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

Kleinert, Hagen; Schmidt, Hans-Jürgen

Gen. Relativ. Gravitation 34, No. 8, 1295-1318 (2002).

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.

MSC:

83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83F05	Relativistic cosmology

Keywords:

cosmology; effective gravitational constant; fourth-order gravity

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