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.
