Cosmological dynamics and dark energy with a quadratic equation of state: Anisotropic models, large-scale perturbations, and cosmological singularities

In standard general-relativistic cosmology, for fluids with a linear equation of state (EoS) $P=w\rho$ or scalar fields, the high isotropy of the universe requires special initial conditions: singularities are velocity dominated and anisotropic in general. In brane world effective 4-dimensional cosmological models an effective term, quadratic in the energy density, appears in the evolution equations, which has been shown to be responsible for the suppression of anisotropy and inhomogeneities at the singularity under reasonable assumptions. Thus in the brane world isotropy is generically built in, and singularities are matter dominated. There is no reason why the effective EoS of matter should be linear at the highest energies, and an effective nonlinear EoS may describe dark energy or unified dark matter (Paper I [K. Ananda and M. Bruni, preceding article, Phys. Rev. D 74, 023523 (2006).]). In view of this, here we investigate the effects of a quadratic EoS in homogenous and inhomogeneous anisotropic cosmological models in general relativity, in order to understand if in this standard context the quadratic EoS can isotropize the universe at early times. With respect to Paper I [K. Ananda and M. Bruni, preceding article, Phys. Rev. D 74, 023523 (2006).], here we use the simplified EoS $P=\alpha \rho +{\rho }^2/{\rho }_c$, which still allows for an effective cosmological constant and phantom behavior, and is general enough to analyze the dynamics at high energies. We first study homogenous and anisotropic Bianchi I and V models, focusing on singularities. Using dynamical systems methods, we find the fixed points of the system and study their stability. We find that models with standard nonphantom behavior are in general asymptotic in the past to an isotropic fixed point IS, i.e. in these models even an arbitrarily large anisotropy is suppressed in the past: the singularity is matter dominated. Using covariant and gauge-invariant variables, we then study linear anisotropic and inhomogeneous perturbations about the homogenous and isotropic spatially flat models with a quadratic EoS. We find that, in the large-scale limit, all perturbations decay asymptotically in the past, indicating that the isotropic fixed point IS is the general asymptotic past attractor for nonphantom inhomogeneous models with a quadratic EoS.

Figure 1

The state space for the Bianchi I reduced planar system (38, 39), showing fixed points at finite values of the energy density and shear. (a) Left panel: the state space for $\alpha >-1$; the only fixed point present represents a Minkowski model. (b) Right panel: the state space for $\alpha <-1$; in this case there is a new fixed point dS which represents a flat expanding de Sitter model with ${\rho }_{\Lambda }=-(1+\alpha ){\rho }_c$. The separatrix represents models with $\rho ={\rho }_{\Lambda }$. Models below this line have phantom behavior.

Figure 2

The compactified state space $(X,Z)$ for the Bianchi I reduced system, including fixed points at infinity. These represent the isotropic singularity (IS) and the Kasner anisotropic singularity (KS). (a) Left panel: the state space for $\alpha >-1$, corresponding to case A, Fig. 1a. (b) Right panel: the state space for $\alpha <-1$. The separatrix corresponds to that of Fig. 1b, with $\rho ={\rho }_{\Lambda }$. Above this line we have models of type B. Models below this line are phantom, of type C, and Kasner-like at the singularity KS. Clearly the nonphantom models of type A and B are asymptotic to the IS fixed point in the past.

Figure 3

The compactified state space $(X,Z,C)$ for the Bianchi V reduced system, including fixed points at infinity. These represent the isotropic singularity (IS), the Kasner anisotropic singularity (KS) and the curvature singularity (CS). (a) Left panel: The state space when $\alpha >-1$. (b) Right panel: the state space when $\alpha <-1$. It is clear that standard (nonphantom) trajectories are asymptotic to the IS fixed point in the past.