Matter power spectrum in a power-law $f(G)$ gravity

Cosmological models based on $f(G)$ gravity are efficient in fitting different observational datasets at both background and perturbation levels. This motivates the current study to take into account dynamical system analysis to investigate the matter power spectrum within the framework of modified Gauss-Bonnet gravity. After defining the dimensionless dynamical system variables for a power-law $f(G)$ model, We derive the full system of equations governing the energy density perturbations for both matter and Gauss-Bonnet fluids using the $1+3$ covariant formalism. After solving the energy density perturbation equations, we compute the matter power spectrum. The importance of studying first order perturbations for the defined $f(G)$ model and the relevance of different initial conditions in computing the matter power spectrum are also stressed. It is reported that matter power spectrum for $f(G)$ gravity, for a particular functional form of $f(G)$ model considered is not scale invariant as the case for General Relativity.