381/28/22/225014. arXiv:1108.1147 [gr-qc]
Pawlowski, T., Pierini, R., Wilson-Ewing, E.: Loop quantum cosmology of a radiation-dominated flat FLRW universe. Phys. Rev. D 90, 123538 (2014). https://doi.org/10.1103/PhysRevD.90.123538. arXiv:1404.4036 [gr-qc]
Rama, S.K.: Some cosmological consequences of nontrivial PPN parameters beta and gamma. Phys. Lett. B 373, 282 (1996). https://doi.org/10.1016/0370-2693(96)00146-3. arXiv:hep-th/9506020
Rama, S.K.: Singularity free (homogeneous isotropic) universe in graviton-dilaton models. Phys. Rev. Lett. 78, 1620 (1997). https://doi.org/10.1103/PhysRevLett.78.1620. arXiv:hep-th/9608026
Rama, S.K.: Early universe evolution in graviton-dilaton models. Phys. Rev. D 56, 6230 (1997). https://doi.org/10.1103/PhysRevD.56.6230. arXiv:hep-th/9611223
Rama, S.K.: Can string theory avoid cosmological singularities? Phys. Lett. B 408, 91 (1997). https://doi.org/10.1016/S0370-2693(97)00795-8. arXiv:hep-th/9701154
Bagchi, A., Rama, S.K.: Cosmology and static spherically symmetric solutions in D-dimensional scalar tensor theories: some novel features. Phys. Rev. D 70, 104030 (2004). https://doi.org/10.1103/PhysRevD.70.104030. arXiv:gr-qc/0408030
Bodendorfer, N., Thiemann, T., Thurn, A.: New variables for classical and quantum gravity in all dimensions I. Hamiltonian analysis. Class. Quant. Gravit. 30, 045001 (2013). https://doi.org/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. Quant. Gravit. 30, 045002 (2013). https://doi.org/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. Quant. Gravit. 30, 045003 (2013). https://doi.org/10.1088/0264-9381/30/4/045003. arXiv:1105.3705 [gr-qc]
Bodendorfer, N.: Black hole entropy from loop quantum gravity in higher dimensions. Phys. Lett. B 726, 887 (2013). https://doi.org/10.1016/j.physletb.2013.09.043. arXiv:1307.5029 [gr-qc]
Rama, S.K., Ghosh, S.: Short distance repulsive gravity as a consequence of nontrivial PPN parameters beta and gamma. Phys. Lett. B 383, 31 (1996). https://doi.org/10.1016/0370-2693(96)00818-0
Rama, S.K., Ghosh, S.: Short distance repulsive gravity as a consequence of nontrivial PPN parameters beta and gamma. Phys. Lett. B 384, 50 (1996). https://doi.org/10.1016/0370-2693(96)00706-X. arXiv:hep-th/9505167
Chamseddine, A.H., Mukhanov, V.: Resolving cosmological singularities. JCAP 03, 009 (2017). https://doi.org/10.1088/1475-7516/2017/03/009. arXiv:1612.05860 [gr-qc]
Bodendorfer, N., Schäfer, A. Schliemann, J.: On the canonical structure of general relativity with a limiting curvature and its relation to loop quantum gravity. arXiv:1703.10670 [gr-qc]
Langlois, D., Liu, H., Noui, K., Wilson-Ewing, E.: Effective loop quantum cosmology as a higher-derivative scalar–tensor theory. Class. Quant. Grav. 34, 225004 (2017). https://doi.org/10.1088/1361-6382/aa8f2f. arXiv:1703.10812 [gr-qc]
Ben Achour, J., Lamy, F., Liu, H., Noui, K.: Non-singular black holes and the limiting curvature mechanism: a Hamiltonian perspective. arXiv:1712.03876 [gr-qc]
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Rama, S.K. Isotropic LQC and LQC-inspired models with a massless scalar field as generalised Brans–Dicke theories. Gen Relativ Gravit 50, 56 (2018). https://doi.org/10.1007/s10714-018-2378-0
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DOI: https://doi.org/10.1007/s10714-018-2378-0