  We study the evolution of matter density perturbations in Galileon cosmology
where the late-time cosmic acceleration can be realized by a field kinetic
energy. We obtain full perturbation equations at linear order in the presence
of five covariant Lagrangians ${cal L}_i$ ($i=1,...,5$) satisfying a Galilean
symmetry in the flat space-time. The equations for a matter perturbation as
well as an effective gravitational potential are derived under a quasi-static
approximation on sub-horizon scales. This approximation can reproduce full
numerical solutions with high accuracy for the wavelengths relevant to
large-scale structures. For the model parameters constrained by the background
expansion history of the Universe the growth rate of matter perturbations is
larger than that in the LCDM model, with the growth index $gamma$ today
typically smaller than 0.4. We also find that, even on very large scales
associated with the Integrated-Sachs-Wolfe (ISW) effect in Cosmic Microwave
Background (CMB) temperature anisotropies, the effective gravitational
potential exhibits a temporal growth during the transition from the matter era
to the epoch of cosmic acceleration. These properties are useful to distinguish
the Galileon model from the LCDM in future high-precision observations.
