Models of universe with a polytropic equation of state: III. The phantom universe

Abstract

We construct models of universe with a generalized equation of state $p=(\alpha \rho +k{\rho }^{1+1/n})c^2$ having a linear component and a polytropic component. The linear equation of state $p=\alpha \rho c^2$ with $-1\le \alpha \le 1$ describes radiation ($\alpha =1/3$), pressureless matter ($\alpha =0$), stiff matter ($\alpha =1$), and vacuum energy ($\alpha =-1$). The polytropic equation of state $p=k{\rho }^{1+1/n}{c}^2$ may be due to Bose-Einstein condensates with repulsive ($k>0$) or attractive ($k<0$) self-interaction, or have another origin. In this paper, we consider the case where the density increases as the universe expands. This corresponds to a "phantom universe" for which $w=p/\rho c^2<-1$ (this requires $k<0$). We complete previous investigations on this problem and analyze in detail the different possibilities. We describe the singularities using the classification of [S. Nojiri, S.D. Odintsov, S. Tsujikawa, Phys. Rev. D {\bf 71}, 063004 (2005)]. We show that for $\alpha >-1$ there is no Big Rip singularity although $w\le -1$. For $n=-1$, we provide an analytical model of phantom bouncing universe "disappearing" at $t=0$. We also determine the potential of the phantom scalar field and phantom tachyon field corresponding to the generalized equation of state $p=(\alpha \rho +k{\rho }^{1+1/n})c^2$.

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