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 power-law potential $V=\frac{K}{\ensuremath{\gamma}\ensuremath{-}1}(\frac{m}{\ensuremath{\hbar}}{)}^{2\ensuremath{\gamma}}|\ensuremath{\varphi}{|}^{2\ensuremath{\gamma}}$ associated polytropic equation of state $P=K{\ensuremath{\rho}}^{\ensuremath{\gamma}}$ (the an isothermal $P=\ensuremath{\rho}{k}_{B}T/m$ is $V=\frac{m{k}_{B}T}{{\ensuremath{\hbar}}^{2}}|\ensuremath{\varphi}{|}^{2}[\mathrm{ln}({m}^{2}|\ensuremath{\varphi}{|}^{2}/{\ensuremath{\rho}}_{*}{\ensuremath{\hbar}}^{2})\ensuremath{-}1]$ and the logotropic $P=A\mathrm{ln}(\ensuremath{\rho}/{\ensuremath{\rho}}_{P})$ $V=\ensuremath{-}A[\mathrm{ln}({m}^{2}|\ensuremath{\varphi}{|}^{2}/{\ensuremath{\hbar}}^{2}{\ensuremath{\rho}}_{P})+1]$). consider fast oscillation regime ``spintessence'' where equations problem can be simplified. study all possible cases arbitrary (positive negative) values constant index. The $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model, Chaplygin gas Bose-Einstein condensate model are recovered as particular our corresponding to ($\ensuremath{\gamma}=0$), inverse square-law ($\ensuremath{\gamma}=\ensuremath{-}1$), quartic ($\ensuremath{\gamma}=2$). also derive two-fluid representation model.