Tolman--Bondi--Lema\^\i tre Spacetime with a Generalised Chaplygin Gas

Abstract

The Tolman-Bondi-Lema\^itre type of inhomogeneous spacetime with generalised Chaplygin gas equation of state given by $p = -\frac{A}{\rho^{\alpha}}$ is investigated where $ \alpha$ is a constant. We get an inhomogeneous spacetime at early stage but at the late stage of universe the inhomogeneity disappear with suitable radial co-ordinate transformation. For the large scale factor our model behaves like $\Lambda$CDM type which is in accord with the recent WMAP studies. We have calculated $\frac{\partial \rho}{\partial r}$ and it is found to be negative for $\alpha > 0$ which is in agreement with the observational analysis. A striking difference with Chaplygin gas ($ \alpha = 1$) lies in the fact that with any suitable co-ordinate transformation our metric cannot be reduced to the Einstein-de Sitter type of homogeneous spacetime as is possible for the Chaplygin gas. We have also studied the effective deceleration parameter and find that the desired feature of \emph{flip} occurs at the late universe. It is seen that the flip time depends explicitly on $\alpha$. We also find that flip is not synchronous occurring earlier at the outer shells, thus offering a natural path against occurrence of wellknown shell crossing singularity. This is unlike the Tolman-Bondi case with perfect gas where one has to impose stringent external conditions to avoid this type of singularity. We further observe that if we adopt separation of variables method to solve the field equations the inhomogeneity in matter distribution disappears. The whole situation is later discussed with the help of Raychaudhury equation and the results compared with previous cases. This work is the generalisation of our previous article where we have taken $\alpha =1$.