## 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.

### MSC:

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

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