Extended family of generalized Chaplygin gas models

TL;DR

This paper introduces an extended family of generalized Chaplygin gas models with three parameters, analyzing their stability, sound speed, and equations of state, including a new logarithmic model, and provides their Lagrangian formulations.

Contribution

It extends the generalized Chaplygin gas framework by adding a new parameter, explores stability conditions, and introduces a novel logarithmic model with a corresponding Lagrangian.

Findings

Stability and sound speed depend on the parameter , with bounds for  ;  ; in the relativistic regime, the standard EOS is recovered only for specific .

In the non-relativistic limit, the standard generalized Chaplygin gas equation of state is always recovered.

A new logarithmic Chaplygin gas model is derived as a limit case, with a specific equation of state and Lagrangian formulation.

Abstract

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.