  We investigate cosmological perturbations of f(G) gravity in the presence of
a scalar field. Using the 1 + 3 covariant formalism, we present the energy
overdensity perturbation equations responsible for large scale structure
formation. After applying harmonic decomposition method together with the
redshift transformation technique, we obtain the fully perturbed equations in
redshift space. The equations are solved to study the growth of matter
overdensities contrast with redshift. For both short- and long-wavelength
modes, we obtain numerical results for particular functional form f(G) models
and scalar field. We find that, for this choice the energy overdensity
perturbations decay with increase in redshift. However, for both short- and
long- wavelength modes, the perturbations which include amplitude effects due
to the f(G) models with a scalar field do differ remarkably from those in
Lambda CDM. The results reduce to GR results in the limit of f(G) tends to G
and in the absence of scalar field.
