Gauss-Bonnet lagrangian G ln G and cosmological exact solutions

TL;DR

This paper investigates the unique properties of the Gauss-Bonnet Lagrangian L = G ln G, deriving its field equations, solving them for flat Friedmann models, and highlighting its scale-invariance and distinction from other F(G) theories.

Contribution

It introduces the L = G ln G Lagrangian, derives its field equations, finds exact cosmological solutions, and demonstrates its unique scale-invariance among F(G) theories.

Findings

Derived closed-form solutions for flat Friedmann models.

Showed L = G ln G is the only non-G^r form with scale-invariance.

Established analogies with 2D f(R)-theories.

Abstract

For the lagrangian L = G ln G where G is the Gauss-Bonnet curvature scalar we deduce the field equation and solve it in closed form for 3-flat Friedman models using a statefinder parametrization. Further we show, that among all lagrangians F(G) this L is the only one not having the form G^r with a real constant r but possessing a scale-invariant field equation. This turns out to be one of its analogies to f(R)-theories in 2-dimensional space-time. In the appendix, we systematically list several formulas for the decomposition of the Riemann tensor in arbitrary dimensions n, which are applied in the main deduction for n=4.