Dynamical systems of modified Gauss–Bonnet gravity: cosmological implications

Abstract In this paper, we derive the field equations of modified Gauss–Bonnet gravity termed as f ( R , G ) gravity for the non-flat Friedmann–Robertson–Walker (FRW) spacetime. We utilize the dynamical system approach to study the cosmic dynamics of two different class of f ( R , G ) models composed of radiation and matter (cold dark matter and baryonic matter). The linear perturbations around the fixed points are studied to explore the corresponding stability of points. The cosmological implications are studied in $$f(R,G)=f_0R^nG^{1-n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and $$f(R,G)=f_0R^\alpha +f_1G^\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>β</mml:mi> </mml:msup> </mml:mrow> </mml:math> models to identify the qualitative evolution of universe with the flat-FRW spacetime. The qualitative differences between the considered class of models are discussed in detail. The fixed points corresponding to the late-time accelerated and radiation phase of the universe will exist in the model but, the existence of fixed point corresponding to the matter dominated phase will depend on the functional form of f ( R , G ). Furthermore, the autonomous systems are utilized to study the cosmographic parameters along with the statefinder diagnostic.