Cosmology with curvature-saturated gravitational Lagrangian \(R/\sqrt{1 + l^4 R^2}\). (English) Zbl 1019.83010

Summary: We argue that the Lagrangian for gravity should remain bounded at large curvature, and interpolate between the weak-field tested Einstein-Hilbert Lagrangian \({\mathcal L}_{EH} = R/16\pi G\) and a pure cosmological constant for large \(R\) with the curvature-saturated ansatz \({\mathcal L}_{cs} = {\mathcal L}_{EH}/\sqrt{1+l^4 R^2}\), where \(l\) is a length parameter expected to be a few orders of magnitude above the Planck length. The curvature-dependent effective gravitational constant defined by \(d{\mathcal L}/dR = 1/16\pi G_{eff}\) is \(G_{eff} = G{\sqrt{1+l^4 R^2}}^3\), and tends to infinity for large \(R\), in contrast to most other approaches where \(G_{eff}\to 0\). The theory possesses neither ghosts nor tachyons, but it fails to be linearization stable. In a curvature saturated cosmology, the coordinates with \(ds^2 = a^2 [da^2/B(a) - dx^2 - dy^2 - dz^2]\) are most convenient since the curvature scalar becomes a linear function of \(B(a)\). Cosmological solutions with a singularity of type \(R\to \pm\infty\) are possible which have a bounded energy-momentum tensor everywhere; such a behaviour is excluded in Einstein’s theory. In synchronized time, the metric is given by \[ ds^2 = dt^2 - t^{6/5}(dx^2 + dy^2+ dz^2). \] On the technical side we show that two different conformal transformations make \({\mathcal L}_{cs}\) asymptotically equivalent to the Gurovich-ansatz \({\mathcal L} = R ^{4/3}\) on the one hand, and to Einstein’s theory with a minimally coupled scalar field with self-interaction on the other.


83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83F05 Relativistic cosmology
Full Text: DOI arXiv


[1] A. Sakharov, (1967
[2] S. Adler,
[3] See, for example J. Hwang, H. N
[4] H.-J. Schmidt, Comparing selfinteracting · Zbl 0614.53071
[5] H.-J. Schmidt, New exact solutions for power-law inflation Friedmann models, gr-qc/0109004 · Zbl 0697.53068
[6] V. Müller and H.-J. Schmidt, · Zbl 0564.53037
[7] H.-J. Schmidt, Stability and Hamiltonian formulation of higher derivative theo
[8] H.-J. Schmidt, A new duality transformation for fourth-order gravity, · Zbl 0883.53071
[9] S. Hawking, T. Hertog, and H. Rea
[10] A. Starobinsk · Zbl 1371.83222
[11] V. Sahni and A. Starobinsky, (200
[12] H. Kleinert and A. Zhuk; unpublished Preprint 1993; Finite-Size and Temperature properties of Matter and Radiation fluctuations in Closed Friedmann Universe; H. Kleinert and A. Zhuk, (1
[13] V. Gurovich,
[14] V. Gurovich, I. Tokareva, · Zbl 1071.83526
[15] J. Fabris, S. Reuter, Quantum cosmology in higher derivative and scalar-tensor gravity, · Zbl 1169.83305
[16] Y. Anini, p. 183 in: Current topics in mathematical cosmology, Eds.: M. Rainer, H.-J. Schmidt (WSPC Singapore 1998); R. Brandenberger, V. Mukhanov, A. Sornborg
[17] K. Stel
[18] L. Buchbinder, S. Odintsov, I. Shapiro, (1992). Effective action in quantum gravity, IOP Bristol.
[19] P. Gilkey · Zbl 0316.53035
[20] I. Jack, D. Jones, N. Mohammed
[21] I. Prigogine, J. Geheniau, E. Gunzig, P. Nardone, · Zbl 0668.53062
[22] G. Siemieniec-Ozieblo, A. Woszczyna, Order-reduction in semiclassical cosmo
[23] L. Parker, A. Raval, Nonperturbative effects of vacuum energy on the recent expansion of the univer
[24] H. Kleinert, · Zbl 1052.81572
[25] G. Bick
[26] Yi-Ping Qin, Preprint gr-qc/0005054.
[27] K. Stelle,
[28] L. Bruna and J. Girbau, Linearization stability of the Einstein equation for Robertson-Walker models · Zbl 0982.83008
[29] D. Anselmi, (20 · Zbl 0969.83023
[30] S. Capozziello, G. Lambiase, H.-J. Schmidt, gr-qc/0101090, p. 697 in ”Fluctuating Paths and Fields,” WSPC Singapore 2001, Eds: W. Janke, A. Pelster, H.-J. Schmidt, M. Bachmann.
[31] M. Nowakowski, (200 · Zbl 1155.83325
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.