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 $V=\frac{K}{\gamma-1}(\frac{m}{\hbar})^{2\gamma}|\varphi|^{2\gamma}$ associated with a polytropic equation of state $P=K\rho^{\gamma}$ (the potential associated with an isothermal equation of state $P=\rho k_B T/m$ is $V=\frac{m k_B T}{\hbar^2}|\varphi|^2 [\ln(m^2|\varphi|^2/\rho_*\hbar^2)-1]$ and the potential associated with a logotropic equation of state $P=A\ln(\rho/\rho_P)$ is $V=-A[\ln(m^2|\varphi|^2/\hbar^2\rho_P)+1]$). 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 $\Lambda$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 ($\gamma=0$), an inverse square-law potential ($\gamma=-1$), and a quartic potential ($\gamma=2$). We also derive the two-fluid representation of the Chaplygin gas model.