On 1 + 3 covariant perturbations of the quasi-Newtonian space-time in modified Gauss-Bonnet gravity

The consideration of a 1 + 3 covariant approach to cold dark matter universe with no shear cosmological dust model with irrotational flows is developed in the context of f (G) gravity theory in the present study. This approach reveals the existence of integrability conditions which do not appear in non-covariant treatments. We constructed the integrability conditions in modified Gauss-Bonnet f (G) gravity basing on the constraints and propagation equations. These integrability conditions reveal the linearized silent nature of quasi-Newtonian models in f (G) gravity. Finally, the linear equations for the overdensity and velocity perturbations of the quasi-Newtonian space-time were constructed in the context of modified f (G) gravity. The application of harmonic decomposition and redshift transformation techniques to explore the behaviour of the overdensity and velocity perturbations using f (G) model were made. On the other hand we applied the quasi-static approximation to study the approximated solutions on small scales which helps to get both analytical and numerical results of the perturbation equations. The analysis of the energy overdensity and velocity perturbations for both short and long wavelength modes in a dust-Gauss-Bonnet fluids were done and we see that both energy overdensity and velocity perturbations decay with redshift for both modes. In the limits to {\Lambda}CDM , it means f (G) = G the considered f (G) model results coincide with {\Lambda}CDM .