Extended family of generalized Chaplygin gas models

The generalized Chaplygin gas is usually defined as a barotropic perfect fluid with an equation of state $p=-A \rho^{-\alpha}$, where $\rho$ and $p$ are the proper energy density and pressure, respectively, and $A$ and $\alpha$ are positive real parameters. It has been extensively studied in the literature as a quartessence prototype unifying dark matter and dark energy. Here, we consider an extended family of generalized Chaplygin gas models parameterized by three positive real parameters $A$, $\alpha$ and $\beta$, which, for two specific choices of $\beta$ [$\beta=1$ and $\beta=\left(1+\alpha\right)/(2\alpha)$], is described by two different Lagrangians previously identified in the literature with the generalized Chaplygin gas. We show that, for $\beta > 1/2$, the linear stability conditions and the maximum value of the sound speed $c_s$ are regulated solely by $\beta$, with $0 \le c_s \le 1$ if $\beta \ge 1$. We further demonstrate that in the non-relativistic regime the standard equation of state $p=-A \rho^{-\alpha}$ of the generalized Chaplygin gas is always recovered, while in the relativistic regime this is true only if $\beta=\left(1+\alpha\right)/(2\alpha)$. We present a regularization of the ($\alpha\rightarrow 0$, $A \rightarrow \infty$) limit of the generalized Chaplygin gas, showing that it leads to a logarithmic Chaplygin gas model with an equation of state of the form $p = {\mathcal A} \ln\left(\rho/\rho_{*}\right)$, where ${\mathcal A}$ is a real parameter and $\rho_*>0$ is an arbitrary energy density. We finally derive its Lagrangian formulation.