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The universe dominated by oscillating scalar with non-minimal derivative coupling to gravity

We study the expansion law of the universe dominated by the oscillating scalar field with non-minimal derivative coupling to gravity as G^{μν} \partial_μ ϕ\partial_ν ϕ. In this system the Hubble parameter oscillates with a frequency of the effective mass of the scalar field, which formerly caused a difficulty in analyzing how the universe expands. We find an analytical solution for power law potentials and interpret the solution in an intuitive way by using a new invariant of the system. As a result, we find marginally accelerated expansion for the quadratic potential and no accelerated expansion for the potential with higher power.