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Cosmological models based on a complex scalar field with a power-law potential associated with a polytropic equation of state

We construct cosmological models based on a complex scalar field with a power-law potential $V=\frac{K}{γ-1}(\frac{m}{\hbar})^{2γ}|φ|^{2γ}$ associated with a polytropic equation of state $P=Kρ^γ$ (the potential associated with an isothermal equation of state $P=ρk_B T/m$ is $V=\frac{m k_B T}{\hbar^2}|φ|^2 [\ln(m^2|φ|^2/ρ_*\hbar^2)-1]$ and the potential associated with a logotropic equation of state $P=A\ln(ρ/ρ_P)$ is $V=-A[\ln(m^2|φ|^2/\hbar^2ρ_P)+1]$). We consider a fast oscillation regime of ``spintessence'' where the equations of the problem can be simplified. We study all possible cases with arbitrary (positive and negative) values of the polytropic constant and polytropic index. The $Λ$CDM model, the Chaplygin gas model and the Bose-Einstein condensate model are recovered as particular cases of our study corresponding to a constant potential ($γ=0$), an inverse square-law potential ($γ=-1$), and a quartic potential ($γ=2$). We also derive the two-fluid representation of the Chaplygin gas model.