Polytropic equation of state and primordial quantum fluctuations

We study the primordial Universe in a cosmological model where inflation is driven by a fluid with a polytropic equation of state $p = \alpha\rho + k\rho^{1 + 1/n}$. We calculate the dynamics of the scalar factor and build a Universe with constant density at the origin. We also find the equivalent scalar field that could create such equation of state and calculate the corresponding slow-roll parameters. We calculate the scalar perturbations, the scalar power spectrum and the spectral index.